Introduction to Constraint Programming...Introduction to Constraint Programming Helmut Simonis ACP...
427
Introduction to Constraint Programming Helmut Simonis ACP Summerschool, June 20th, 2016
Transcript of Introduction to Constraint Programming...Introduction to Constraint Programming Helmut Simonis ACP...
Introduction to ConstraintProgramming
Helmut Simonis
ACP Summerschool, June 20th, 2016
Licence
This work is licensed under the Creative CommonsAttribution-Noncommercial-Share Alike 3.0 Unported License.To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ orsend a letter to Creative Commons, 171 Second Street, Suite300, San Francisco, California, 94105, USA.
Insight Centre for Data Analytics Slide 2June 20th, 2016
Acknowledgments
The author is partially supported by Science Foundation Ireland(Grant Number 05/IN/I886). This material was developed aspart of the ECLiPSe ELearning course:http://4c.ucc.ie/~hsimonis/ELearning/index.htmSupport from Cisco Systems and the Silicon Valley CommunityFoundation is gratefully acknowledged.
Insight Centre for Data Analytics Slide 3June 20th, 2016
Objectives
• Overview of Core Constraint Programming• Three Main Concepts
• Constraint Propagation• Global Constraints• Customizing Search
• Based on Examples, not Formal Description
Insight Centre for Data Analytics Slide 4June 20th, 2016
Outline
• Why Constraint Programming?
• Constraint Propagation
• Global Constraints
• Customizing Search
• What is missing?
Insight Centre for Data Analytics Slide 5June 20th, 2016
Examples in ECLiPSe
• Open sourced constraint programming language
• Development goes back to 1985
• ECRC, ICL, IC-Parc, PTL, Cisco
• http://www.eclipse-clp.org/
• Specialities• Develop new solvers for specific domains• Integration with MIP
Insight Centre for Data Analytics Slide 6June 20th, 2016
ECLiPSe ELearning Course
• Self-study course in constraint programming
• Supported by Cisco Systems and Silicon ValleyCommunity Foundation
• Multi-media format, video lectures, slides, handout etc
• http://4c.ucc.ie/~hsimonis/ELearning/index.htm
Insight Centre for Data Analytics Slide 7June 20th, 2016
Constraint Programming - in a nutshell
• Declarative description of problems with• Variableswhich range over (finite) sets of values• Constraints over subsets of variables which restrict
possible value combinations• A solution is a value assignment which satisfies all
constraints
• Constraint propagation/reasoning• Removing inconsistent values for variables• Detect failure if constraint can not be satisfied• Interaction of constraints via shared variables• Incomplete
• Search• User controlled assignment of values to variables• Each step triggers constraint propagation
• Different domains require/allow different methods
Insight Centre for Data Analytics Slide 8June 20th, 2016
Basic Process
Problem
Human
Model
Constraint Solver/Search
Solution
Insight Centre for Data Analytics Slide 9June 20th, 2016
More Realistic
Problem
Human
Model
Constraint Solver/Search
Hangs Solution Wrong Solution
Insight Centre for Data Analytics Slide 10June 20th, 2016
Dual Role of Model
• Allows Human to Express Problem• Close to Problem Domain• Constraints as Abstractions
• Allows Solver to Execute• Variables as Communication Mechanism• Constraints as Algorithms
Insight Centre for Data Analytics Slide 11June 20th, 2016
Modelling Frameworks
• MiniZinc (NICTA, Australia)
• NumberJack (Insight, Ireland)
• Essence (UK)
• Allow use of multiple back-end solvers
• Compile model into variants for each solver
• A priori solver independent model(CP, MIP, SAT)
Insight Centre for Data Analytics Slide 12June 20th, 2016
Framework Process
Problem
Human
Model
Compile/Reformulate
CP MIP SAT Other
Solution Solution Solution Solution
Insight Centre for Data Analytics Slide 13June 20th, 2016
Part I
Basic Constraint Propagation
Insight Centre for Data Analytics Slide 14June 20th, 2016
Example 1: SEND+MORE=MONEY
• Example of Finite Domain Constraint Problem
• Models and Programs
• Constraint Propagation and Search
• Some Basic Constraints: linear arithmetic, alldifferent,disequality
• A Built-in search
• Visualizers for variables, constraints and search
Insight Centre for Data Analytics Slide 15June 20th, 2016
Outline
Problem
Program
Constraint Setup
Search
Points to Remember
Insight Centre for Data Analytics Slide 16June 20th, 2016
Problem Definition
A Crypt-Arithmetic PuzzleWe begin with the definition of the SEND+MORE=MONEYpuzzle. It is often shown in the form of a hand-writtenaddition:
S E N D+ M O R EM O N E Y
Insight Centre for Data Analytics Slide 17June 20th, 2016
Rules
S E N D+ M O R EM O N E Y
• Each character stands for a digit from 0 to 9.
• Numbers are built from digits in the usual, positionalnotation.
• Repeated occurrence of the same character denote thesame digit.
• Different characters denote different digits.
• Numbers do not start with a zero.
• The equation must hold.
Insight Centre for Data Analytics Slide 18June 20th, 2016
Outline
Problem
Program
Constraint Setup
Search
Points to Remember
Insight Centre for Data Analytics Slide 19June 20th, 2016
Model
• Each character is a variable, which ranges over the values0 to 9.
• An alldifferent constraint between all variables, whichstates that two different variables must have differentvalues. This is a very common constraint, which we willencounter in many other problems later on.
• Two disequality constraints (variable X must be differentfrom value V ) stating that the variables at the beginningof a number can not take the value 0.
• An arithmetic equality constraint linking all variables withthe proper coefficients and stating that the equationmust hold.
Insight Centre for Data Analytics Slide 20June 20th, 2016
Program Sendmory
:- module(sendmory).é Define Module
:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).é Make predicate visible
:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).é Use ic library
sendmory(L):-L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-é Predicate definition
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y], é Define listL :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,é Define integer domain 0..9
alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,
alldifferent(L), é Digits must be differentS #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
S E N D+ M O R EM O N E Y
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),
S #\= 0, M #\= 0,é Numbers don’t start with 01000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
S E N D+ M O R EM O N E Y
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,
1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,
labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,
labeling(L).é built-in search routine
Insight Centre for Data Analytics Slide 21June 20th, 2016
Program Sendmory
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 21June 20th, 2016
General Program Structure
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],é VariablesL :: 0..9,
alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,labeling(L).
Insight Centre for Data Analytics Slide 22June 20th, 2016
General Program Structure
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,
alldifferent(L),é ConstraintsS #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,
labeling(L).
Insight Centre for Data Analytics Slide 22June 20th, 2016
General Program Structure
:- module(sendmory).:- export(sendmory/1).:- lib(ic).sendmory(L):-
L = [S,E,N,D,M,O,R,Y],L :: 0..9,alldifferent(L),S #\= 0, M #\= 0,1000*S + 100*E + 10*N + D +1000*M + 100*O + 10*R + E #=10000*M + 1000*O + 100*N + 10*E + Y,
labeling(L).é Search
Insight Centre for Data Analytics Slide 22June 20th, 2016
Choice of Model
• This is onemodel, not themodel of the problem
• Many possible alternatives• Choice often depends on your constraint system
• Constraints available• Reasoning attached to constraints
• Not always clear which is the bestmodel
• Often: Not clear what is the problem
Insight Centre for Data Analytics Slide 23June 20th, 2016
Running the program
• To run the program, we have to enter the query• sendmory:sendmory(L).
• Result• L = [9, 5, 6, 7, 1, 0, 8, 2]• yes (0.00s cpu, solution 1, maybe more)
Insight Centre for Data Analytics Slide 24June 20th, 2016
Example in NumberJack
from Numberjack import *def get_model():
model = Model()s, m = (Variable(1, 9) for val in range(2))e, n, d, o, r, y =
(Variable(0, 9) for val in range(6))model.add( s*1000 + e*100 + n*10 + d +
m*1000 + o*100 + r*10 + e ==m*10000 + o*1000 + n*100 + e*10 + y)
model.add(AllDiff((s, e, n, d, m, o, r, y)))return s, e, n, d, m, o, r, y, model
def solve(param):s, e, n, d, m, o, r, y, model = get_model()solver = model.load(param[’solver’])solver.setVerbosity(param[’verbose’])solver.solve()
Insight Centre for Data Analytics Slide 25June 20th, 2016
MiniZinc
include "alldifferent.mzn";
var 1..9: S;var 0..9: E;var 0..9: N;var 0..9: D;var 1..9: M;var 0..9: O;var 0..9: R;var 0..9: Y;constraint 1000 * S + 100 * E + 10 * N + D
+ 1000 * M + 100 * O + 10 * R + E= 10000 * M + 1000 * O + 100 * N + 10 * E + Y;
constraint alldifferent([S,E,N,D,M,O,R,Y]);
solve satisfy;
Insight Centre for Data Analytics Slide 26June 20th, 2016
JSR331 (Standardized JAVA Solver Interface)
public class SendMoreMoney extends Problem {public void define() {Var S = variable( "S",1, 9);Var E = variable( "E",0, 9);Var N = variable( "N",0, 9);Var D = variable( "D",0, 9);Var M = variable( "M",1, 9);Var O = variable( "O",0, 9);Var R = variable( "R",0, 9);Var Y = variable( "Y",0, 9);Var[] vars = new Var[] { S, E, N, D, M, O, R, Y };postAllDiff(vars);int coef[] = { 1000, 100, 10, 1, 1000, 100, 10, 1,
-10000, -1000, -100, -10, -1 };Var[] sendmoremoney = new Var[] { S, E, N, D,
M, O, R, E, M, O, N, E, Y};post(coef, sendmoremoney, "=", 0);
}Insight Centre for Data Analytics Slide 27June 20th, 2016
JSR331 (II)
package org.jcp.jsr331.samples;import javax.constraints.Solution;import javax.constraints.SolutionIterator;import javax.constraints.Var;import javax.constraints.impl.Problem;
public void solve() {Solution s = getSolver().findSolution();
}
public static void main(String[] args) {SendMoreMoney p = new SendMoreMoney();p.define();p.solve();
}}
Insight Centre for Data Analytics Slide 28June 20th, 2016
Question
• But how did the program come up with this solution?
Insight Centre for Data Analytics Slide 29June 20th, 2016
Outline
Problem
Program
Constraint SetupDomain DefinitionAlldifferent ConstraintDisequality ConstraintsEquality Constraint
Search
Points to Remember
Insight Centre for Data Analytics Slide 30June 20th, 2016
Domain Definition
L = [S,E,N,D,M,O,R,Y],L :: 0..9,
[S,E ,N,D,M,O,R,Y ] ∈ {0..9}
Insight Centre for Data Analytics Slide 31June 20th, 2016
Domain Visualization
0 1 2 3 4 5 6 7 8 9SENDMORY
Insight Centre for Data Analytics Slide 32June 20th, 2016
Domain Visualization
Rows =Variables
0 1 2 3 4 5 6 7 8 9SENDMORY
Insight Centre for Data Analytics Slide 32June 20th, 2016
Domain Visualization
Columns = Values0 1 2 3 4 5 6 7 8 9
SENDMORY
Insight Centre for Data Analytics Slide 32June 20th, 2016
Domain Visualization
Cells= State
0 1 2 3 4 5 6 7 8 9SENDMORY
Insight Centre for Data Analytics Slide 32June 20th, 2016
Alldifferent Constraint
alldifferent(L),
• Built-in of ic library
• No initial propagation possible
• Suspends, waits until variables are changed
• When variable is fixed, remove value from domain ofother variables
• Forward checking
Insight Centre for Data Analytics Slide 33June 20th, 2016
Alldifferent Visualization
Uses the same representation as the domain visualizer
0 1 2 3 4 5 6 7 8 9SENDMORY
Insight Centre for Data Analytics Slide 34June 20th, 2016
Disequality Constraints
S #\= 0, M#\= 0,
Remove value from domain
S ∈ {1..9},M ∈ {1..9}
Constraints solved, can be removed
Insight Centre for Data Analytics Slide 35June 20th, 2016
Domains after Disequality
0 1 2 3 4 5 6 7 8 9SENDMORY
Insight Centre for Data Analytics Slide 36June 20th, 2016
Equality Constraint
• Normalization of linear terms• Single occurence of variable• Positive coefficients
• Propagation
Insight Centre for Data Analytics Slide 37June 20th, 2016
Normalization
1000*S+ 100*E+ 10*N+ D+1000*M+ 100*O+ 10*R+ E
10000*M+ 1000*O+ 100*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ 10*N+ D+1000*M+ 100*O+ 10*R+ E
10000*M+ 1000*O+ 100*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ 10*N+ D+ 100*O+ 10*R+ E
9000*M+ 1000*O+ 100*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ 10*N+ D+ 100*O+ 10*R+ E
9000*M+ 1000*O+ 100*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ 10*N+ D+ 10*R+ E
9000*M+ 900*O+ 100*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ 10*N+ D+ 10*R+ E
9000*M+ 900*O+ 100*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ D+ 10*R+ E
9000*M+ 900*O+ 90*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 100*E+ D+ 10*R+ E
9000*M+ 900*O+ 90*N+ 10*E+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Normalization
1000*S+ 91*E+ D+ 10*R
9000*M+ 900*O+ 90*N+ Y
Insight Centre for Data Analytics Slide 38June 20th, 2016
Simplified Equation
1000∗S+91∗E +10∗R+D = 9000∗M +900∗O+90∗N +Y
Insight Centre for Data Analytics Slide 39June 20th, 2016
Propagation
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9 =
9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9
Insight Centre for Data Analytics Slide 40June 20th, 2016
Propagation
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸1000..9918
=
9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..89919
Insight Centre for Data Analytics Slide 40June 20th, 2016
Propagation
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸9000..9918
=
9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..9918
Insight Centre for Data Analytics Slide 40June 20th, 2016
Propagation
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸9000..9918
=
9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..9918
Deduction:M = 1,S = 9,O ∈ {0..1}
Insight Centre for Data Analytics Slide 40June 20th, 2016
Propagation
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸9000..9918
=
9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..9918
Deduction:M = 1,S = 9,O ∈ {0..1}
Why? Skip
Insight Centre for Data Analytics Slide 40June 20th, 2016
Consider lower bound for S
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸9000..9918
= 9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..9918
• Lower bound of equation is 9000
• Rest of lhs (left hand side) (91 ∗ E0..9 + 10 ∗ R0..9 + D0..9)is atmost 918
• S must be greater or equal to 9000−9181000 = 8.082
• otherwise lower bound of equation not reached by lhs
• S is integer, therefore S ≥ d9000−9181000 e = 9
• S has upper bound of 9, so S = 9
Insight Centre for Data Analytics Slide 41June 20th, 2016
Consider upper bound of M
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸9000..9918
= 9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..9918
• Upper bound of equation is 9918
• Rest of rhs (right hand side) 900 ∗O0..9 + 90 ∗N0..9 + Y 0..9
is at least 0
• M must be smaller or equal to 9918−09000 = 1.102
• M must be integer, therefore M ≤ b9918−09000 c = 1
• M has lower bound of 1, so M = 1
Insight Centre for Data Analytics Slide 42June 20th, 2016
Consider upper bound of O
1000 ∗ S1..9 + 91 ∗ E0..9 + 10 ∗ R0..9 + D0..9︸ ︷︷ ︸9000..9918
= 9000 ∗M1..9 + 900 ∗O0..9 + 90 ∗ N0..9 + Y 0..9︸ ︷︷ ︸9000..9918
• Upper bound of equation is 9918
• Rest of rhs (right hand side) 9000 ∗ 1 + 90 ∗ N0..9 + Y 0..9 isat least 9000
• O must be smaller or equal to 9918−9000900 = 1.02
• O must be integer, therefore O ≤ b9918−9000900 c = 1
• O has lower bound of 0, so O ∈ {0..1}
Insight Centre for Data Analytics Slide 43June 20th, 2016
Propagation of equality: Result
0 1 2 3 4 5 6 7 8 9S - - - - - - - - Y
ENDM Y - - - - - - - -O 6 6 6 6 6 6 6 6
RY
Insight Centre for Data Analytics Slide 44June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9S - - - - - - - - Y
ENDM Y - - - - - - - -O 6 6 6 6 6 6 6 6
RY
Insight Centre for Data Analytics Slide 45June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9S Y
E |N |D |M Y
OR |Y |
Insight Centre for Data Analytics Slide 45June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SE |N |D |M Y
O |R |Y |
Insight Centre for Data Analytics Slide 45June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SENDMO Y
RY
Insight Centre for Data Analytics Slide 45June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SE |N |D |MO Y
R |Y |
Insight Centre for Data Analytics Slide 45June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SENDMORY
O = 0, [E ,R,D,N,Y ] ∈ {2..8}
Insight Centre for Data Analytics Slide 45June 20th, 2016
Waking the equality constraint
• Triggered by assignment of variables
• or update of lower or upper bound
Insight Centre for Data Analytics Slide 46June 20th, 2016
Removal of constants
1000 ∗ 9 + 91 ∗ E2..8 + 10 ∗ R2..8 + D2..8 =
9000 ∗ 1 + 900 ∗ 0 + 90 ∗ N2..8 + Y 2..8
Insight Centre for Data Analytics Slide 47June 20th, 2016
Removal of constants
1000 ∗ 9 + 91 ∗ E2..8 + 10 ∗ R2..8 + D2..8 =
9000 ∗ 1 + 900 ∗ 0 + 90 ∗ N2..8 + Y 2..8
Insight Centre for Data Analytics Slide 47June 20th, 2016
Removal of constants
91 ∗ E2..8 + 10 ∗ R2..8 + D2..8 = 90 ∗ N2..8 + Y 2..8
Insight Centre for Data Analytics Slide 47June 20th, 2016
Propagation of equality (Iteration 1)
91 ∗ E2..8 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸204..816
= 90 ∗ N2..8 + Y 2..8︸ ︷︷ ︸182..728
Insight Centre for Data Analytics Slide 48June 20th, 2016
Propagation of equality (Iteration 1)
91 ∗ E2..8 + 10 ∗ R2..8 + D2..8 = 90 ∗ N2..8 + Y 2..8︸ ︷︷ ︸204..728
Insight Centre for Data Analytics Slide 48June 20th, 2016
Propagation of equality (Iteration 1)
91 ∗ E2..8 + 10 ∗ R2..8 + D2..8 = 90 ∗ N2..8 + Y 2..8︸ ︷︷ ︸204..728
N ≥ 3 = d204− 890
e,E ≤ 7 = b728− 2291
c
Insight Centre for Data Analytics Slide 48June 20th, 2016
Propagation of equality (Iteration 2)
91 ∗ E2..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N3..8 + Y 2..8
Insight Centre for Data Analytics Slide 49June 20th, 2016
Propagation of equality (Iteration 2)
91 ∗ E2..7 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸204..725
= 90 ∗ N3..8 + Y 2..8︸ ︷︷ ︸272..728
Insight Centre for Data Analytics Slide 49June 20th, 2016
Propagation of equality (Iteration 2)
91 ∗ E2..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N3..8 + Y 2..8︸ ︷︷ ︸272..725
Insight Centre for Data Analytics Slide 49June 20th, 2016
Propagation of equality (Iteration 2)
91 ∗ E2..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N3..8 + Y 2..8︸ ︷︷ ︸272..725
E ≥ 3 = d272− 8891
e
Insight Centre for Data Analytics Slide 49June 20th, 2016
Propagation of equality (Iteration 3)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N3..8 + Y 2..8
Insight Centre for Data Analytics Slide 50June 20th, 2016
Propagation of equality (Iteration 3)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸295..725
= 90 ∗ N3..8 + Y 2..8︸ ︷︷ ︸272..728
Insight Centre for Data Analytics Slide 50June 20th, 2016
Propagation of equality (Iteration 3)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N3..8 + Y 2..8︸ ︷︷ ︸295..725
Insight Centre for Data Analytics Slide 50June 20th, 2016
Propagation of equality (Iteration 3)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N3..8 + Y 2..8︸ ︷︷ ︸295..725
N ≥ 4 = d295− 890
e
Insight Centre for Data Analytics Slide 50June 20th, 2016
Propagation of equality (Iteration 4)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N4..8 + Y 2..8
Insight Centre for Data Analytics Slide 51June 20th, 2016
Propagation of equality (Iteration 4)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸295..725
= 90 ∗ N4..8 + Y 2..8︸ ︷︷ ︸362..728
Insight Centre for Data Analytics Slide 51June 20th, 2016
Propagation of equality (Iteration 4)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N4..8 + Y 2..8︸ ︷︷ ︸362..725
Insight Centre for Data Analytics Slide 51June 20th, 2016
Propagation of equality (Iteration 4)
91 ∗ E3..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N4..8 + Y 2..8︸ ︷︷ ︸362..725
E ≥ 4 = d362− 8891
e
Insight Centre for Data Analytics Slide 51June 20th, 2016
Propagation of equality (Iteration 5)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N4..8 + Y 2..8
Insight Centre for Data Analytics Slide 52June 20th, 2016
Propagation of equality (Iteration 5)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸386..725
= 90 ∗ N4..8 + Y 2..8︸ ︷︷ ︸362..728
Insight Centre for Data Analytics Slide 52June 20th, 2016
Propagation of equality (Iteration 5)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N4..8 + Y 2..8︸ ︷︷ ︸386..725
Insight Centre for Data Analytics Slide 52June 20th, 2016
Propagation of equality (Iteration 5)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N4..8 + Y 2..8︸ ︷︷ ︸386..725
N ≥ 5 = d386− 890
e
Insight Centre for Data Analytics Slide 52June 20th, 2016
Propagation of equality (Iteration 6)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N5..8 + Y 2..8
Insight Centre for Data Analytics Slide 53June 20th, 2016
Propagation of equality (Iteration 6)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸386..725
= 90 ∗ N5..8 + Y 2..8︸ ︷︷ ︸452..728
Insight Centre for Data Analytics Slide 53June 20th, 2016
Propagation of equality (Iteration 6)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N5..8 + Y 2..8︸ ︷︷ ︸452..725
Insight Centre for Data Analytics Slide 53June 20th, 2016
Propagation of equality (Iteration 6)
91 ∗ E4..7 + 10 ∗ R2..8 + D2..8 = 90 ∗ N5..8 + Y 2..8︸ ︷︷ ︸452..725
N ≥ 5 = d452− 890
e,E ≥ 4 = d452− 8891
e
No further propagation at this point
Insight Centre for Data Analytics Slide 53June 20th, 2016
Domains after setup
0 1 2 3 4 5 6 7 8 9SENDMORY
Insight Centre for Data Analytics Slide 54June 20th, 2016
Outline
Problem
Program
Constraint Setup
SearchStep 1Step 2Further StepsSolution
Points to Remember
Insight Centre for Data Analytics Slide 55June 20th, 2016
labeling built-in
labeling([S,E,N,D,M,O,R,Y])
• Try variable is order given
• Try values starting from smallest value in domain
• When failing, backtrack to last open choice
• Chronological Backtracking
• Depth First search
Insight Centre for Data Analytics Slide 56June 20th, 2016
Search Tree Step 1
Variable S already fixed
S
E9
Insight Centre for Data Analytics Slide 57June 20th, 2016
Step 2, Alternative E = 4
Variable E ∈ {4..7}, first value tested is 4
S
E4
9
Insight Centre for Data Analytics Slide 58June 20th, 2016
Assignment E = 4
0 1 2 3 4 5 6 7 8 9SE Y - - -NDMORY
Insight Centre for Data Analytics Slide 59June 20th, 2016
Propagation of E = 4, equality constraint
91 ∗ 4 + 10 ∗ R2..8 + D2..8 = 90 ∗ N5..8 + Y 2..8
Insight Centre for Data Analytics Slide 60June 20th, 2016
Propagation of E = 4, equality constraint
91 ∗ 4 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸386..452
= 90 ∗ N5..8 + Y 2..8︸ ︷︷ ︸452..728
Insight Centre for Data Analytics Slide 60June 20th, 2016
Propagation of E = 4, equality constraint
91 ∗ 4 + 10 ∗ R2..8 + D2..8 = 90 ∗ N5..8 + Y 2..8︸ ︷︷ ︸452
Insight Centre for Data Analytics Slide 60June 20th, 2016
Propagation of E = 4, equality constraint
91 ∗ 4 + 10 ∗ R2..8 + D2..8 = 90 ∗ N5..8 + Y 2..8︸ ︷︷ ︸452
N = 5,Y = 2,R = 8,D = 8
Insight Centre for Data Analytics Slide 60June 20th, 2016
Result of equality propagation
0 1 2 3 4 5 6 7 8 9SEN Y - - -D - - - - - - Y
MOR - - - - - - Y
Y Y - - - - - -
Insight Centre for Data Analytics Slide 61June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SEN Y - - -D - - - - - - Y
MOR - - - - - - Y
Y Y - - - - - -
Insight Centre for Data Analytics Slide 62June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SEN Y - -D - - - - - - Y
MOR - - - - - - Y
Y Y - - - - -
Alldifferent fails!
Insight Centre for Data Analytics Slide 62June 20th, 2016
Step 2, Alternative E = 5
Return to last open choice, E , and test next value
S
E4
N5
9
Insight Centre for Data Analytics Slide 63June 20th, 2016
Assignment E = 5
0 1 2 3 4 5 6 7 8 9SE - Y - -NDMORY
Insight Centre for Data Analytics Slide 64June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SE - Y - -NDMORY
Insight Centre for Data Analytics Slide 65June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SE Y
N |D |MOR |Y |
Insight Centre for Data Analytics Slide 65June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SENDMORY
N 6= 5,N ≥ 6
Insight Centre for Data Analytics Slide 65June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ R2..8 + D2..8 = 90 ∗ N6..8 + Y 2..8
Insight Centre for Data Analytics Slide 66June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ R2..8 + D2..8︸ ︷︷ ︸477..543
= 90 ∗ N6..8 + Y 2..8︸ ︷︷ ︸542..728
Insight Centre for Data Analytics Slide 66June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ R2..8 + D2..8 = 90 ∗ N6..8 + Y 2..8︸ ︷︷ ︸542..543
Insight Centre for Data Analytics Slide 66June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ R2..8 + D2..8 = 90 ∗ N6..8 + Y 2..8︸ ︷︷ ︸542..543
N = 6,Y ∈ {2,3},R = 8,D ∈ {7..8}
Insight Centre for Data Analytics Slide 66June 20th, 2016
Result of equality propagation
0 1 2 3 4 5 6 7 8 9SEN Y - -D 6 6 6 6
MOR - - - - - Y
Y 6 6 6 6
Insight Centre for Data Analytics Slide 67June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SEN Y - -D 6 6 6 6
MOR - - - - - Y
Y 6 6 6 6
Insight Centre for Data Analytics Slide 68June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SEND |MOR Y
Y
Insight Centre for Data Analytics Slide 68June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SEND Y
MORY
Insight Centre for Data Analytics Slide 68June 20th, 2016
Propagation of alldifferent
0 1 2 3 4 5 6 7 8 9SENDMORY
D = 7
Insight Centre for Data Analytics Slide 68June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ 8 + 7 = 90 ∗ 6 + Y 2..3
Insight Centre for Data Analytics Slide 69June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ 8 + 7︸ ︷︷ ︸542
= 90 ∗ 6 + Y 2..3︸ ︷︷ ︸542..543
Insight Centre for Data Analytics Slide 69June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ 8 + 7 = 90 ∗ 6 + Y 2..3︸ ︷︷ ︸542
Insight Centre for Data Analytics Slide 69June 20th, 2016
Propagation of equality
91 ∗ 5 + 10 ∗ 8 + 7 = 90 ∗ 6 + Y 2..3︸ ︷︷ ︸542
Y = 2
Insight Centre for Data Analytics Slide 69June 20th, 2016
Last propagation step
0 1 2 3 4 5 6 7 8 9SENDMORY Y -
Insight Centre for Data Analytics Slide 70June 20th, 2016
Further Steps: Nothing more to do
S
E4
N5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Further Steps: Nothing more to do
S
E4
N
D6
5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Further Steps: Nothing more to do
S
E4
N
D
M7
6
5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Further Steps: Nothing more to do
S
E4
N
D
M
O1
7
6
5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Further Steps: Nothing more to do
S
E4
N
D
M
O
R0
1
7
6
5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Further Steps: Nothing more to do
S
E4
N
D
M
O
R
Y8
0
1
7
6
5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Further Steps: Nothing more to do
S
E4
N
D
M
O
R
Y2
8
0
1
7
6
5
9
Insight Centre for Data Analytics Slide 71June 20th, 2016
Complete Search Tree
S
E4
N
D
M
O
R
Y2
8
0
1
7
6
5 6 7
9
Insight Centre for Data Analytics Slide 72June 20th, 2016
Solution
9 5 6 7+ 1 0 8 51 0 6 5 2
Insight Centre for Data Analytics Slide 73June 20th, 2016
Outline
Problem
Program
Constraint Setup
Search
Points to Remember
Insight Centre for Data Analytics Slide 74June 20th, 2016
Points to Remember
• Constraint models are expressed by variables andconstraints.
• Problems can have many different models, which canbehave quite differently. Choosing the best model is anart.
• Constraints can take many different forms.
• Propagation deals with the interaction of variables andconstraints.
• It removes some values that are inconsistent with aconstraint from the domain of a variable.
• Constraints only communicate via shared variables.
Insight Centre for Data Analytics Slide 75June 20th, 2016
Points to Remember
• Propagation usually is not sufficient, search may berequired to find a solution.
• Propagation is data driven, and can be quite complexeven for small examples.
• The default search uses chronological depth-firstbacktracking, systematically exploring the completesearch space.
• The search choices and propagation are interleaved, afterevery choice some more propagation may further reducethe problem.
Insight Centre for Data Analytics Slide 76June 20th, 2016
Part II
Global Constraints
Insight Centre for Data Analytics Slide 77June 20th, 2016
Example 2: Sudoku
• Global Constraints• Powerful modelling abstractions• Non-trivial propagation• Different consistency levels
• Example: Sudoku puzzle
Insight Centre for Data Analytics Slide 78June 20th, 2016
Outline
Problem
Initial Propagation (Forward Checking)
Improved Reasoning
Search
Insight Centre for Data Analytics Slide 79June 20th, 2016
Problem Definition
SudokuFill in numbers from 1 to 9 so that each row, column and blockcontain each number exactly once
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Insight Centre for Data Analytics Slide 80June 20th, 2016
Problem Definition
SudokuFill in numbers from 1 to 9 so that each row, column and blockcontain each number exactly once
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
4 2 8 5 6 3 1 7 93 5 9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6 3 9 8 2 5 75 9 2 7 4 1 3 8 68 3 7 6 2 5 9 4 12 7 4 9 5 6 8 1 36 8 3 2 1 4 7 9 59 1 5 8 3 7 6 2 4
Insight Centre for Data Analytics Slide 80June 20th, 2016
Model
• A variable for each cell, ranging from 1 to 9
• A 9x9 matrix of variables describing the problem
• Preassigned integers for the given hints
• alldifferent constraints for each row, column and 3x3block
Insight Centre for Data Analytics Slide 81June 20th, 2016
Reminder: alldifferent
• Argument: list of variables
• Meaning: variables are pairwise different• Reasoning: Forward Checking (FC)
• When variable is assigned to value, remove the value fromall other variables
• If a variable has only one possible value, then it is assigned• If a variable has no possible values, then the constraint
fails• Constraint is checked whenever one of its variables is
assigned• Equivalent to decomposition into binary disequality
constraints
Insight Centre for Data Analytics Slide 82June 20th, 2016
Main Program
model(Matrix):-Matrix[1..9,1..9] :: 1..9,(for(I,1,9),param(Matrix) do
alldifferent(Matrix[I,1..9]),alldifferent(Matrix[1..9,I])
),(multifor([I,J],[1,1],[7,7],[3,3]),param(Matrix) do
alldifferent(flatten(Matrix[I..I+2,J..J+2]))),flatten_array(Matrix,List),labeling(List).
Insight Centre for Data Analytics Slide 83June 20th, 2016
Domain Visualizer
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
• Problem shown as matrix
• Each cell corresponds to a variable
• Instantiated: Shows integer value (large)
• Uninstantiated: Shows values in domain
Insight Centre for Data Analytics Slide 84June 20th, 2016
Outline
Problem
Initial Propagation (Forward Checking)
Improved Reasoning
Search
Insight Centre for Data Analytics Slide 85June 20th, 2016
Initial State (Forward Checking)
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Insight Centre for Data Analytics Slide 86June 20th, 2016
Propagation Steps (Forward Checking)
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 2 35 6
7 9 81 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 2 35 6
7 9 81 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 91 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 35 6
7 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 2 35 6
7 9 81 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
2 34 5 68 9
2 34 5 68 9 1 7
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 35 6
7 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5 68 9 1 7
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 35 6
7 8 9
1 24 5 68 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5 68 9 1 7
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5 67 9
14 5 67 9 8
14 5 67 9
14 5 67 9 3 2
1 35 6
7 8 9
1 24 5 68 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
9
14 5 67 9 8
14 5 67 9
14 5 67 9 3 2
1 35 6
7 8 9
1 24 5 68 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
9
14 5 67 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
1 347 9
1 347 9 8 2 5
1 347 9
1 35 6
7 8 9 91 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
1 347 9 8 2 5
1 347 9
1 35 6
7 8 9 91 2 34 5
9
2 34 57 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
1 347 9 8 2 5
1 347 9
1 35 6
7 91 2 34 5
2 34 57
1 2 34 5 67
1 2 34 5 67
1 2 34 5 67 8
1 2 34 5 67
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 34 5 67
1 2 34 5 67 8
1 2 34 5 67
1 35 6
7 8 9 3 7 62 3
4 69
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 34 5 67
1 2 34 5 67 8
1 2 34 5 67
158 3 7 6
24
1 24 58 9
1 24 58
1 24 58
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 2 34 5 67 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 2 34 5
9
2 34 57 8 9 5
1 2 35 6
7 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 2 34 5 67 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 34
9
348 9 5
1 369
1 34 68 9
1 34 68 9
1 34 68 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 34
9
348 9 5
1 369
1 348
1 34 68 9
1 34 68 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 34 57 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 34
9
348 9 5
1 369
1 348
1 34 68 9
1 34 68 9
35 6
7 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
2 35 6
7 8 9
2 35 6
7 8 91 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
24 6
9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24
1 24 58
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 34 68 9
35 6
7 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
2 35 6
7 8 91 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 24 67 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
24 6
9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24
1 24 58
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 34 68 9
35 6
7 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
2 35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
24 6
9
35 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 3
7 91 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7158 3 7 6
24
1 25 9
1 24
158
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 93
5 67 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 35 6
7 93
5 69
25 6
2 359 1 7
2 35 69
34 58
24 6
9
35 68 9
15 6
7 9
15 6
1594 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 3
7 91 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7158 3 7 6
24
1 25 9
1 24
158
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 93
5 67 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
2 369
2 35 69
1 35
7
1 26
7 9
1 35 6
7 93
5 69
25 6
2 359 1 7
2 35 69
34 58
24 6
9
35 68 9
15 6
7 9
15 6
1594 5
9 8 5 69
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 3
7 91 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7158 3 7 6
24
1 25 9
1 24
158
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 93
5 67 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
2 369
2 35 69
15
7
16
7 9
15 6
7 93
5 69
25 6
2 359 1 7
2 35 694 58
4 69
5 68 9
15 6
7 9
15 6
1594 5
9 8 5 69
14 57 3 2
1 4 63
47 9
34
9 8 2 51 3
7 9
5 9 22 3
4 57
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7
8 3 7 62
41 25 9
1 24
158
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 93
5 67 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
2 369
2 35 69
15
7
16
7 9
15 6
7 9369
25 6
359 1 7
2 35 694 58
4 69
5 68 9
67 9
15 6
1594 5
9 8 5 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7 9
5 9 23
47
34 6
1 36
7
1 347 8
1 36
7
8 3 7 62
41 25 9
1 24
15
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 936
7 9
25 68
359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 93
7 9
1 258
1 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
2 369
2 35 69
15
7
16
7 9
15 6
7 9369
25 6
359 1 7
2 35 694 58
4 69
5 68 9
67 9
15 6
1594 5
9 8 5 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7 9
5 9 23
47
34
1 3
7
1 347 8
1 36
7
8 3 7 6 2 5 9 4 12 7
1 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 936
7 9
25 68
359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 93
7 9
1 258
1 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
348 9 5
1 369
1 348
169
36
8 936
7 9
25 68
359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 93
7 9
1 258
1 359
2 35
7 8 9
3
9
1 2 3
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
348 9 5
1 369
1 348
169
36
8 9369
5 68
359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 93
9
158
1 359
2 35
7 8 9
3
9
1 2 3
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 348
169
36
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
26
7 9
35 6
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 87June 20th, 2016
Propagation Steps (Forward Checking)
41 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
Back to Start
Insight Centre for Data Analytics Slide 87June 20th, 2016
After Setup (Forward Checking)
41 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
Insight Centre for Data Analytics Slide 88June 20th, 2016
Outline
Problem
Initial Propagation (Forward Checking)
Improved ReasoningDomain ConsistencyComparison
Search
Insight Centre for Data Analytics Slide 89June 20th, 2016
Can we do better?
• The alldifferent constraint is missing propagation• How can we do more propagation?• Do we know when we derive all possible information from
the constraint?
• Constraints only interact by changing domains ofvariables
Insight Centre for Data Analytics Slide 90June 20th, 2016
A Simpler Example
:-lib(ic).
top:-X :: 1..2,Y :: 1..2,Z :: 1..3,alldifferent([X,Y,Z]),writeln([X,Y,Z]).
Insight Centre for Data Analytics Slide 91June 20th, 2016
Using Forward Checking
• No variable is assigned
• No reduction of domains
• But, values 1 and 2 can be removed from Z
• This means that Z is assigned to 3
Insight Centre for Data Analytics Slide 92June 20th, 2016
Visualization of alldifferent as Graph
X
Y
Z
1
2
3• Show problem as graph with two types of nodes
• Variables on the left• Values on the right
• If value is in domain of variable, show link between them
• This is called a bipartite graph
Insight Centre for Data Analytics Slide 93June 20th, 2016
A Simpler Example
Value Graph for
X :: 1..2,
Y :: 1..2,
Z :: 1..3
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 94June 20th, 2016
A Simpler Example
Check interval [1,2]X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 94June 20th, 2016
A Simpler Example
• Find variables completelycontained in interval
• There are two: X and Y
• This uses up the capacity of theinterval
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 94June 20th, 2016
A Simpler Example
No other variable can use thatinterval
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 94June 20th, 2016
A Simpler Example
Only one value left in domain of Z,this can be assigned
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 94June 20th, 2016
Idea (Hall Intervals)
• Take each interval of possible values, say size N• Find all K variables whose domain is completelycontained in interval
• If K > N then the constraint is infeasible
• If K = N then no other variable can use that interval
• Remove values from such variables if their bounds change
• If K < N do nothing
• Re-check whenever domain bounds change
Insight Centre for Data Analytics Slide 95June 20th, 2016
Implementation
• Problem: Too many intervals (O(n2)) to consider• Solution:
• Check only those intervals which update bounds• Enumerate intervals incrementally• Starting from lowest(highest) value• Using sorted list of variables
• Complexity: O(n log(n)) in standard implementations
• Important: Only looks at min/max bounds of variables
Insight Centre for Data Analytics Slide 96June 20th, 2016
Bounds Consistency
DefinitionA constraint achieves bounds consistency, if for the lower andupper bound of every variable, it is possible to find values forall other variables between their lower and upper boundswhich satisfy the constraint.
Insight Centre for Data Analytics Slide 97June 20th, 2016
Can we do better?
• Bounds consistency only considers min/max bounds
• Ignores ‘‘holes’’ in domain
• Sometimes we can improve propagation looking at thoseholes
Insight Centre for Data Analytics Slide 98June 20th, 2016
Another Simple Example
:-lib(ic).
top:-X :: [1,3],Y :: [1,3],Z :: 1..3,alldifferent([X,Y,Z]),writeln([X,Y,Z]).
Insight Centre for Data Analytics Slide 99June 20th, 2016
Another Simple Example
Value Graph for
X :: [1,3],
Y :: [1,3],
Z :: 1..3
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
• Check interval [1,2]
• No domain of a variablecompletely contained in interval
• No propagation
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
• Check interval [2,3]
• No domain of a variablecompletely contained in interval
• No propagation
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
But, more propagation is possible,there are only two solutions
X
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
Solution 1: assignment in blueX
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
Solution 2: assignment in greenX
Y
Z
1
2
3
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
Solution 1 Solution 2 CombinedX
Y
Z
1
2
3
X
Y
Z
1
2
3
X
Y
Z
1
2
3
Combining solutions shows that Z=1 andZ=3 are not possible.
Insight Centre for Data Analytics Slide 100June 20th, 2016
Another Simple Example
Can we deduce this without enumeratingsolutions?
Insight Centre for Data Analytics Slide 100June 20th, 2016
Solutions and maximal matchings
• AMatching is subset of edges which do not coincide inany node
• No matching can have more edges than number ofvariables
• Every solution corresponds to amaximal matching andvice versa
• If a link does not belong to some maximal matching, thenit can be removed
Insight Centre for Data Analytics Slide 101June 20th, 2016
Implementation
• Possible to compute all links which belong to somematching
• Without enumerating all of them!
• Enough to compute one maximal matching
• Requires algorithm for strongly connected components
• Extra work required if more values than variables
• All links (values in domains) which are not supported canbe removed
• Complexity: O(n1.5d)
Insight Centre for Data Analytics Slide 102June 20th, 2016
Domain Consistency
DefinitionA constraint achieves domain consistency, if for every variableand for every value in its domain, it is possible to find values inthe domains of all other variables which satisfy the constraint.
• Also called generalized arc consistency (GAC)
• or hyper arc consistency
Insight Centre for Data Analytics Slide 103June 20th, 2016
Can we still do better?
• NO! This extracts all information from this one constraint
• We could perhaps improve speed, but not propagation
• But possible to use different model
• Or model interaction of multiple constraints
Insight Centre for Data Analytics Slide 104June 20th, 2016
Should all constraints achieve domain consistency?
• Domain consistency is usually more expensive thanbounds consistency
• Overkill for simple problems• Nice to have choices
• For some constraints achieving domain consistency isNP-hard
• We have to live with more restricted propagation
Insight Centre for Data Analytics Slide 105June 20th, 2016
Initial State (Domain Consistency)
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Insight Centre for Data Analytics Slide 106June 20th, 2016
Propagation Steps (Domain Consistency)
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 2 35 6
7 9 81 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 7 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 2 35 6
7 9 81 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 91 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 7
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 35 6
7 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 2 35 6
7 9 81 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
2 34 5 68 9
2 34 5 68 9 1 7
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 35 6
7 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5 68 9 1 7
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9 3 2
1 35 6
7 8 9
1 24 5 68 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5 68 9 1 7
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5 67 9
14 5 67 9 8
14 5 67 9
14 5 67 9 3 2
1 35 6
7 8 9
1 24 5 68 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5 67 8 9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
9
14 5 67 9 8
14 5 67 9
14 5 67 9 3 2
1 35 6
7 8 9
1 24 5 68 6
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8 2 5
1 2 34 5 67 8 9
1 35 6
7 8 9 91 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
9
14 5 67 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
1 347 9
1 347 9 8 2 5
1 347 9
1 35 6
7 8 9 91 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
1 2 34 5 67 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
1 347 9 8 2 5
1 347 9
1 35 6
7 8 9 91 2 34 5
9
2 34 57 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 8
1 2 34 5 67 8 9
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
1 347 9 8 2 5
1 347 9
1 35 6
7 91 2 34 5
2 34 57
1 2 34 5 67
1 2 34 5 67
1 2 34 5 67 8
1 2 34 5 67
1 35 6
7 8 9 3 7 61 2 34 5 67 8 9
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 34 5 67
1 2 34 5 67 8
1 2 34 5 67
1 35 6
7 8 9 3 7 62 3
4 69
1 2 34 5 67 8 9 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 3
4 5 68 9
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
14 5 67 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 34 5 67
1 2 34 5 67 8
1 2 34 5 67
158 3 7 6
24
1 24 58 9
1 24 58
1 24 58
2 71 2 34 5
9
2 34 57 8 9 5
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 34 5 67 8 9 6
1 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 2 34 5 67 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 2 34 5
9
2 34 57 8 9 5
1 2 35 6
7 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
2 34 5 68 9
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 5 67 9 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 2 34 5 67 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 34
9
348 9 5
1 369
1 34 68 9
1 34 68 9
1 34 68 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 34
9
348 9 5
1 369
1 348
1 34 68 9
1 34 68 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9 1 4
1 34 57 8
1 2 34 5 67 8 9
1 2 34 5 67 8 9
1 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 2 35 6
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
2 34 5 68 9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24 58
1 24 58
2 71 34
9
348 9 5
1 369
1 348
1 34 68 9
1 34 68 9
35 6
7 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
2 35 6
7 8 9
2 35 6
7 8 91 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 2 34 5 67 8 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
24 6
9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24
1 24 58
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 34 68 9
35 6
7 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
2 35 6
7 8 91 35 6
7 8 9
1 24 5 68
1 2 34 5
9
2 34 57 8 9
2 34 6
9
1 2 35 6
7 9 61 24 67 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 2 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
24 6
9
2 34 5 68 9
15 6
7 9
14 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9
14 6
347 9
34
9 8 2 51 347 9
1 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 2 34 5 67
158 3 7 6
24
1 25 9
1 24
1 24 58
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 34 68 9
35 6
7 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
2 35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 35 6
7 93
5 68 9
24 5 68
2 34 5
9 1 72 35 69
34 58
24 6
9
35 68 9
714 5 6
14 5
94 57 9 8
15 6
7 9
14 57 3 2
1 3
7 9 4 63
47 9
34
9 8 2 51 3
7 91 35 6
7 91 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7158 3 7 6
24
1 25 9
1 24
158
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 93
5 67 8 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 35
7 8 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 35
7 9
2 369
1 2 35 6
7 9
1 35
7
1 26
7 9
1 35 6
7 93
5 69
25 6
2 359 1 7
2 35 69
34 58
24 6
9
35 68 9
715 6
159 4 8
15 69
14 5 3 2
1 3
9 4 63
7 9
3
9 8 2 51 3
7 91 35 6 9
1 2 34 5
2 34 57
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7158 3 7 6
24
1 25 9
1 24
158
2 71 34
9
348 9 5
1 369
1 348
14 6
9
1 36
8 93
5 68 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 358 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 359
2 369
2 35 69
1 35
7
1 26
7 9
1 35 6
7 93
5 69
25 6
2 359 1 7
2 35 69
34 58
24 6
9
35 68 9
715 6
159 4 8 5 6
9
15 3 2
1 3
9 4 63
7 9
3
9 8 2 51 3
7 91 35 6 9
1 2 34 5
2 35
7
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7158 3 7 6
24
1 25 9
1 24
158
2 71 34
9
3
8 9 51 3
69
1 348
14 6
9
1 36
8 93
5 68 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 358 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
41 25 6 8
2 359
2 369
2 35 69
15
7
16
7 9
15 6
7 93
5 69
25 6
2 359 1 7
2 35 694 58
4 69
5 68 9
715 6
159 4 8 5 6
9
15 3 2
1 4 63
7 9
3
9 8 2 51 3
7 9
5 9 22 35
7
2 34 6
1 2 35 6
7
1 34 57 8
1 35 6
7
8 3 7 62
41 25 9
1 24
158
2 7 43
8 9 51 3
69
1 348
14 6
9
1 36
8 93
5 68 9
25 68
2 359
2 35
7 8 9 1 43
57 8
26
7 9
35 6
7 8 91 358 9
1 258
1 2 359
2 35
7 8 9
2 3
9
1 2 35
7 9 61 2
7 9 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
4 2 8 5 6 3 11
67 9
15 6
7 9369 5
359 1 7 2 4 6 8
7 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7 9
5 9 23
7 4 11 347 8 6
8 3 7 6 2 5 9 4 12 7 4
3
8 9 5 6 8 11 3
68 9
6 83
59 2 1 4
35
7 8
26
7 9 53
9 1 5 82 3
9 7 6 2 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7 9
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 107June 20th, 2016
Propagation Steps (Domain Consistency)
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Back to Start
Insight Centre for Data Analytics Slide 107June 20th, 2016
After Setup (Domain Consistency)
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Insight Centre for Data Analytics Slide 108June 20th, 2016
Comparison
Forward Checking Bounds Consistency Domain Consistency4
1 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
41 2
8 5 62 3 1
7
1
7 9 7 93
9
25
3
9 1 72 3
4 58
69
58 9
7 615 4 8 9
15 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
7
34 1
3
7 8 68 3 7 6 2 5 9 4 12 7 4
3
8 9 5 61 3
8
1
9
3
8 9
6 58
3
9
2 3
8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
15
2
8
3
9 7 61 2
4
4 2 8 5 6 3 1 7 9 7 93
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
Insight Centre for Data Analytics Slide 109June 20th, 2016
Typical?
• This does not always happen
• Sometimes, two methods produce same amount ofpropagation
• Possible to predict in certain special cases
• In general, tradeoff between speed and propagation
• Not always fastest to remove inconsistent values early
• But often required to find a solution at all
Insight Centre for Data Analytics Slide 110June 20th, 2016
Outline
Problem
Initial Propagation (Forward Checking)
Improved Reasoning
Search
Insight Centre for Data Analytics Slide 111June 20th, 2016
Simple search routine
• Enumerate variables in given order
• Try values starting from smallest one in domain
• Complete, chronological backtracking
Insight Centre for Data Analytics Slide 112June 20th, 2016
Search Tree (Forward Checking)4
1 25 6 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
41
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 2
369
2 369
57
67 9
5 67 9
369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
4
52
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 2
369
369
57
67 9
5 67 9
369
25 6
359 1 7
3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
3
2
4
53
2
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 2
369
369
57
67 9
5 67 9
369
25 6
359 1 7
3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
6
2
4
53 6
2
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 2
369
369
57
67 9
5 67 9
369
25 6
359 1 7
3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
9
2
4
53 6 9
2
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 3
369
2 369
57
67 9
5 67 9
369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
4
53 6 9
2
53
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 3 6
9
269
57
67 9
5 67 9
369
25 6
359 1 7
2694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 6 7 9
3
9 8 2 53
7
5 9 2 47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9 8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
2
7 8 9
3
9
2 3
7 9 61 2
7 9 4
6
2
4
53 6 9
2
56
3
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8 3 6
9
269
57
67 9
5 67 9
369
25 6
359 1 7
2694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 6 7 9
3
9 8 2 53
7
5 9 2 47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9 8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
2
7 8 9
3
9
2 3
7 9 61 2
7 9 4
9
2
4
53 6 9
2
56 9
3
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8
2 359
369
2 369
57
67 9
5 67 9
369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
5
2
4
53 6 9
2
56 9
3 5
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 1 8
2 359
369
2 369
57
67 9
5 67 9
369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
5 6 594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
958
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
9
2
4
53 6 9
2
56 9
3 5 9
1
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8
2 359
369
2 369
15
7
16
7 95 6
7 9369
25 6
359 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
4
53 6 9
2
56 9
3 5 9
1
42
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 3
369
369
15
7
16
7 95 6
7 9369
5 63
59 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
4
53 6 9
2
56 9
3 5 9
1
4
53
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 3 6
969
15
7
16
7 95 6
7 9369
5 63
59 1 7
2694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 6 7 9
3
9 8 2 53
7
5 9 2 47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9 8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2
7 8 9
3
9
2 3
7 9 61 2
7 9 4
6
2
4
53 6 9
2
56 9
3 5 9
1
4
56
3
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 3 6
969
15
7
16
7 95 6
7 9369
5 63
59 1 7
2694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 6 7 9
3
9 8 2 53
7
5 9 2 47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9 8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2
7 8 9
3
9
2 3
7 9 61 2
7 9 4
9
2
4
53 6 9
2
56 9
3 5 9
1
4
56 9
3
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 5
369
369
15
7
16
7 95 6
7 9369
5 63
59 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594 5
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
2
4
53 6 9
2
56 9
3 5 9
1
4
56 9
3
55
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 5
369
369
1
7
16
7 96
7 9369
5 63
59 1 7
2 3694 58
69
5 68 9
67 9
15 6
1594
9 8 69
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34
1 3
7
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5369
1 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9
2 3
7 9 61 2
7 9 4
3
2
4
53 6 9
2
56 9
3 5 9
1
4
56 9
3
53
5
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 5 6 3
1
7
16
7 96
7 9369
5 63
59 1 7 2 4 5
869
5 68 9
67 9
15 6
159 4 8 9
14 57 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
47
34 1
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5 61 3
8
1
9
3
8 9369
5 68
359
2 3
7 8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
7 8 9
3
9 7 61 2
7 9 4
2
4
53 6 9
2
56 9
3 5 9
1
4
56 9
3
53
76
5
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 5 6 3 1
1
7 9 7 9369
5 63
59 1 7 2 4 5
869
5 68 9
67
15 6
15 4 8 9
15
7 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7
34 1
3
7 836
7
8 3 7 6 2 5 9 4 12 7
1 34
9
3
8 9 5 61 3
8
1
9
3
8 9369
5 68
359
2 3
8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
1 359
2 3
8 9
3
9 7 61 2
9 4
2
4
53 6 9
2
56 9
3 5 9
1
4
56 9
3
53
7
81
6
5
2
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Forward Checking)4 2 8 5 6 3 1 7 93 5 9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6 3 9 8 2 5 75 9 2 7 4 1 3 8 68 3 7 6 2 5 9 4 12 7 4 9 5 6 8 1 36 8 3 2 1 4 7 9 59 1 5 8 3 7 6 2 4
2
4
53 6 9
2
56 9
3 5 9
1
4
56 9
3
53
7
87
1
6
5
2
Back to StartInsight Centre for Data Analytics Slide 113June 20th, 2016
Search Tree (Bounds Consistency)4
1 2
8 5 62 3 1
7
1
7 9 7 93
9
25
3
9 1 72 3
4 58
69
58 9
7 615 4 8 9
15 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
7
34 1
3
7 8 68 3 7 6 2 5 9 4 12 7 4
3
8 9 5 61 3
8
1
9
3
8 9
6 58
3
9
2 3
8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
15
2
8
3
9 7 61 2
4
2
Skip AnimationInsight Centre for Data Analytics Slide 114June 20th, 2016
Search Tree (Bounds Consistency)4
1 2
8 5 62 3 1
7
1
7 9 7 93
9
25
3
9 1 72 3
4 58
69
58 9
7 615 4 8 9
15 3 2
1 4 63
7 9
3
9 8 2 53
7
5 9 23
7
34 1
3
7 8 68 3 7 6 2 5 9 4 12 7 4
3
8 9 5 61 3
8
1
9
3
8 9
6 58
3
9
2 3
8 9 1 43
57 8
2
7 9
35
7 8 93
9
158
15
2
8
3
9 7 61 2
4
1
21
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 114June 20th, 2016
Search Tree (Bounds Consistency)4 2 8 5 6 3 1
1
7 9 7 93
9 53
9 1 7 2 4 6 58 9
7 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7
34 1
3
7 8 68 3 7 6 2 5 9 4 12 7 4
3
8 9 5 6 8 13
8 9
6 83
9 2 1 43
57 8
2
7 9 53
9 1 5 83
9 7 6 2 4
21
82
Back to Start Skip AnimationInsight Centre for Data Analytics Slide 114June 20th, 2016
Search Tree (Bounds Consistency)4 2 8 5 6 3 1 7 93 5 9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6 3 9 8 2 5 75 9 2 7 4 1 3 8 68 3 7 6 2 5 9 4 12 7 4 9 5 6 8 1 36 8 3 2 1 4 7 9 59 1 5 8 3 7 6 2 4
21
87
2
Back to StartInsight Centre for Data Analytics Slide 114June 20th, 2016
Search Tree (Domain Consistency)4 2 8 5 6 3 1 7 9 7 9
3
9 53
9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6
3
7 9
3
9 8 2 53
7
5 9 23
7 4 13
7 8 68 3 7 6 2 5 9 4 12 7 4
3
9 5 6 8 13
9
6 83
9 2 1 43
7 7 9 53
9 1 5 83
9 7 6 2 4
8
Skip AnimationInsight Centre for Data Analytics Slide 115June 20th, 2016
Search Tree (Domain Consistency)4 2 8 5 6 3 1 7 93 5 9 1 7 2 4 6 87 6 1 4 8 9 5 3 21 4 6 3 9 8 2 5 75 9 2 7 4 1 3 8 68 3 7 6 2 5 9 4 12 7 4 9 5 6 8 1 36 8 3 2 1 4 7 9 59 1 5 8 3 7 6 2 4
87
Back to StartInsight Centre for Data Analytics Slide 115June 20th, 2016
Global Constraint Catalog
• http://www.emn.fr/z-info/sdemasse/gccat/index.html
• Description of 354 global constraints, 2800 pages
• Not all of them are widely used
• Detailed, meta-data description of constraints in Prolog
Insight Centre for Data Analytics Slide 116June 20th, 2016
Families of Global Constraints
• Value Counting• alldifferent, global cardinality
• Scheduling• cumulative
• Properties of Sequences• sequence, no_valley
• Graph Properties• circuit,tree
Insight Centre for Data Analytics Slide 117June 20th, 2016
Common Algorithmic Techniques
• Flow Based Algorithms
• Automata
• Task Intervals
• Reduced Cost Filtering
• Decomposition
Insight Centre for Data Analytics Slide 118June 20th, 2016
Part III
Customizing Search
Insight Centre for Data Analytics Slide 119June 20th, 2016
What we want to introduce
• Importance of search strategy, constraints alone are notenough
• Dynamic variable ordering exploits information frompropagation
• Variable and value choice
• Hard to find strategy which works all the time
• search builtin, flexible search abstraction
• Different way of improving stability of search routine
Insight Centre for Data Analytics Slide 120June 20th, 2016
Example Problem
• N-Queens puzzle
• Rather weak constraint propagation
• Many solutions, limited number of symmetries
• Easy to scale problem size
Insight Centre for Data Analytics Slide 121June 20th, 2016
Outline
Problem
Program
Naive Search
Improvements
Insight Centre for Data Analytics Slide 122June 20th, 2016
Problem Definition
8-QueensPlace 8 queens on an 8× 8 chessboard so that no queenattacks another. A queen attacks all cells in horizontal, verticaland diagonal direction. Generalizes to boards of size N × N .
Insight Centre for Data Analytics Slide 123June 20th, 2016
Problem Definition
8-QueensPlace 8 queens on an 8× 8 chessboard so that no queenattacks another. A queen attacks all cells in horizontal, verticaland diagonal direction. Generalizes to boards of size N × N .
Solution for board size 8× 8
Insight Centre for Data Analytics Slide 123June 20th, 2016
Outline
Problem
ProgramModel
Naive Search
Improvements
Insight Centre for Data Analytics Slide 124June 20th, 2016
Basic Model
• Cell based Model• A 0/1 variable for each cell to say if it is occupied or not• Constraints on rows, columns and diagonals to enforce
no-attack• N2 variables, 6N − 2 constraints
• Column (Row) based Model• A 1..N variable for each column, stating position of queen
in the column• Based on observation that each column must contain
exactly one queen• N variables, N2/2 binary constraints
Insight Centre for Data Analytics Slide 125June 20th, 2016
Model
assign [X1,X2, ...XN ]
s.t.
∀1 ≤ i ≤ N : Xi ∈ 1..N∀1 ≤ i < j ≤ N : Xi 6= Xj
∀1 ≤ i < j ≤ N : Xi 6= Xj + i − j∀1 ≤ i < j ≤ N : Xi 6= Xj + j − i
Insight Centre for Data Analytics Slide 126June 20th, 2016
Outline
Problem
Program
Naive Search
Improvements
Insight Centre for Data Analytics Slide 127June 20th, 2016
Default Strategy
1
Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
21
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
33
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
46
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42
6
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
47
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
52
7
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54
2
7
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
48
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42
8
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
34
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
42
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45
2
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7
2
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
46
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43
6
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
47
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
53
7
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56
3
7
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3
7
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
48
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43
8
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
35
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
42
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
2
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
58
2
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53
8
2
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
47
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
52
7
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54
2
7
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
48
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
52
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54
2
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
56
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
5
63
6
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
5
6
77
3
6
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
5
6
7
82
7
3
6
8
5
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 128June 20th, 2016
Default Strategy
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
5
6
7
84
2
7
3
6
8
5
1
Back to Start
Insight Centre for Data Analytics Slide 128June 20th, 2016
First Solution
1
2
35
42 8
6
4
54 8
2
7
42 6
8
3
3
45 7 8
2
43 8
6
4
56 8
3 5
7
43 5
8
4
3
46
53 6
8
2
4
54 6
2
7
4
54 7
2
5
6
7
84
2
7
3
6
8
5
1
Insight Centre for Data Analytics Slide 129June 20th, 2016
Observations
• Even for small problem size, tree can become large
• Not interested in all details
• Ignore all automatically fixed variables
• For more compact representation abstract failedsub-trees
Insight Centre for Data Analytics Slide 130June 20th, 2016
Compact Representation
Number inside triangle: Number of choicesNumber under triangle: Number of failures1
2
4
7
3
6
10
4
3
2
3
2
1
2
7
4
1
2
2 6
8
5
1
Insight Centre for Data Analytics Slide 131June 20th, 2016
Exploring other board sizes
• How stable is the model?
• Try all sizes from 4 to 100
• Timeout of 100 seconds
Insight Centre for Data Analytics Slide 132June 20th, 2016
Naive Stategy, Problem Sizes 4-100
0
5
10
15
20
25
0 5 10 15 20 25 30
Tim
e[s]
Problem Size
"naive/all.txt"
Insight Centre for Data Analytics Slide 133June 20th, 2016
Observations
• Time very reasonable up to size 20
• Sizes 20-30 times very variable
• Not just linked to problem size
• No size greater than 30 solved within timeout
Insight Centre for Data Analytics Slide 134June 20th, 2016
Outline
Problem
Program
Naive Search
ImprovementsDynamic Variable ChoiceImproved HeuristicsMaking Search More Stable
Insight Centre for Data Analytics Slide 135June 20th, 2016
Possible Improvements
• Better constraint reasoning• Remodelling problem with 3 alldifferent constraints• Global reasoning as described before• Not explored here
• Better control of search• Static vs. dynamic variable ordering• Better value choice• Not using complete depth-first chronological backtracking
Insight Centre for Data Analytics Slide 136June 20th, 2016
Static vs. Dynamic Variable Ordering
• Heuristic Static Ordering• Sort variables before search based on heuristic• Most important decisions• Smallest initial domain
• Dynamic variable ordering• Use information from constraint propagation• Different orders in different parts of search tree• Use all information available
Insight Centre for Data Analytics Slide 137June 20th, 2016
First Fail strategy
• Dynamic variable ordering
• At each step, select variable with smallest domain
• Idea: If there is a solution, better chance of finding it
• Idea: If there is no solution, smaller number ofalternatives
• Needs tie-breaking method
Insight Centre for Data Analytics Slide 138June 20th, 2016
Modification of Program
:-module(nqueen).:-export(top/0).:-lib(ic).
top:-nqueen(8,L), writeln(L).
nqueen(N,L):-length(L,N),L :: 1..N,alldifferent(L),noattack(L),labeling(L).é replace with
Insight Centre for Data Analytics Slide 139June 20th, 2016
Modification of Program
:-module(nqueen).:-export(top/0).:-lib(ic).
top:-nqueen(8,L), writeln(L).
nqueen(N,L):-length(L,N),L :: 1..N,alldifferent(L),noattack(L),search(L,0,first_fail,indomain,complete,[]).
Insight Centre for Data Analytics Slide 139June 20th, 2016
Variable Choice
• Determines the order in which variables are assigned
• input_order assign variables in static order given
• first_fail select variable with smallest domain first
• most_constrained like first_fail, tie break based onnumber of constraints in which variable occurs
• Others, including programmed selection
Insight Centre for Data Analytics Slide 140June 20th, 2016
Value Choice
• Determines the order in which values are tested forselected variables
• indomain Start with smallest value, on backtracking trynext larger value
• indomain_max Start with largest value
• indomain_middle Start with value closest to middle ofdomain
• indomain_random Choose values in random order
Insight Centre for Data Analytics Slide 141June 20th, 2016
Comparison
• Board size 16x16
• Naive (Input Order) Strategy
• First Fail variable selection
Insight Centre for Data Analytics Slide 142June 20th, 2016
Naive (Input Order) Strategy (Size 16)
1
2
3
4
5
145
209
4
188
322
8
197
358
9
173
292
10
200
326
11
184
282
12
6
7
4
8
4
8
13
6
9
15
12
8
2
3
4
94
1
2
8
104
118
127
16
6
15
12
14
9
13
2
5
3
1
Insight Centre for Data Analytics Slide 143June 20th, 2016
FirstFail Strategy (Size 16)
1
Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
21
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
33
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
65
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
84
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
137
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
116
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
102
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
913
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
715
13
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711
15
13
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
13
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
716
13
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711
16
13
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
914
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
14
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
1416
14
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
14
78
16
14
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
14
710
8
16
14
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
14
710 13
8
16
14
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
14
710 13
8
129
16
14
2
6
7
4
5
3
1
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 144June 20th, 2016
FirstFail Strategy (Size 16)
1
2
3
6
8
13
11
10
9
711 14
15
711 15
16
13
912
14
710 13
8
128
9
16
14
2
6
7
4
5
3
1
Back to Start
Insight Centre for Data Analytics Slide 144June 20th, 2016
Comparing Solutions
Naive First Fail
Insight Centre for Data Analytics Slide 145June 20th, 2016
Comparing Solutions
Naive First Fail
Solutions are different!
Insight Centre for Data Analytics Slide 145June 20th, 2016
FirstFail, Problem Sizes 4-100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 10 20 30 40 50 60 70 80 90
Tim
e[s]
Problem Size
"first_fail/all.txt"
Insight Centre for Data Analytics Slide 146June 20th, 2016
Observations
• This is much better
• But some sizes are much harder
• Timeout for sizes 88, 91, 93, 97, 98, 99
Insight Centre for Data Analytics Slide 147June 20th, 2016
Can we do better?
• Improved initial ordering• Queens on edges of board are easier to assign• Do hard assignment first, keep simple choices for later• Begin assignment in middle of board
• Matching value choice• Values in the middle of board have higher impact• Assign these early at top of search tree• Use indomain_middle for this
Insight Centre for Data Analytics Slide 148June 20th, 2016
Start fromMiddle (Size 16)
9
Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
88
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
1210
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
59
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
146
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
65
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
44
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
311
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1513
11
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515
13
11
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
11
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
153
11
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157
3
11
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
1616
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
711
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
1115
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
15
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
1014
15
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
102
14
15
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
102 3
14
15
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
102 3
14
15
151
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
102 3
14
15
15
1115
1
11
16
4
5
6
9
10
8
Back to Start Skip Animation
Insight Centre for Data Analytics Slide 149June 20th, 2016
Start fromMiddle (Size 16)
9
8
12
5
14
6
412
3
1515 7
13
157 1
3
11
16
77
113
102 3
14
15
15
113
15
1
11
16
4
5
6
9
10
8
Back to Start
Insight Centre for Data Analytics Slide 149June 20th, 2016
Comparing Solutions
Naive First Fail Middle
Insight Centre for Data Analytics Slide 150June 20th, 2016
Comparing Solutions
Naive First Fail Middle
Again, solutions are different!
Insight Centre for Data Analytics Slide 150June 20th, 2016
Middle, Problem Sizes 4-100
0
5
10
15
20
25
30
35
0 10 20 30 40 50 60 70 80 90 100
Tim
e[s]
Problem Size
"middle/all.txt"
Insight Centre for Data Analytics Slide 151June 20th, 2016
Observations
• Not always better than first fail
• For size 16, trees are similar size
• Timeout only for size 94
• But still, one strategy does not work for all problem sizes
• There are ways to resolve this!
Insight Centre for Data Analytics Slide 152June 20th, 2016
Approach 1: Heuristic Portfolios
• Try multiple strategies for the same problem
• With multi-core CPUs, run them in parallel
• Only one needs to be successful for each problem
Insight Centre for Data Analytics Slide 153June 20th, 2016
Approach 2: Restart with Randomization
• Only spend limited number of backtracks for a searchattempt
• When this limit is exceeded, restart at beginning
• Requires randomization to explore new search branch
• Randomize variable choice by random tie break
• Randomize value choice by shuffling values
• Needs strategy when to restart
Insight Centre for Data Analytics Slide 154June 20th, 2016
Approach 3: Partial Search
• Abandon depth-first, chronological backtracking
• Don’t get locked into a failed sub-tree
• A wrong decision at a level is not detected, and we haveto explore the complete subtree below to undo thatwrong choice
• Explore more of the search tree
• Spend time in promising parts of tree
Insight Centre for Data Analytics Slide 155June 20th, 2016
Example: Credit Search
• Explore top of tree completely, based on credit
• Start with fixed amount of credit
• Each node consumes one credit unit
• Split remaining credit amongst children
• When credit runs out, start bounded backtrack search
• Each branch can use only K backtracks
• If this limit is exceeded, jump to unexplored top of tree
Insight Centre for Data Analytics Slide 156June 20th, 2016
Credit, Search Tree Problem Size 94
48
47
51
44
53
39
59
35
28
68
25
62
32
65
29
66
27
26
61
63
31
64
76
70
77
74
19
20
18
78
79
24
80
17
21
82
81
15
75
11
84
14
16
12
6
9
30
71
83
92
93
69
5
3
72
1
73
2
22
4
85
8
94
13
10
86
87
88
77
8989 17
84
86
7
8987 17
7
8989 17
5
3
75 84
88
83
8888
77
8989 17
3
86
4 87
18
87
88
77 84
86
77 5
3 88
17
7
91
8818 16
89 15
7
91
8818 16
91 15
5
4
75
8818 16
84
87
82
7
87
91
8818 16
89
83
7
5
13
81
78
12
80
9
79
72
27
76
20
19
70
75
74
69
24
22
23
28
67
21
71
25
73
29
65
68
61
26
31
30
66
64
32
63
33
34
62
35
60
58
36
38
59
57
37
56
39
53
55
40
29
66
27
26
61
63
67
70
19
72
77
76
74
17
18
78
16
15
21
75
81
12
20
79
36
22
14
80
13
71
10
83
84
94
82
92
91
23
93
24
90
4
73
2
11
5
3
85
9
8
86
87
88
784 10
4
784 85
91 3
82
88
784 10
83
784 2
91
710 2
3
5
88
784 2
4
784 85
83 3
90
88
72 85
91
710 85
83
4
80
87
6
8882 83
85 1
5
883 4
82
89
6
81
15
7
11
16
72
21
17
74
24
22
70
71
25
75
66
20
28
78
67
23
27
73
61
69
68
29
26
32
31
63
30
65
64
34
62
59
33
60
36
57
58
35
38
37
56
39
53
40
55
54
65
29
66
30
71
70
83
77
86
82
80
79
84
55
26
76
31
22
49
27
19
20
73
75
13
34
74
78
12
11
9
10
5
8
6
69
72
81
4
23
3
85
14
7
1
2
21
87
24
88
18
16
15
89
9085 17
5
91
90
92
93
94
17
33
56
67
61
54
50
52
58
36
46
41
42
43
40
37
452
38
57
60
63
647
3
91
5
2
93
92
9
89
8
86
84
6
90
11
85
80
83
87
14
10
94
13
88
78
4
17
1
15
77
82
18
12
74
81
20
16
76
19
79
22
21
70
23
75
73
71
67
26
72
25
65
69
27
24
30
66
68
28
33
29
64
63
62
61
37
32
31
36
60
34
35
59
39
57
53
38
58
56
40
54
55
41
42
50
52
43
51
46
44
45
48
49
47
Insight Centre for Data Analytics Slide 157June 20th, 2016
Credit, Problem Sizes 4-200
0
0.5
1
1.5
2
2.5
0 50 100 150 200
Tim
e[s]
Problem Size
"credit/all.txt"
Insight Centre for Data Analytics Slide 158June 20th, 2016
Points to Remember
• Choice of search can have huge impact on performance
• Dynamic variable selection can lead to large reduction ofsearch space
• Packaged search can do a lot, but programming searchadds even more
• Depth-first chronologicial backtracking not always bestchoice
• How to control this explosion of search alternatives?
Insight Centre for Data Analytics Slide 159June 20th, 2016
Part IV
What is missing?
Insight Centre for Data Analytics Slide 160June 20th, 2016
Many Specialized Topics
• How to design efficient core engine
• Hybrids with LP/MIP tools
• Hybrids with SAT
• Symmetry breaking
• Use of MDD/BDD to encode sets of solutions
• High level modelling tools
• Debugging/visualization
Insight Centre for Data Analytics Slide 161June 20th, 2016
Reformulation
• Just because the user has modelled it this way, it doesn’tmean we have to solve it that way
• Replace some constraint(s) by other, equivalentconstraints
• Because we don’t have that constraint in our system• For performance
Insight Centre for Data Analytics Slide 162June 20th, 2016
Learning
• While solving the problem we can learn how tostrengthen the model/search
• Understand which constraints/method contribute topropagation and change schedule
• Learn no-good constraints by explaining failure• Adapt search strategy based on search experience
Insight Centre for Data Analytics Slide 163June 20th, 2016
Refined Process
Problem
Human
User Model
Reformulation
Implementation Model
Constraint Solver/Search
Solution
Insight Centre for Data Analytics Slide 164June 20th, 2016
Exercises
• Install Constraint Solver of your Choice
• MiniZinc (MiningZinc) will be used in later sessions
• Run Sendmoremoney, Sudoku, Queens examples
• Can you choose which propagator to use for alldifferent?
• Can you use different search strategies?
Insight Centre for Data Analytics Slide 165June 20th, 2016
