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

Normal distribution Random walk

With bread crumbs one can find exit in time proportional to V+E DFS/BFS Hansel & Gretel

Breadth First Search

F A BCG DE H

F A BCG DE H Queue: A get 0 distance from A visit(A)

Breadth First Search F A BCG DE H Queue: 0 F 1 F discovered

Breadth First Search F A BCG DE H Queue: F 0 1 B 1 B discovered

Breadth First Search F A BCG DE H Queue: F B C 1 C discovered

Breadth First Search F A BCG DE H Queue: F B C G 1 G discovered

Breadth First Search F A BCG DE H Queue: F B C G get A finished

Breadth First Search F A BCG DE H Queue: B C G A already visited

Breadth First Search F A BCG DE H Queue: B C G D 2 D discovered

Breadth First Search F A BCG DE H Queue: B C G D E 2 E discovered

Breadth First Search F A BCG DE H Queue: B C G D E get F finished

Breadth First Search F A BCG DE H Queue: C G D E

Breadth First Search F A BCG DE H Queue: C G D E A already visited

Breadth First Search F A BCG DE H Queue: C G D E get B finished

Breadth First Search F A BCG DE H Queue: G D E A already visited

Breadth First Search F A BCG DE H Queue: G D E get C finished

Breadth First Search F A BCG DE H Queue: D E A already visited

Breadth First Search F A BCG DE H Queue: D E E already visited

Breadth First Search F A BCG DE H Queue: D E get G finished

Breadth First Search F A BCG DE H Queue: E E already visited

Breadth First Search F A BCG DE H Queue: E F already visited

Breadth First Search F A BCG DE H Queue: E get D finished

Breadth First Search F A BCG DE H Queue: D already visited

Breadth First Search F A BCG DE H Queue: F already visited

Breadth First Search F A BCG DE H Queue: G already visited

Breadth First Search F A BCG DE H Queue: H 3 H discovered

Breadth First Search F A BCG DE Queue: H get H 3 E finished

Breadth First Search F A BCG DE H Queue: E already visited

Breadth First Search F A BCG DE H Queue: STOP H finished

Breadth First Search F A BCG DE H distance from A

Breadth-First Search

b c a d a c d b v

Rod Steiger Martin Sheen Donald Pleasence #1 #2 #3 #876 Kevin Bacon Barabasi

Why Kevin Bacon? Measure the average distance between Kevin Bacon and all other actors. 876 Kevin Bacon Barabasi

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

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