DIRECTED GRAPHS BBM ALGORITHMS DEPT. OF COMPUTER ENGINEERING ERKUT ERDEM. Apr. 3, 2013

Transcription

1 BBM - ALGORIHMS DEP. O COMPUER ENGINEERING ERKU ERDEM DIRECED GRAPHS Apr., Acknowledgement: he course slides are adapted from the slides prepared by R. Sedgewick and K. Wayne of Princeton Uniersity.

2 Directed Graphs Digraph API Digraph search opological sort Strong components ODAY

3 Directed graphs Digraph. Set of ertices connected pairwise by directed edges. ertex of outdegree and indegree directed path from to directed cycle

4 Road network Vertex = intersection; edge = one-way street. Google - Map data Sanborn, NAVEQ - erms of Use

5 Political blogosphere graph Vertex = political blog; edge = link. he Political Blogosphere and the U.S. Election: Diided hey Blog, Adamic and Glance, igure : Community structure of political blogs (expanded set), shown using utilizing a GEM layout [] in the GUESS[] isualization and analysis tool. he colors reflect political orientation,

6 Oernight interbank loan graph Vertex = bank; edge = oernight loan. GSCC GWCC GIN GOU DC endril he opology of the ederal unds Market, Bech and Atalay,

7 Implication graph Vertex = ariable; edge = logical implication. ~x x if x is true, then x is true x ~x x ~x ~x x x ~x x ~x ~x x

8 Combinational circuit Vertex = logical gate; edge = wire.

9 WordNet graph Vertex = synset; edge = hypernym relationship. eent happening occurrence occurrent natural_eent change altera on modi ca on miracle miracle act human_ac on human_ac ity group_ac on damage harm.. impairment transi on increase forfeit forfeiture ac on resistance opposi on transgression leap jump salta on jump leap change demo on mo on moement moe aria on locomo on trael descent run running jump parachu ng dash sprint

10 he McChrystal Afghanistan PowerPoint slide

11 Digraph applications digraph ertex directed edge transportation street intersection one-way street web web page hyperlink food web species predator-prey relationship WordNet synset hypernym scheduling task precedence constraint financial bank transaction cell phone person placed call infectious disease person infection game board position legal moe citation journal article citation object graph object pointer inheritance hierarchy class inherits from control flow code block jump

12 Some digraph problems Path. Is there a directed path from s to t? Shortest path. What is the shortest directed path from s to t? opological sort. Can you draw the digraph so that all edges point upwards? Strong connectiity. Is there a directed path between all pairs of ertices? ransitie closure. or which ertices and w is there a path from to w? PageRank. What is the importance of a web page? s t

13 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS

14 Digraph API public class Digraph Digraph(int V) Digraph(In in) create an empty digraph with V ertices create a digraph from input stream oid addedge(int, int w) add a directed edge w Iterable<Integer> adj(int ) ertices pointing from int V() int E() number of ertices number of edges Digraph reerse() reerse of this digraph String tostring() string representation In in = new In(args[]); Digraph G = new Digraph(in);! for (int = ; < G.V(); ++) for (int w : G.adj()) StdOut.println( + "->" + w); read digraph from input stream print out each edge (once)

15 Digraph API V tinydg.txt E % jaa Digraph tinydg.txt -> -> -> -> -> -> -> -> -> -> -> - In in = new In(args[]); Digraph G = new Digraph(in);! for (int = ; < G.V(); ++) for (int w : G.adj()) StdOut.println( + "->" + w); read digraph from input stream print out each edge (once)

16 Adjacency-lists digraph representation Maintain ertex-indexed array of lists. DG.txt E adj[]

17 Adjacency-lists graph representation: Jaa implementation public class Graph { priate final int V; priate final Bag<Integer>[] adj;! public Graph(int V) { this.v = V; adj = (Bag<Integer>[]) new Bag[V]; for (int = ; < V; ++) adj[] = new Bag<Integer>(); }! public oid addedge(int, int w) { adj[].add(w); adj[w].add(); }! public Iterable<Integer> adj(int ) { return adj[]; } } adjacency lists create empty graph with V ertices add edge w iterator for ertices adjacent to

18 Adjacency-lists digraph representation: Jaa implementation public class Digraph { priate final int V; priate final Bag<Integer>[] adj;! public Digraph(int V) { this.v = V; adj = (Bag<Integer>[]) new Bag[V]; for (int = ; < V; ++) adj[] = new Bag<Integer>(); }! public oid addedge(int, int w) { adj[].add(w);! }! public Iterable<Integer> adj(int ) { return adj[]; } } adjacency lists create empty digraph with V ertices add edge w iterator for ertices pointing from

19 Digraph representations In practice. Use adjacency-lists representation. Algorithms based on iterating oer ertices pointing from. Real-world digraphs tend to be sparse. huge number of ertices, small aerage ertex degree representation space insert edge from to w edge from to w? iterate oer ertices pointing from? list of edges E E E adjacency matrix V V adjacency lists E + V outdegree() outdegree() disallows parallel edges

20 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS

21 Reachability Problem. ind all ertices reachable from s along a directed path. s

22 Depth-first search in digraphs Same method as for undirected graphs. Eery undirected graph is a digraph (with edges in both directions). DS is a digraph algorithm. DS (to isit a ertex ) Mark as isited. Recursiely isit all unmarked ertices w pointing from.

23 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. a directed graph

24 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] a directed graph

25 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

26 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

27 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

28 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

29 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

30 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

31 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

32 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

33 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

34 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

35 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

36 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

37 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

38 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] isit

39 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

40 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

41 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] done

42 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] reachable from ertex reachable from

43 Depth-first search (in undirected graphs) Recall code for undirected graphs. public class DepthirstSearch { priate boolean[] marked;! public DepthirstSearch(Graph G, int s) { marked = new boolean[g.v()]; dfs(g, s); }! priate oid dfs(graph G, int ) { marked[] = true; for (int w : G.adj()) if (!marked[w]) dfs(g, w); }! public boolean isited(int ) { return marked[]; } } true if path to s constructor marks ertices connected to s recursie DS does the work client can ask whether any ertex is connected to s

44 Depth-first search (in directed graphs) Code for directed graphs identical to undirected one. [substitute Digraph for Graph] public class DirectedDS { priate boolean[] marked;! public DirectedDS(Digraph G, int s) { marked = new boolean[g.v()]; dfs(g, s); }! priate oid dfs(digraph G, int ) { marked[] = true; for (int w : G.adj()) if (!marked[w]) dfs(g, w); }! public boolean isited(int ) { return marked[]; } } true if path from s constructor marks ertices reachable from s recursie DS does the work client can ask whether any ertex is reachable from s

45 Reachability application: program control-flow analysis Eery program is a digraph. Vertex = basic block of instructions (straight-line program). Edge = jump. Dead-code elimination. ind (and remoe) unreachable code. Infinite-loop detection. Determine whether exit is unreachable.

46 Reachability application: mark-sweep garbage collector Eery data structure is a digraph. Vertex = object. Edge = reference. Roots. Objects known to be directly accessible by program (e.g., stack). Reachable objects. Objects indirectly accessible by program (starting at a root and following a chain of pointers). roots

47 Reachability application: mark-sweep garbage collector Mark-sweep algorithm. [McCarthy, ] Mark: mark all reachable objects. Sweep: if object is unmarked, it is garbage (so add to free list). Memory cost. Uses extra mark bit per object (plus DS stack). roots

48 Depth-first search in digraphs summary DS enables direct solution of simple digraph problems. Reachability. Path finding. opological sort. Directed cycle detection. Basis for soling difficult digraph problems. -satisfiability. Directed Euler path. Strongly-connected components. SIAM J. COMPU. Vol., No., June DEPH-IRS SEARCH AND LINEAR GRAPH ALGORIHMS* ROBER ARJAN" Abstract. he alue of depth-first search or "bacltracking" as a technique for soling problems is illustrated by two examples. An improed ersion of an algorithm for finding the strongly connected components of a directed graph and ar algorithm for finding the biconnected components of an undirect graph are presented. he space and time requirements of both algorithms are bounded by k V + ke d- k for some constants kl, k, and k a, where Vis the number of ertices and E is the number of edges of the graph being examined.

49 Breadth-first search in digraphs Same method as for undirected graphs. Eery undirected graph is a digraph (with edges in both directions). BS is a digraph algorithm. s BS (from source ertex s) Put s onto a IO queue, and mark s as isited. Repeat until the queue is empty: - remoe the least recently added ertex - for each unmarked ertex pointing from : add to queue and mark as isited. Proposition. BS computes shortest paths (fewest number of edges) from s to all other ertices n a digraph in time proportional to E+V.

50 Multiple-source shortest paths Multiple-source shortest paths. Gien a digraph and a set of source ertices, find shortest path from any ertex in the set to each other ertex. Ex. S={,, }. Shortest path to is. Shortest path to is. Shortest path to is. Q. How to implement multi-source constructor? A. Use BS, but initialize by enqueuing all source ertices.

51 Breadth-first search in digraphs application: web crawler Goal. Crawl web, starting from some root web page, say Solution. BS with implicit graph. BS. Choose root web page as source s. Maintain a Queue of websites to explore. Maintain a SE of discoered websites. Dequeue the next website and enqueue websites to which it links (proided you haen't done so before). Q. Why not use DS? How many strong components are there in this digraph?

52 Bare-bones web crawler: Jaa implementation Queue<String> queue = new Queue<String>(); SE<String> discoered = new SE<String>();! String root = " queue.enqueue(root); discoered.add(root);! while (!queue.isempty()) { String = queue.dequeue(); StdOut.println(); In in = new In(); String input = in.readall();! String regexp = " Pattern pattern = Pattern.compile(regexp); Matcher matcher = pattern.matcher(input); while (matcher.find()) { String w = matcher.group(); if (!discoered.contains(w)) { discoered.add(w); queue.enqueue(w); } } } queue of websites to crawl set of discoered websites start crawling from root website read in raw html from next website in queue use regular expression to find all URLs in website of form [crude pattern misses relatie URLs] if undiscoered, mark it as discoered and put on queue

53 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS

54 Precedence scheduling Goal. Gien a set of tasks to be completed with precedence constraints, in which order should we schedule the tasks?! Digraph model. ertex = task; edge = precedence constraint.. Algorithms. Complexity heory. Artificial Intelligence. Intro to CS. Cryptography. Scientific Computing. Adanced Programming tasks precedence constraint graph feasible schedule

55 opological sort DAG. Directed acyclic graph.! opological sort. Redraw DAG so all edges point upwards.!!!!!!!!!! directed edges DAG! Solution. DS. What else? topological order

56 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. a directed acyclic graph

57 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

58 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

59 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

60 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

61 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

62 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

63 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

64 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

65 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

66 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

67 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

68 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

69 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

70 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

71 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

72 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder check

73 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder check

74 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

75 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

76 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

77 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

78 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

79 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

80 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

81 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder isit

82 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder done

83 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder check

84 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder check

85 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder check

86 opological sort algorithm Run depth-first search. Return ertices in reerse postorder. postorder topological order done

87 Depth-first search order public class DepthirstOrder { priate boolean[] marked; priate Stack<Integer> reersepost;! public DepthirstOrder(Digraph G) { reersepost = new Stack<Integer>(); marked = new boolean[g.v()]; for (int = ; < G.V(); ++) if (!marked[]) dfs(g, ); }! priate oid dfs(digraph G, int ) { marked[] = true; for (int w : G.adj()) if (!marked[w]) dfs(g, w); reersepost.push(); } } public Iterable<Integer> reersepost() { return reersepost; } returns all ertices in reerse DS postorder

88 opological sort in a DAG: correctness proof Proposition. Reerse DS postorder of a DAG is a topological order. Pf. Consider any edge w. When dfs() is called: Case : dfs(w) has already been called and returned. hus, w was done before. Case : dfs(w) has not yet been called. dfs(w) will get called directly or indirectly by dfs() and will finish before dfs(). hus, w will be done before. Case : dfs(w) has already been called, but has not yet returned. Can t happen in a DAG: function call stack contains path from w to, so w would complete a cycle. Ex: case case dfs() dfs() dfs() done done dfs() done dfs() check done done check check dfs() check check check dfs() done done check check check done all ertices pointing from are done before is done, so they appear after in topological order

89 Directed cycle detection Proposition. A digraph has a topological order iff no directed cycle. Pf. If directed cycle, topological order impossible. If no directed cycle, DS-based algorithm finds a topological order. a digraph with a directed cycle Goal. Gien a digraph, find a directed cycle. Solution. DS. What else? See textbook.

90 Directed cycle detection application: precedence scheduling Scheduling. Gien a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks?!!!!!!!! Remark. A directed cycle implies scheduling problem is infeasible.

91 Directed cycle detection application: cyclic inheritance he Jaa compiler does cycle detection. public class A extends B {... } public class B extends C {... } % jaac A.jaa A.jaa:: cyclic inheritance inoling A public class A extends B { } ^ error public class C extends A {... }

92 Directed cycle detection application: spreadsheet recalculation Microsoft Excel does cycle detection (and has a circular reference toolbar!)

93 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS

94 Strongly-connected components Def. Vertices and w are strongly connected if there is a directed path from to w and a directed path from w to. Key property. Strong connectiity is an equialence relation: is strongly connected to. If is strongly connected to w, then w is strongly connected to. If is strongly connected to w and w to x, then is strongly connected to x. Def. A strong component is a maximal subset of strongly-connected ertices.

95 Connected components s. strongly-connected components and w are connected if there is a path between and w and w are strongly connected if there is a directed path from to w and a directed path from w to A graph and its connected components connected components strongly-connected components connected component id (easy to compute with DS) cc[] strongly-connected component id (how to compute?) scc[] public int connected(int, int w) { return cc[] == cc[w]; } public int stronglyconnected(int, int w) { return scc[] == scc[w]; } constant-time client connectiity query constant-time client strong-connectiity query

96 Strong component application: ecological food webs ood web graph. Vertex = species; edge = from producer to consumer.!!!!!!!!!!!!! Strong component. Subset of species with common energy flow.

97 Strong component application: software modules Software module dependency graph. Vertex = software module. Edge: from module to dependency.!!!!!!!! irefox Internet Explorer Strong component. Subset of mutually interacting modules. Approach. Package strong components together. Approach. Use to improe design!

98 Strong components algorithms: brief history s: Core OR problem. Widely studied; some practical algorithms. Complexity not understood. : linear-time DS algorithm (arjan). Classic algorithm. Leel of difficulty: Algs++. Demonstrated broad applicability and importance of DS. s: easy two-pass linear-time algorithm (Kosaraju-Sharir). orgot notes for lecture; deeloped algorithm in order to teach it! Later found in Russian scientific literature (). s: more easy linear-time algorithms. Gabow: fixed old OR algorithm. Cheriyan-Mehlhorn: needed one-pass algorithm for LEDA.

99 Kosaraju's algorithm: intuition Reerse graph. Strong components in G are same as in G R. Kernel DAG. Contract each strong component into a single ertex. Idea. Compute topological order (reerse postorder) in kernel DAG. Run DS, considering ertices in reerse topological order. how to compute? first ertex is a sink (has no edges pointing from it) Kernel DAG in reerse topological order digraph G and its strong components kernel DAG of G (in reerse topological order)

100 DS in reerse graph DS in original graph KOSARAJU'S ALGORIHM

101 Kosaraju-Sharir Phase. Compute reerse postorder in G R. digraph G

102 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] reerse digraph G R

103 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

104 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

105 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

106 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

107 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

108 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

109 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

110 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

111 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

112 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

113 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

114 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

115 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

116 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

117 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

118 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

119 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

120 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

121 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

122 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

123 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

124 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

125 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

126 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

127 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

128 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

129 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

130 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

131 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

132 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

133 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

134 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

135 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

136 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

137 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] isit

138 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] done

139 Kosaraju-Sharir Phase. Compute reerse postorder in G R. marked[] check

140 Kosaraju-Sharir Phase. Compute reerse postorder in G R. reerse digraph G R

141 Kosaraju's algorithm Simple (but mysterious) algorithm for computing strong components. Run DS on G R to compute reerse postorder. Run DS on G, considering ertices in order gien by first DS. DS in reerse digraph G R check unmarked ertices in the order reerse postorder for use in second dfs() dfs() dfs() dfs() check done dfs() done done dfs() dfs() dfs() dfs() dfs() check dfs() check done done check check done...

142 DS in reerse graph DS in original graph KOSARAJU'S ALGORIHM

143 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] original digraph G

144 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

145 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

146 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] strong component:

147 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

148 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

149 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

150 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

151 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

152 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

153 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

154 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

155 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

156 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

157 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

158 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

159 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

160 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

161 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] strong component:

162 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

163 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

164 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

165 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

166 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

167 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

168 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

169 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

170 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

171 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

172 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

173 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

174 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

175 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

176 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] strong component:

177 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

178 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

179 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

180 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

181 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

182 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

183 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

184 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

185 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

186 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

187 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] strong component:

188 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

189 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] isit

190 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

191 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] strong component:

192 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] check

193 Kosaraju-Sharir Phase. Run DS in G, isiting unmarked ertices in reerse postorder of G R. scc[] done

194 Kosaraju's algorithm Simple (but mysterious) algorithm for computing strong components. Run DS on G R to compute reerse postorder. Run DS on G, considering ertices in order gien by first DS.! DS in original digraph G!!!!!!!! check unmarked ertices in the order!!! dfs() done dfs() dfs() dfs() dfs() check dfs() check check done done check done done check done check check check check dfs() check dfs() dfs() check dfs() check done done done done check check check Proposition. Second DS gies strong components. (!!) dfs() check check dfs() check done check done dfs() check check done check

195 Connected components in an undirected graph (with DS) public class CC { priate boolean marked[]; priate int[] id; priate int count;! public CC(Graph G) { marked = new boolean[g.v()]; id = new int[g.v()];! for (int = ; < G.V(); ++) { if (!marked[]) { dfs(g, ); count++; } } }! priate oid dfs(graph G, int ) { marked[] = true; id[] = count; for (int w : G.adj()) if (!marked[w]) dfs(g, w); }! public boolean connected(int, int w) { return id[] == id[w]; } }

196 Strong components in a digraph (with two DSs) public class KosarajuSCC { priate boolean marked[]; priate int[] id; priate int count;! public KosarajuSCC(Digraph G) { marked = new boolean[g.v()]; id = new int[g.v()]; DepthirstOrder dfs = new DepthirstOrder(G.reerse()); for (int : dfs.reersepost()) { if (!marked[]) { dfs(g, ); count++; } } }! priate oid dfs(digraph G, int ) { marked[] = true; id[] = count; for (int w : G.adj()) if (!marked[w]) dfs(g, w); }! public boolean stronglyconnected(int, int w) { return id[] == id[w]; } }

197 Digraph-processing summary: algorithms of the day single-source reachability DS topological sort (DAG) DS strong components Kosaraju DS (twice)