2. 233-234233-234 Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005)

3. Adjacency lists

4. 1. Birds eat the bread crumbs 2. They don’t random walk DFS/BFS Hansel & Gretel

6. Diffusion equation

7. Normal distribution Random walk

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

10. Breadth First Search

16. F A BCG DE H

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

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

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

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

21. Breadth First Search F A BCG DE H Queue: F B C 0 1 1 1 G 1 G discovered

22. Breadth First Search F A BCG DE H Queue: F B C G get 0 1 1 1 1 A finished

23. Breadth First Search F A BCG DE H Queue: B C G 0 1 1 1 1 A already visited

24. Breadth First Search F A BCG DE H Queue: B C G 0 1 1 1 1 D 2 D discovered

25. Breadth First Search F A BCG DE H Queue: B C G D 0 1 1 1 1 2 E 2 E discovered

26. Breadth First Search F A BCG DE H Queue: B C G D E get 0 1 1 1 1 2 2 F finished

27. Breadth First Search F A BCG DE H Queue: C G D E 0 1 1 1 1 2 2

28. Breadth First Search F A BCG DE H Queue: C G D E 0 1 1 1 1 2 2 A already visited

29. Breadth First Search F A BCG DE H Queue: C G D E get 0 1 1 1 1 2 2 B finished

30. Breadth First Search F A BCG DE H Queue: G D E 0 1 1 1 1 2 2 A already visited

31. Breadth First Search F A BCG DE H Queue: G D E get 0 1 1 1 1 2 2 C finished

32. Breadth First Search F A BCG DE H Queue: D E 0 1 1 1 1 2 2 A already visited

33. Breadth First Search F A BCG DE H Queue: D E 0 1 1 1 1 2 2 E already visited

34. Breadth First Search F A BCG DE H Queue: D E get 0 1 1 1 1 2 2 G finished

35. Breadth First Search F A BCG DE H Queue: E 0 1 1 1 1 2 2 E already visited

36. Breadth First Search F A BCG DE H Queue: E 0 1 1 1 1 2 2 F already visited

37. Breadth First Search F A BCG DE H Queue: E get 0 1 1 1 1 2 2 D finished

38. Breadth First Search F A BCG DE H Queue: 0 1 1 1 1 2 2 D already visited

39. Breadth First Search F A BCG DE H Queue: 0 1 1 1 1 2 2 F already visited

40. Breadth First Search F A BCG DE H Queue: 0 1 1 1 1 2 2 G already visited

41. Breadth First Search F A BCG DE H Queue: 0 1 1 1 1 2 2 H 3 H discovered

42. Breadth First Search F A BCG DE Queue: H get 0 1 1 1 1 2 2 H 3 E finished

43. Breadth First Search F A BCG DE H Queue: 0 1 1 1 1 2 2 3 E already visited

44. Breadth First Search F A BCG DE H Queue: STOP 0 1 1 1 1 2 2 3 H finished

45. Breadth First Search F A BCG DE H 0 1 1 1 1 2 2 3 distance from A

46. Breadth-First Search

47. b c a d a c d b v

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

52. Why Kevin Bacon? Measure the average distance between Kevin Bacon and all other actors. 876 Kevin Bacon 2.786981 46 1811 Barabasi

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

57. size of dominating set

58. Expected size of dominating set Assume each node has at least d neighbors Naïve algorithm still n/2 in worst case Simple probabilistic algorithm:

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

60. 2. Color blue any non-dominated vertex

61. X= number of red nodes Y= number of blue nodes Size of dominating set = X+Y

66. Expected size of dominating set S =

67. Markov’s inequality proof j= k E|S|

68. Probability that is < 1/2 Run algorithm 10 times and keep smallest S with probability > 0.999