Informed Search and Exploration. COSC 159 - Fundamentals of AI2 Overview Heuristic Search Best-First...

Informed Search and Exploration

COSC 159 - Fundamentals of AI

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Overview Heuristic Search

Best-First Search Approach Greedy A* Heuristic Functions

Local Search and Optimization Hill-climbing Simulated Annealing Local Beam Genetic Algorithms


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Informed Searching An informed search strategy uses

knowledge beyond the definition of the problem

The knowledge is embodied in an evaluation function f(n) Estimates the distance of node n from

the goal


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Best-First Search An instance of tree or graph search Fringe is ordered by f(n) Several best-first algorithms

Greedy, A*, … Key is the heuristic function h(n)

Heuristic measures distance from n to goal based solely on the state at n

Problem specific Constraint: h(n) = 0 if n is a goal node


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Greedy Best-First Search Expands node estimated to be

closest to the goal f(n) = h(n)

Consider the route finding problem. Can we use additional information to

avoid costly paths that lead nowhere? Consider using the straight line

distance (SLD)


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Route Finding

374

253366

329


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Route Finding

Arad f(n) = 366

TimisoaraSibiu Zerind 374329253

Arad Rimnicu VilceaOradeaFagaras366 176 380 193

Bucharest Sibiu0 253


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Exercise

So is Arad->Sibiu->Fagaras->Bucharest optimal?


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Greedy Best-First Search Not optimal. Not complete.

Could go down a path and never return to try another.

Time Complexity O(bm)

Space Complexity O(bm)


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A* Search The greedy best-first search does

not consider how costly it was to get to a node.

Combine g(n), the cost to reach node n, with h(n) f(n) = g(n) + h(n) estimated cost of cheapest solution

through n


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Route Finding


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A* Search

Arad f(n) = 0 + 366

TimisoaraSibiu Zerind 449447393=140+253


Things are different

now!


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A* Search Continued


591

SibiuBucharest

450

Pitesti SibiuCraiova526 417 553

Craiova615

Rimnicu Vilcea

607

Bucharest

418


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A* Search When used with tree search Optimal

Yes! If h(n) is an admissible heuristic Complete

Yes! Same as above. Time and Space Complexity

Depends on h(n) Subexponential if

where h*(n) is true cost of getting from n to goal Optimally efficient

))((log)()( ** nhOnhnh


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Admissible Heuristics A heuristic function h(n) is admissible if it

never overestimates the cost to reach the goal from n

Is SLD an admissible heuristic? Another property of heuristic functions is

consistency h(n)c(n,a,n’)+h(n’) where c(n,a,n’) is the

cost to get to n’ from n using action a. Graph search is optimal if h(n) is consistent.

Is SLD consistent?


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Heuristic Functions

h1 = number of misplaced tiles h2 = sum of distances of tiles to goal

position.


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Heuristic Functions Is h1

admissible? consistent?

Is h2 admissible? consistent?


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Heuristic Functions Heuristics are often obtained from

relaxed problem Simplify the original problem by

removing constraints The cost of an optimal solution to a

relaxed problem is an admissible heuristic.


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8-Puzzle Original

A tile can move from A to B if A is horizontally or vertically adjacent to B and B is blank.

Relaxations Move from A to B if A is adjacent to B

• h2 by moving each tile in turn to destination Move from A to B if B is blank Move from A to B

• h1 by simply moving each tile directly to destination.


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8-Puzzle What heuristic is obtained from

Move from A to B if B is blank?

Gaschnig’s heuristic (1979)


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More Complicated Monkey and Banana Suppose now that the bananas are hung

from the ceiling and the monkey needs to climb on a box to get the bananas. In addition, walls exist in the environment.

Box, monkey, and bananas placed randomly

Monkey can see the entire world. Actions: N,S,E,W,Grab,Drop,Climb,Eat


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Monkey and Banana Describe the problem as a search

problem. Initial state, successor function, goal,

path cost In what way(s) can we relax the

problem to obtain a heuristic?


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Dominating Heuristics If h2(n) h1(n) for all n, then h2 dominates

h1

h2 is better for use with searching

For the 8-puzzle, h2 does dominate h1

If a tile is out of place, it is at least a distance of 1 away from its correct position. Since h1 counts the number of out of place tiles, and h2 totals all of the distances, h2(n) h1(n) for all n.


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Optimization Problems Idea is to find the best state. We don’t really care how to get to the

best state, just that we get there. The best state is defined according to an

objective function Measures the “fitness” of a state.

Problem: Find the optimal state The one that maximizes (or minimizes) the

objective function.


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State Space LandscapesObjective Function

State Space

shoulder

global max

local max


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State Space Landscapes


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Problem Formulation Complete-state formulation

Start with an approximate solution and perturb

n-queens problem Place n queens on a board so that no

queen is attacking another queen.


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Problem Formulation Initial State: n queens placed

randomly on the board, one per column.

Successor function: States that obtained by moving one queen to a new location in its column.

Heuristic/objective function: The number of pairs of attacking queens.


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n-Queens

5

5

4

3

2

3

4

5

2

4

3

4

3

3

2 3

5

4

2

3

1

4

2

2


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Phasor Measurement Unit Problem formulation?


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Local Search Algorithms Hill climbing Simulated annealing Local beam search Genetic Algorithms


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Hill Climbing (or Descent)Objective Function

State Space


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Hill Climbing Pseudo-code


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Hill Climbing ProblemsObjective Function

State Space


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Other Problematic Spaces

http://www.cs.qub.ac.uk/~M.Sullivan/ga/optimization_pictures.html


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n-Queens

5

5

4

3

2

3

4

5

2

4

3

4

3

3

2 3

5

4

2

3

1

4

2

2

What happens if we move 3rd queen?


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Possible Improvements Stochastic hill climbing

Choose at random from uphill moves Probability of move could be influenced by steepness

First-choice hill climbing Generate successors at random until one is better than

current. Random-restart

Execute hill climbing several times, choose best result. If p is probability of a search succeeding, then expected

number of restarts is 1/p.


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Simulated Annealing Similar to stochastic hill climbing

Moves are selected at random If a move is an improvement, accept Otherwise, accept with probability less

than 1. Probability gets smaller as time

passes and by the amount of “badness” of the move.


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Simulated Annealing Algorithm

Success


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Traveling Salesperson Problem (TSP) Tour of cities Visit each one exactly once Minimize distance/cost/etc.


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TSP State: Permutation of cities Successors: Any permutation

reached by swapping two cities Could be others.

Objective function: Total distance/cost/etc.


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TSP By Simulated Annealing Key is identifying proper schedule

Requires experimentation One possible schedule

Start T at some value k After 1000*N successes, let T=.90*T,

where N is the number of cities.


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Local Beam Search Keep k states in memory instead of

just one Generate successors of all k states If one is a goal, return the goal Otherwise, take k best successors

and repeat.


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Local Beam Search

Concentrates on promising paths


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Local Beam Search Initial k states may not be diverse

enough Could have clustered around a local

max. Improvement is stochastic beam

search Choose k states at random, with

probability of choice an increasing function of its value.


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Genetic Algorithms Variant of stochastic beam search Successor states are generated by

combining two parent states Hopefully improves diversity

Start with k states, the population Each state, or individual, represented as a

string over a finite alphabet (e.g. DNA) Each state is rated by a fitness function Select parents for reproduction using the

fitness function


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Genetic Algorithms

Taken from http://www.cs.qub.ac.uk/~M.Sullivan/ga/ga_index.html


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A Genetic Algorithm


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Genetic Algorithm for PMU Each state is represented as a

binary string 1 in location n represents PMU on node

n Fitness function

(nodes+edges observed)/# of PMUs


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Just Some Chess Pieces

Informed Search and Exploration. COSC 159 - Fundamentals of AI2 Overview Heuristic Search Best-First...

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