Analysis of Algorithms Prof. Karen Daniels

Transcription

1 UMass Lowell omputer Science nalysis of lgorithms Prof. Karen aniels Spring, 2013 hapter 22: raph lgorithms & rief Introduction to Shortest Paths [Source: ormen et al. textbook except where noted]

2 Overview: raph lgorithms hapter 22: lementary raph lgorithms Introductory oncepts raph raversals: epth-irst Search readth-irst Search opological Sort Strongly onnected omponents (P figure) rief Introduction to Shortest Paths

3 hapter 22 raph lgorithms Introductory oncepts epth-irst Search readth-irst Search opological Sort [Source: ormen et al. textbook except where noted]

4 Introductory raph oncepts = (V,) Vertex egree Self-Loops dges may have weights (see hapters 23-24) Undirected raph No Self-Loops djacency is symmetric irected raph (digraph) egree: in/out Self-Loops allowed his treatment follows textbook ormen et al. Some definitions differ slightly from other graph literature.

5 Introductory raph oncepts: Motivation Networks are often modeled using graphs.

6 Introductory raph oncepts: Representations Undirected raph irected raph (digraph) djacency Matrix djacency List djacency Matrix djacency List his treatment follows textbook ormen et al. Some definitions differ slightly from other graph literature.

7 Introductory raph oncepts: Paths, ycles Path: length: number of edges simple: all vertices distinct ycle: irected raph: <v 0,v 1,...,v k > forms cycle if v 0 =v k and k>=1 simple cycle: v 1,v 2..,v k also distinct self-loop is cycle of length 1 Undirected raph: <v 0,v 1,...,v k > forms (simple) cycle if v 0 =v k and k>=3 simple cycle: v 1,v 2..,v k also distinct path <,,> simple cycle <,,,> simple cycle <,,,>= <,,,> his treatment follows textbook ormen et al. Some definitions differ slightly from other graph literature.

8 Introductory raph oncepts: onnectivity Undirected raph: connected every pair of vertices is connected by a path one connected component connected components: equivalence classes under is reachable from relation irected raph: strongly connected every pair of vertices is reachable from each other one strongly connected component strongly connected components: equivalence classes under mutually reachable relation maximal sized vertex subset inducing strongly connected subgraph not strongly connected connected 2 connected components strongly connected component his treatment follows textbook ormen et al. Some definitions differ slightly from other graph literature.

9 epth-irst Search (S) & readth-irst Search (S) xamples Vertex olor hanges dge lassification Using the Results of S & S Running ime nalysis

10 epth-irst Search (S)

11 xample: S of irected raph =(V,) dge lassification Legend: : tree edge : back edge : forward edge : cross edge djacency List: :,, :, : - :, :, :, :, Source: raph is from omputer lgorithms: Introduction to esign and nalysis by aase and elder.

12 S Pseudoode S() 1 for each vertex u V[] 2 do color[u] WHI 3 for each vertex u V[] 4 do if color[u] is WHI 5 then S_VISI(, u) S_VISI(, u) 1 color[u] RY 2 for each vertex v djlist[u] 3 do if color[v] is WHI 4 then S_VISI(,v) 5 color[u] LK see web handout for full details

13 Vertex olor hanges Vertex is WHI if it has not yet been encountered during the search. Vertex is RY if it has been encountered but has not yet been fully explored. Vertex is LK if it has been fully explored.

14 dge lassification ach edge of the original graph is classified during the search produces information needed to: build S or S spanning forest of trees detect cycles (S) or find shortest paths (S) When vertex u is being explored, edge e = (u,v) is classified based on the color of v when the edge is first explored: e is a tree edge if v is WHI [for S and S] e is a back edge if v is RY [for S only] for S this means v is an ancestor of u in the S tree e is a forward edge if v is LK and [for S only] v is a descendent of u in the S tree e is a cross edge if v is LK and [for S only] there is no ancestor or descendent relationship between u and v in the S tree Note that: or S we ll only consider tree edges. or S we consider all 4 edge types. In S of an undirected graph, every edge is either a tree edge or a back edge.

15 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

16 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

17 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

18 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

19 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

20 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

21 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

22 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

23 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

24 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

25 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

26 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

27 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

28 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

29 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

30 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

31 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

32 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

33 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

34 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

35 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

36 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

37 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

38 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

39 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

40 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

41 xample: (continued) S of irected raph djacency List: :,, :, : - :, :, :, :,

42 xample: (continued) S of irected raph xample: (continued) S of irected raph S ree 1 S ree 2

43 xample: S of Undirected raph =(V,) djacency List: :,,, :,, :,,,, :,,, :, :, :,

44 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

45 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

46 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

47 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

48 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

49 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

50 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

51 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

52 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

53 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

54 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

55 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

56 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

57 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

58 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

59 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

60 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

61 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

62 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

63 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

64 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

65 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

66 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

67 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

68 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :,

69 xample: (continued) S of Undirected raph xample: (continued) S of Undirected raph S ree

70 lementary raph lgorithms: S Review problem: RU or LS? he tree shown below on the right can be a S tree for some adjacency list representation of the graph shown below on the left. ack dge ree dge ree dge ree dge ree dge ross dge ree dge

71 readth-irst Search (S)

72 S Pseudoode See handout on web

73 xample: S of irected raph =(V,) dge lassification Legend: djacency List: :,, :, : - :, :, :, :, : tree edge only tree edges are used Source: raph is from omputer lgorithms: Introduction to esign and nalysis by aase and elder.

74 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

75 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

76 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

77 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

78 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

79 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

80 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

81 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

82 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

83 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue: -

84 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

85 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

86 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue:

87 xample: S of irected raph djacency List: :,, :, : - :, :, :, :, Queue: -

88 xample: (continued) S of irected raph Shortest path distance from : to = 1 to = 1 to = 1 to = 2 S ree 1 Shortest path distance from : to = 1 S ree 2

89 xample: S of Undirected raph =(V,) dge lassification Legend: : tree edge only tree edges are used djacency List: :,,, :,, :,,,, :,,, :, :, :,

90 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

91 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

92 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

93 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

94 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

95 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

96 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

97 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

98 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

99 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

100 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

101 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

102 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue:

103 xample: S of Undirected raph djacency List: :,,, :,, :,,,, :,,, :, :, :, Queue: -

104 xample: (continued) S of Undirected raph Shortest path distance from : to = 1 to = 2 to = 1 to = 2 to = 1 to = 1 S ree

105 epth-irst Search (S) & readth-irst Search (S) Using the Results of S & S Running ime nalysis

106 Using the Results of S & S Using S to etect ycles: directed graph is acyclic if and only if a epth-irst Search of yields no back edges. see p. 614 of text for proof Note: S can also be used to detect cycles in undirected graphs if notion of cycle is defined appropriately. S can also be used to identify strongly connected components in directed graphs and topologically sort. Using S for Shortest Paths: readth-irst Search of yields shortest path information: or each readth-irst Search tree, the path from its root u to a vertex v yields the shortest path from u to v in. see p of text for proof

107 lementary raph lgorithms: SRHIN: S, S for unweighted directed or undirected graph =(V,) ime: O( V + ) adj list O( V 2 ) adj matrix predecessor subgraph = forest of spanning trees readth-irst-search (S): Shortest Path istance rom source to each reachable vertex Record during traversal oundation of many shortest path algorithms Vertex color shows status: not yet encountered encountered, but not yet finished finished epth-irst-search (S): ncountering, finishing times well-formed nested (( )( ) ) structure S of undirected graph produces only back edges or tree edges irected graph is acyclic if and only if S yields no back edges See S, S Handout for Pseudoode

108 Running ime nalysis Key ideas in the analysis are similar for S and S. In both cases we assume an djacency List representation. Let s examine S. Let t be number of S trees generated by S Outer loop in S() executes t times each execution contains call: S_Visit(,u) each such call constructs a S tree by visiting (recursively) every node reachable from vertex u t ime: time to construct S tree i i 1 Now, let r i be the number of vertices in S tree i ime to construct S tree i: S() 1 for each vertex u V[] 2 do color[u] WHI 3 for each vertex u V[] 4 do if color[u] is WHI 5 then S_VISI(, u) S_VISI(, u) 1 color[u] RY 2 for each vertex v djlist[u] r i j 1 djlist [ jth vertex in S tree i] 3 do if color[v] is WHI 4 then S_VISI(,v) continued on next slide... 5 color[u] LK

109 otal S time: Running ime nalysis i t i 1 r j 1 djlist vertex S Now, consider this expression for the extreme values of t: if t=1, all edges are in one S tree and the expression simplifies to O( ) if t= V, each vertex is its own (degenerate) S tree with no edges so the expression simplifies to O( V ) O( V + ) is therefore an upper bound on the time for the extreme cases or values of t in between 1 and V we have these contributions to running time: 1 for each vertex that is its own (degenerate) S tree with no edges upper bound on this total is O( V ) djlist[u] for each vertex u that is a node of a non-degenerate S tree upper bound on this total is O( ) [ jth tree otal time for values of t in between 1 and V is therefore also O( V + ) otal time= O( V ) Note that for an djacency Matrix representation, we would need to scan an entire matrix row (containing V entries) each time we examined the vertices adjacent to a vertex. his would make the running time O( V 2 ) instead of O( V + ). in i]

110 opological Sort Source: Previous instructors

111 efinition: irected cyclic raph often abbreviated s used in many applications to indicate precedence among events. If S of a directed graph yields no back edges, then the graph contains no cycles [Lemma in text] his graph has more than one cycle. an you find them all? his graph has no cycles, so it is a.

112 efinition: opological Sort topological sort of a = (V, ) is a linear ordering of all its vertices such that if contains an edge (u, v), then u appears before v in the ordering. If the graph is not acyclic, then no linear ordering is possible. topological sort of a graph can be viewed as an ordering of its vertices along a horizontal line so that all directed edges go from left to right. opological sorting is thus different from the usual kind of "sorting".

113 lementary raph lgorithms: opological Sort for irected, cyclic raph () =(V,) OPOLOIL-SOR() 1 S() computes finishing times for each vertex 2 as each vertex is finished, insert it onto front of list 3 return list Produces linear ordering of vertices. or edge (u,v), u is ordered before v. source: textbook ormen et al.

114 opological Sort he following algorithm topologically sorts a : OPOLOIL-SOR() call S() to compute finishing times f[v] for each vertex v (this is equal to the order in which vertices change color from gray to black) as each vertex is finished (turns black), insert it onto the front of a linked list return the linked list of vertices

115 xample or this : S produces this result: this contains 2 S trees Vertices are blackened in the following order:,,,,,,

116 xample Vertices are added to front of a linked list in the blackening order. inal result is shown below Note that all tree edges and non-tree edges point to the right

117 opological Sort We can perform a topological sort in time (V + ), since depth-first search takes (V + ) time and it takes 0(1) time to insert each of the V vertices onto the front of the linked list.

118 opological Sort heorem 23.11: OPOLOIL-SOR() produces a topological sort of a directed acyclic graph. Proof: Suppose that S is run on a given dag = (V, ) to determine finishing times for its vertices. It suffices to show that for any pair of distinct vertices u,v Î V, if there is an edge in from u to v, then f[v] < f[u]. onsider any edge (u,v) explored by S(). When this edge is explored, v cannot be gray, since then v would be an ancestor of u and (u,v) would be a back edge, contradicting Lemma herefore, v must be either white or black. If v is white, it becomes a descendant of u, and so f[v] < f[u]. If v is black, then f[v] < f[u] as well. hus, for any edge (u,v) in the dag, we have f[v] < f[u], proving the theorem.

119 S as a asis for Shortest Path lgorithms for unweighted, undirected graph =(V,) S ime: O( V + ) adj list O( V 2 ) adj matrix Problem: iven 2 vertices u, v, find the shortest path in from u to v. Problem: iven a vertex u, find the shortest path in from u to each vertex. Source/Sink Shortest Path Single-Source Shortest Paths Solution: S starting at u. Stop at v. Solution: S starting at u. ull S tree. Problem: ind the shortest path in from each vertex u to each vertex v. ll-pairs Shortest Paths Solution: or each u: S starting at u; full S tree. ime: O( V ( V + )) adj list O( V 3 ) adj matrix source: based on Sedgewick, raph lgorithms