BFS Dijkstra. Oct abhi shelat

Transcription

1 4102 BFS Dijkstra Oct abhi shelat

2 breadth first search

3 bfs(g, a) 1 2 a b 1 2 d c e f g 2 h

4 bfs theorem

5 Theorem 1 (CLRS, p. 599) Let G =(V, E) be a graph and suppose that BFS is run on G from vertex s V. Then, when BFS terminates, d v = δ(s, v) for all v V where δ(s, v) = if there is no path from s to v in G.

6 claim:

7 lemma: when bfs terminates,

8 lemma: then suppose is enqueued before during bfs

9 breadth first search

10 Theorem 1 (CLRS, p. 599) Let G =(V, E) be a graph and suppose that BFS is run on G from vertex s V. Then, when BFS terminates, d v = δ(s, v) for all v V where δ(s, v) = if there is no path from s to v in G.

11

12 breadth first search

13 dfsall(g =(V, E)) 1 for each v V set d v 2 Set t 0 and S while s V such that d s = 4 do push(s, s) 5 while S is not empty 6 do u pop(s) 7 if d u = or u=s 8 then 9 t t d u t 11 for each v Adj(u) such that d v = 12 do push(s, v)

14 dfs(g, a)

15 partial order

16

17

18 strongly connected component definition: maximal set of vertices such that for every pair (a) u is connected to v (b) v is connected to u

19 example

20 dfs allows one to compute the strongly connected component of a graph.

21 unit edge weight

22 shortest paths b 8 d 8 g 10 2 a 9 e 5 9 i c 1 f 6 h image: thefranciscofamily.org,

23 bfs b 8 d 8 g e 5 9 i 11 a 12 c 1 f 6 h

24 b a c

25 b a c

26 b a c

27 shortest paths b 8 d 8 g 10 2 a 9 e 5 9 i c 1 f 6 h image: thefranciscofamily.org,

28

29 shortest paths b 8 d 8 g 10 2 a 9 e 5 9 i c 1 f 6 h alarmclock model image: thefranciscofamily.org,

30 new data structure

31 binary heap full tree, key value <= to key of children

32 binary heap full tree, key value <= to key of children

33 binary heap full tree, key value <= to key of children

34 binary heap full tree, key value <= to key of children

35 binary heap full tree, key value <= to key of children how to extractmin?

36 binary heap full tree, key value <= to key of children how to extractmin?

37 binary heap full tree, key value <= to key of children how to extractmin?

38 binary heap full tree, key value <= to key of children how to extractmin?

39 binary heap full tree, key value <= to key of children how to decreasekey?

40 binary heap full tree, key value <= to key of children how to decreasekey?

41 implementation use a priority queue to keep track of light edges insert: makequeue: extractmin: decreasekey:

42 @/!(=, 1, ))!"# "%42 % = {)} $" >A%B >A)B C < 9DE:9:9(<, )) %&'() < = $" % F9E:9:9(<)!"# "%42 v 7>?A%B $" '! >AvB = *&)+ >AvB >A%B + 5 9DE:9:9(<,v)

43 algorithm

44 Dijkstra(G =(V,E),s) 1 for all v V 2 do d u π u nil 4 d s 0 5 Q makequeue(v ) use d u as key 6 while Q = 7 do u extractmin(q) 8 for each v Adj (u) 9 do if d v >d u + w(u, v) 10 then d v d u + w(u, v) 11 π v u 12 decreasekey(q, v)

45 @/!(=, 1, ))!"# "%42 % = {)} $" >A%B >A)B C < 9DE:9:9(<, )) %&'() < = $" % F9E:9:9(<)!"# "%42 v 7>?A%B $" '! >AvB = *&)+ >AvB >A%B + 5 9DE:9:9(<,v) Dijkstra(G =(V,E),s) 1 for all v V 2 do d u π u nil 4 d s 0 5 Q makequeue(v ) use d u as key 6 while Q = 7 do u extractmin(q) 8 for each v Adj (u) 9 do if d v >d u + w(u, v) 10 then d v d u + w(u, v) 11 π v u 12 decreasekey(q, v)

46 b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

47 b 8 d 8 g a 9 0 e 5 9 i c 1 f 6 h

48 b 10 8 d 8 g a 9 0 e 5 9 i c 1 12 f 6 h

49 b 10 8 d 18 8 g a 9 0 e 5 9 i c 1 12 f 6 h

50 b 10 8 d 18 8 g a 9 0 e i c 1 12 f 1 6 h

51 b 10 8 d 18 8 g a 9 0 e i c 1 12 f h

52 b 10 8 d 18 8 g a 9 0 e i c 1 12 f h

53 b 10 8 d 18 8 g a 9 0 e i c 1 12 f h

54 b 10 8 d 18 8 g a 9 0 e i c 1 12 f h

55 running time Dijkstra(G =(V,E),s) 1 for all v V 2 do d u π u nil 4 d s 0 5 Q makequeue(v ) use d u as key 6 while Q = 7 do u extractmin(q) 8 for each v Adj (u) 9 do if d v >d u + w(u, v) 10 then d v d u + w(u, v) 11 π v u 12 decreasekey(q, v)

56

57 why does dijkstra work?

58 steps along the way triangle inequality: (u, v) E, δ(s, v) δ(s, u)+w(u, v) upper bound: d v δ(s, v) no paths: δ(s, v) = = d v =