233-234

233-234233-234233-234233-234

Sedgewick & Wayne (2004); Chazelle (2005)Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005)Sedgewick & Wayne (2004); Chazelle (2005)

Adjacency lists

1. Birds eat the bread crumbs

2. They don’t

random walk

DFS/BFS

Hansel & Gretel

Diffusion equation

Diffusion equation

Normal distribution

Random walk

With bread crumbs one canfind exit in time proportional

to V+E DFS/BFS

Hansel & Gretel

Breadth First Search

Breadth First Search

F

A

B C G

D E

H

Breadth First Search

F

A

B C G

D E

H

Queue: A

get

0distance from A

visit(A)

Breadth First Search

F

A

B C G

D E

H

Queue:

0

F1

F discovered

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F

A

B C G

D E

H

Queue: F

0

1

B 1B discovered

Breadth First Search

F

A

B C G

D E

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Queue: F B

0

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1 C 1C discovered

Breadth First Search

F

A

B C G

D E

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Queue: F B C

0

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1

G discovered

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F

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B C G

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Queue: F B C G

get

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1 1

1

A finished

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F

A

B C G

D E

H

Queue: B C G

0

1

1 1

1A already

visited

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A

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D E

H

Queue: B C G

0

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1

D2

D discovered

Breadth First Search

F

A

B C G

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Queue: B C G D

0

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1 1

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2 E

2

E discovered

Breadth First Search

F

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get

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1

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F finished

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1

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2

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A alreadyvisited

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A

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get

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2

B finished

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A

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0

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A alreadyvisited

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get

0

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1

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2

C finished

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A

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0

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2

A alreadyvisited

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A

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0

1

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1

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2

E alreadyvisited

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F

A

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get

0

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1

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2

G finished

Breadth First Search

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A

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0

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2

2

E alreadyvisited

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F

A

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0

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1 1

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2

2

F alreadyvisited

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F

A

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get

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1

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2

D finished

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F

A

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0

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1 1

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2

2

D alreadyvisited

Breadth First Search

F

A

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H

Queue:

0

1

1 1

1

2

2

F alreadyvisited

Breadth First Search

F

A

B C G

D E

H

Queue:

0

1

1 1

1

2

2

G alreadyvisited

Breadth First Search

F

A

B C G

D E

H

Queue:

0

1

1 1

1

2

2

H 3

H discovered

Breadth First Search

F

A

B C G

D E

Queue: H

get

0

1

1 1

1

2

2

H 3

E finished

Breadth First Search

F

A

B C G

D E

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Queue:

0

1

1 1

1

2

2

3

E alreadyvisited

Breadth First Search

F

A

B C G

D E

H

Queue:

STOP

0

1

1 1

1

2

2

3

H finished

Breadth First Search

F

A

B C G

D E

H

0

1

1 1

1

2

2

3

distance from A

Breadth-First Search

b ca da

cdb

v

Rod Steiger

Martin Sheen

Donald Pleasence

#1

#2

#3

#876Kevin Bacon

Barabasi

Rank NameAveragedistance

# ofmovies

# oflinks

1 Rod Steiger 2.537527 112 25622 Donald Pleasence 2.542376 180 28743 Martin Sheen 2.551210 136 35014 Christopher Lee 2.552497 201 29935 Robert Mitchum 2.557181 136 29056 Charlton Heston 2.566284 104 25527 Eddie Albert 2.567036 112 33338 Robert Vaughn 2.570193 126 27619 Donald Sutherland 2.577880 107 2865

10 John Gielgud 2.578980 122 294211 Anthony Quinn 2.579750 146 297812 James Earl Jones 2.584440 112 3787…

876 Kevin Bacon 2.786981 46 1811…

Why Kevin Bacon?

Measure the average distance between Kevin Bacon and all other actors.

876 Kevin Bacon 2.786981 46 1811Barabasi

Langston et al., A combinatorial approach to the analysis of differential gene expression data….

Minimum Dominating Set

Minimum Dominating Set

Minimum Dominating Set

size of dominating set

Expected size of dominating set

Assume each node has at least d neighbors

Naïve algorithm still n/2 in worst case

Simple probabilistic algorithm:

1. For each vertex v, color v red with probability p

1. For each vertex v, color v red with probability p

2. Color blue any non-dominated vertex

X= number of red nodes Y= number of blue nodes

Size of dominating set = X+Y

Expected size of dominating set S =

Markov’s inequality

proof

j= k E|S|

Probability that is < 1/2

Run algorithm 10 times and keep smallest S

with probability > 0.999