Convex Hull(35.3)• Convex Hull, CH(X), is the smallest convex polygon

containing all points from X, |X|=n• Different methods:

– incremental: moving from left to right updates CH(x1..xi). the runtime O(nlogn)

– divide-and-conquer divides into two subsets left and right in O(n), then combine into one convex hull in O(n)

– prune-and-search O(n logh), where h is the # points in CH uses pruning as for finding median to find upper chain

Graham’s Scan (35.3)

O(nlogn)

Finding the Closest Pair(35.4)• Brute-force: O(n2)• Divide-and-conquer algorithm with recurrence

T(n)=2T(n/2)+O(n)• Divide: divide into almost equal parts by a vertical

line which divides given x-sorted array X into 2 sorted subarrays

• Conquer: Recursively find the closest pair in each half of X. Let = min{left, right}

• Combine: The closest pair is either in distance or a pair of points from different halves.

Combine in D-a-C (35.4)• Subarray Y’ (y-sorted) of Y with points in 2 strip pY’ find all in Y’ which are closer than in

– no more than 8 in 2 rectangle– no more than 7 points can be closer than in

• If the closest in the strip closer then it is the answer

2

left right

2

Voronoi Graph• Voronoi region Vor(p) (p in set S)

– the set of points on the plane that are closer to p than to any othe rpoint in S

• Voronoi Graph VOR(S)– dual to voronoi region graph– two points are adjacent if their voronoi regions have common

contiguous boundary (segment)

Voronoi Graph• Voronoi Graph in the rectilinear plane• Rectilinear distance: p = (x, y); p’=(x’,y’)

a

b

c

bc

ac

ab Voronoi region of b

NP-Completeness (36.4-5)

NPC

P

NP-hard

• P: yes and no in pt

• NP: yes in pt

• NPH NPC

NP

Independent Set• Independent set in a graph G:

pairwise nonadjacent vertices• Max Independent Set is NPC• Is there independent set of size k?

– Construct a graph G:

• literal -> vertex

• two vertices are adjacent iff

– they are in the same clause

– they are negations of each other

– 3-CNF with k clauses is satisfiable iff G has independent k-set

• assign 1’s to literals-vertices of independent set

• Example: f = (x+z+y’) & (x’+z’+a) & (a’+x+y)

x

z

y’

x’

z’

a

a’

x

y

x, z’, y independent

F is satisfiable:f = 1 if x = z’ = y = 1

MAX Clique• Max Clique (MC):

– Find the maximum number of pairwise adjacent vertices

• MC is in NP– for the answer yes there is certificate of polynomial length = clique which

can be checked in polynomial time

• MC is in NPC– Polynomial time reduction from MIS:

• For any graph G any independent set in G 1-1 corresponds to clique in the complement graph G’

red independent set red clique

G Complement G’noedge edge edge noedge

Minimum Vertex Cover• Vertex Cover:

– the set of vertices which has at least one endpoint in each edge

• Minimum Vertex Cover (MVC):– the set of vertices which has at least one endpoint in each edge

• MIN Vertex Cover is NPC – If C is vertex cover, then V - C is an independent set

red independent set

blue vertex cover

Set Cover• Given: a set X and a family F of subsets of X, F 2X, s.t. X covered by F

• Find : subfamily G of F such that G covers X and |G| is minimize

• Set Cover is NPC – reduction from Vertex Covert

• Graph representation:

red elements of ground set X

blue subsets in family F

edge between set A F and element x X means x A

A

a b c d

A = {a,b,c}, B = {c,d}

B

Intermediate Classes

NPC

NP-hard

P NP

Dense Set Coveris NP but not in P neither in NPC

Dense Set Cover: Each element of X belongs to at least half of all sets in F

Runtime Complexity Classes• Runtime order:

– constant

– almost constant

– logarithmic

– sublinear

– linear

– pseudolinear

– quadratic

– polynomial

– subexponential

– exponential

– superexponential

• Example– adding an element in a queue/stack

– inverse Ackerman function = O(loglog…log n)

n times

– extracting minimum from binary heap

– n1/2

– traversing binary search tree, list

– O(n log n) sorting n numbers, closest pair, MST, Dijkstra shortest paths

– adding two nn matrices

– e n ^ (1/2)

– e n, , n!

– Ackerman function

2 . . . 2

2 n tim

es

Hamiltonian Cycle and TSP

• Hamiltonian Cycle:– given an undirected graph G

– find a tour which visits each point exactly once

• Traveling Salesperson Problem– given a positive weighted undirected graph G

(with triangle inequality = can make shortcuts)

– find a shortest tour which visits all the vertices

• HC and TSP are NPC

• NPC problems: SP, ISP, MCP, VCP, SCP, HC, TSP

Approximation Algorithms (37.0)

• When problem is in NPC try to find approximate solution in polynomial-time

• Performance Bound = Approximation Ratio (APR) (worst-case performance)

– Let I be an instance of a minimization problem

– Let OPT(I) be cost of the minimum solution for instance I

– Let ALG(I) be cost of solution for instance I given by approximate algorithm ALG

APR(ALG) = max I {ALG(I) / OPT(I)}

• APR for maximization problem =

max I {ALG(I) / OPT(I)}

Vertex Cover Problem (37.1)

• Find the least number of vertices covering all edges

• Greedy Algorithm:– while there are edges

• add the vertex of maximum degree• delete all covered edges

• 2-VC Algorithm:– while there are edges

• add the both ends of an edge• delete all covered edges

• APR of 2-VC is at most 2– e1, e2, ..., ek - edges chosen by 2-VC– the optimal vertex cover has 1 endpoint of ei– 2-VC outputs 2k vertices while optimum k

2-approximation TSP (37.2)

• Given a graph G with positive weights

Find a shortest tour which visits all vertices

• Triangle inequality w(a,b) + w(b,c) w(a,c)

• 2-MST algorithm:– Find the minimum spanning tree MST(G)– Take MST(G) twice: T = 2 MST(G) – The graph T is Eulerian - we can traverse it visiting each

edge exactly once– Make shortcuts

• APR of 2-MST is at most 2– MST weight weight of optimum tour

• any tour is a spanning tree, MST is the minimum

3/2-approximation TSP (Manber)

• Matching Problem (in P)– given weighted complete (all edges) graph with even # vertecies

– find a matching (pairwise disjoint edges) of minimum weight

• Christofides’s Algorithm (ChA)– find MST(G)

– for odd degree vertices find minimum matching M

– output shortcutted T = MST(G) + M

• APR of ChA is at most 3/2– |MST| OPT

– |M| OPT/2

– |T| (3/2) OPT odd

3/2-approximation TSP• Christofides’s Algorithm (ChA)

– find MST(G)

– for odd degree vertices find minimum matching M

– output shortcutted T = MST(G) + M

• The worst case for Christofides heuristic in Euclidean plane:

1

1

1

1 2 3 4 … k

k+1 k+2 … 2k-1k+3

- Minimum Spanning Tree length = 2k - 2- Minimum Matching of 2 odd degree nodes = k - 1- Christofides heuristic length = 3k - 3- Optimal tour length = 2k - 1- Approximation Ratio of Christofides = 3/2-1/(k-1/2)

Non-approximable TSP (37.2)

• Approximating TSP w/o triangle inequality is NPC– any c-approximation algorithm can solve Hamiltonian

Cycle Problem in polynomial time

• Take an instance of HCP = graph G

• Assign weight 0 to any edge of G

• Complete G up to complete graph G’

• Assign weight 1 to each new edge

• c-approximate tour can use only 0-edges -

so it gives Hamiltonian cycle of G

Steiner Tree Problem

Given: A set S of points in the plane = terminals

Find: Minimum-cost tree spanning S = minimum Steiner tree

1

1Cost = 2 Steiner PointCost = 31Terminals

1

1

Euclidean metric

11

1

1

11

1

11

1

Cost = 6 Cost = 4

Rectilinear metric

Steiner Tree Problem in Graphs

• Given a graph G=(V,E,cost) and terminals S in V Find minimum-cost tree spanning all terminals

• MST algorithm (does not use Steiner points):

– find G(S) = complete graph on terminals

• edge cost = shortest path cost

– find T(S) = MST of G(S)

– replace each edge of T(S) with the path in G

– output T(S)

MST -Heuristic

Theorem: MST-heuristic is a 2-approximation in graphs

Proof: MST < Shortcut Tour Tour = 2 • OPTIMUM

Approximation Ratios

• Euclidean Steiner Tree Problem– approximation ratio = 2/3

• Rectilinear Steiner Tree Problem– approximation ratio = 3/2

• Steiner Tree Problem in graphs– approximation ratio = 2

Steiner Point Opt Cost = k

12

3

k

5

4

MST Cost = 2k-2

Approximation ratio = 2-2/k 2

The Set Cover Problem

• Sets Ai cover a set X if X is a union of Ai

• Weighted Set Cover Problem

Given: – A finite set X (the ground set X)

– A family of F of subsets of X, with weights w: F +

Find:– sets S F, such that

• S covers X, X = {s | s S} and

• S has the minimum total weight {w(s) | s S}

• If w(s) =1 (unweighted), then minimum # of sets

Greedy Algorithm for SCP

• Greedy Algorithm:– While X is not empty

• find s F minimizing w(s) / |s X| • X = X - s• C = C + s

– Return C

6

3 4 5

2

1

Analysis of Greedy Algorithm

• Th: APR of the Greedy Algorithm is at most 1+ln k

• Proof:

iii nXS ||

11 iii nkk

ii

i

k

opt

n

w

opt

kwn iii

opt

wkk

opt

wkk i

iii

ii 1111

|| 1 ii Sk

opt

wkk i

last 10

opt

w

opt

w

k

k ii

last

1lnln 0xx )1ln(

opt

approx

k

k

last

0ln1

Approximation Complexity

• Approximation algorithm

= polynomial time approximation algorithm • PTAS = a series of approximation algorithms

s.t. for any > 0 there is pt (1+)-approximation– There is PTAS fro subset sum

• Remarkable progress in 90’s (assuming P NP).– No PTAS for Vertex Cover

– No clog k-approximation for Set Cover for k < 1

• k is the size of the ground set X

– No n1- approximation for Independent Set

• n is the number of vertices