DIRECTED GRAPHS BBM ALGORITHMS TODAY DEPT. OF COMPUTER ENGINEERING ERKUT ERDEM. Apr. 3, Directed graphs. Road network
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1 BBM - ALGORIHMS ODAY DEP. O COMPUER ENGINEERING ERKU ERDEM Directed Graphs Digraph API Digraph search opological sort Strong components DIRECED GRAPHS Apr., Acknowledgement: he course slides are adapted from the slides prepared by R. Sedgewick and K. Wayne of Princeton Uniersity. Directed graphs Digraph. Set of ertices connected pairwise by directed edges. Road network Vertex = intersection; edge = one-way street. ertex of outdegree and indegree directed path from to directed cycle Google - Map data Sanborn, NAVEQ - erms of Use
2 Political blogosphere graph Oernight interbank loan graph Vertex = political blog; edge = link. Vertex = bank; edge = oernight loan. GSCC GWCC GIN GOU DC endril 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, red for conseratie, and blue for liberal. Orange links go from liberal to conseratie, and purple ones from conseratie to liberal. he size of each blog reflects the number of other blogs that link to it. Implication graph x ~x x "#$%&%'$( HI& -)&., * -&/% A*- & *%/"/"-&) "-/ * A++&A/"-, )".M"-/.&/. A*++&) )".A--&A/&) A-&-/. " # " "; HI& -)&. "/I"- &*AI )".A--&A/&) A-&-/ ) -/ I*N& +"-. / %,% -)&. "- *-@ /I&% A-&-/ ";&; #"# $"# $# "" $ " $ " % $ $ "& # ' ", % (# % ; HI& A-&-/ "/I /I& +*%#&./ -$&%, -)&. ". %&,&%%&) / *. /I& )%*$& +"*,-. /'$$"/&" /''$"$& O<=>>P; C- /I&% %). /I& <=>> ". /I& +*%#&./ A-&-/, /I& -&/% "- I"AI *++ -)&. A--&A/ / &*AI /I&% N"* $-)"%&A/&) */I.; HI& %&*"-"-# )".A--&A/&) A-&-/. OB>.P *%&.*++&% A-&-/.,% I"AI /I&.*& ". /%$&; C- &"%"A*+./$)"&. /I& <=>> ".,/&-,$-) / &.&N&%*+ %)&%., *#-"/$)& +*%#&% /I**-@, /I& B>. O.&& Q%)&% "& *- OPP; HI& <=>> A-."./., * )%*$& &'$)-. /'$$"/&" /''$"$& O<>>P * )%*$& '&/''$"$& O<GHP * )%*$& %$/''$"$& O<CDP *-) &"$%- O.&& "#$%& 'P; HI& <>> A%".&. *++ -)&. /I*/ A*- %&*AI &N&%@ /I&% -)& "- /I& <>> /I%$#I * )"%&A/&) */I; R -)& ". "- /I& <GH ", "/ I*. * */I,% /I& <>> $/ -/ / /I& <>>; C- A-/%*./ * -)& ". "- /I& <CD ", "/ I*. * */I / /I& <>> $/ -/,% "/; R -)& ". "- * /&-)%"+ ", "/ )&. -/ %&.")& - * )"%&A/&) */I / %,% /I& <>>;S %/%'$( C- /I& -&/%, *@&-/..&-/ N&% &)"%& %*U" "& *- O:P /I& <>> ". /I& +*%#&./ A-&-/; - *N&%*#& *+./ %&', /I& -)&. "- /I*/ -&/% &+-# / /I& <>>; CA-/%*./ /I& <>> ". $AI.*++&%,% /I&,&)&%*+,$-). -&/%; C- : -+@ (&' ) (', /I& -)&. &+-# / /I". A-&-/; Q@,*% /I& +*%#&./ A-&-/ ". /I& <CD; C- : )%' ) )', /I& -)&. &%& "- /I". A-&-/; HI& <GH A-/*"-&) (*' ) +', *++ -)&. &% )*@ I"+& /I&%& &%& (+' ),', /I& -)&. +A*/&) "- /I& /&-)%"+.;SS V&.. /I*- -' ) (', /I& -)&. &%& "- /I& %&*"-"-# )".A--&A/&) A-&-/. O.&& H*+& JP; longer existed, or had moed to a different location. When looking at the front page of a blog we did not make a distinction between blog references made in blogrolls (blogroll links) from those made in posts (post citations). his had the disadantage of not differentiating between blogs that were actiely mentioned in a post on that day, from blogroll links that remain static oer many weeks []. Since posts usually contain sparse references to other blogs, and blogrolls usually contain dozens of blogs, we assumed that theedge network=obtained crawling the front page of each blog would strongly Vertex = ariable; logicalbyimplication. reflect blogroll links. blogs had blogrolls through blogrolling.com, while many others simply maintained a list of links to their faorite blogs. We did not include blogrolls placed on a secondary page. We constructed a citation network by identifying whether a URL present on the page of one blog if xais page true, link to references another political blog. We called a link found anywhere on a blog s page, distinguish it from a post citation, a link to another blog that occurs strictly within a post. then x is trueigure shows the unmistakable diision between the~x liberal and xconseratie political (blogo)spheres. In fact, % of the links originating within either the conseratie or liberal communities stay within that community. An effect that may not be as apparent from the isualization is that een though we started with a balanced set of blogs, conseratie blogs show a greater tendency to link. % of conseratie blogs link to at least receie a link. In contrast, % of x one other ~xblog, andx% liberal blogs link to another blog, while only % are linked to by another blog. So oerall, we see a slightly higher tendency for conseratie blogs to link. Liberal blogs linked to. blogs on aerage, while conseratie blogs linked to an aerage of., and this difference is almost entirely due to the higher proportion of liberal blogs with no links at all. ~x may not link as generously ~x x x liberal blogs, Although liberal blogs on aerage, the most popular Daily Kos and Eschaton (atrios.blogspot.com), had and links from our single-day snapshot ~x "#$%& '( &)&%*+,$-). -&/%,% &/&&% ' :; <=>>? #"*-/ &*+@ A--&A/&) A-&-/ he opology of? the ederal unds Market, A-&-/ Bech and Atalay, B>? )".A--&A/&) A-&-/ <>> #"*-/./%-#+@ A--&A/&) <CD? #"*-/ "-EA-&-/ <GH? #"*-/ $/E A-&-/; - /I". )*@ /I&%& &%& JK -)&. "- /I& <>> LL -)&. "- /I& <CD :K -)&. "- <GH J -)&. "- /I& /&-)%"+. *-) -)&. "- * )".A--&A/&) A-&-/; Combinational circuit Vertex = logical gate; edge = wire. S HI& /&-)%"+. *@ *+. & )"W&%&-/"*/&) "-/ /I%&&.$A-&-/.( *.&/, -)&. /I*/ *%& - * */I &*-*/"-#,% <CD *.&/, -)&. /I*/ *%& - * */I +&*)"-# / <GH *-) *.&/, -)&. /I*/ *%& - * */I /I*/ &#"-. "- <CD *-) &-). "- <GH; S S "#, -)&. &%& "- X,%E<CDY /&-)%"+. $%#, -)&. &%& "- /I& X/E<GHY /&-)%"+. *-) "&#, -)&. &%& "X/$&.Y,% <CD / <GH; S ~x
3 WordNet graph he McChrystal Afghanistan PowerPoint slide Vertex = synset; edge = hypernym relationship. eent happening occurrence occurrent natural_eent miracle act human_ac on human_ac ity change altera on modi ca on miracle group_ac on damage harm.. impairment transi on increase forfeit forfeiture ac on resistance opposi on leap jump salta on transgression jump leap change demo on aria on mo on moement moe locomo on trael run running descent jump parachu ng dash sprint Digraph applications Some digraph problems 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 Path. Is there a directed path from s to t? s 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? t 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?
4 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS Digraph API public class Digraph Digraph(int V) create an empty digraph with V ertices Digraph(In in) 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) Digraph API Adjacency-lists digraph representation V tinydg.txt E In in = new In(args[]); Digraph G = new Digraph(in); for (int = ; < G.V(); ++) for (int w : G.adj()) StdOut.println( + "->" + w); % jaa Digraph tinydg.txt -> -> -> -> -> -> -> -> -> -> -> - read digraph from input stream print out each edge (once) V tinydg.txt E Maintain ertex-indexed array of lists. V tinydg.txt E adj[] adj[]
5 Adjacency-lists graph representation: Jaa implementation Adjacency-lists digraph 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 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 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 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS 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
6 Reachability Problem. ind all ertices reachable from s along a directed path. s 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. Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] a directed graph a directed graph
7 Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] isit isit Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] isit isit
8 Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] isit isit Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] isit done
9 Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] done isit Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] done done
10 Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] isit isit Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] done done
11 Depth-first search Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices pointing from. marked[] edgeo[] marked[] edgeo[] reachable from ertex done reachable from Depth-first search (in undirected graphs) Recall code for undirected graphs. Depth-first search (in directed graphs) Code for directed graphs identical to undirected one. [substitute Digraph for Graph] 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 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
12 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. 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 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). 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. roots 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.
13 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. Breadth-first search in digraphs application: web crawler Multiple-source shortest paths Multiple-source shortest paths. Gien a digraph and a set of source ertices, find shortest path from tinydg.txt any ertex in the set to each other ertex. V E 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. Bare-bones web crawler: Jaa implementation 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? 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 How many strong components are there in this digraph?
14 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS 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 opological sort opological sort algorithm DAG. Directed acyclic graph. opological sort. Redraw DAG so all edges point upwards. Return ertices in reerse. directed edges DAG Solution. DS. What else? a directed acyclic graph topological order
15 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit isit opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit done
16 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit done opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit isit
17 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. done isit opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit isit
18 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. done isit opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. done check
19 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. check isit opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit isit
20 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit isit opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit done
21 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. isit done opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. check check
22 opological sort algorithm Return ertices in reerse. opological sort algorithm Return ertices in reerse. topological order check done Depth-first search order opological sort in a DAG: correctness proof 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 Proposition. Reerse DS 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
23 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. 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. Directed cycle detection application: cyclic inheritance Directed cycle detection application: spreadsheet recalculation he Jaa compiler does cycle detection. Microsoft Excel does cycle detection (and has a circular reference toolbar) public class A extends B... % jaac A.jaa A.jaa:: cyclic inheritance inoling A public class A extends B ^ error public class B extends C... public class C extends A...
24 Digraph API Digraph search opological sort Strong components DIRECED GRAPHS 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. Connected components s. strongly-connected components Strong component application: ecological food webs and w are connected if there is a path between and w A graph and its connected components connected components connected component id (easy to compute with DS) cc[] public int connected(int, int w) return cc[] == cc[w]; constant-time client connectiity query and w are strongly connected if there is a directed path from to w and a directed path from w to strongly-connected components strongly-connected component id (how to compute?) scc[] public int stronglyconnected(int, int w) return scc[] == scc[w]; constant-time client strong-connectiity query ood web graph. Vertex = species; edge = from producer to consumer. Strong component. Subset of species with common energy flow.
25 Strong component application: software modules Strong components algorithms: brief history 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 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 (). 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. Kosaraju's algorithm: intuition KOSARAJU'S ALGORIHM Reerse graph. Strong components in G are same as in G R. Kernel DAG. Contract each strong component into a single ertex. DS in reerse graph DS in original graph Idea. Compute topological order (reerse ) 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)
26 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] digraph G reerse digraph G R Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit
27 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit done Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit
28 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] done done Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit
29 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit
30 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] done done
31 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] done done
32 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit isit
33 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit done Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit done
34 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] done isit Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] done done
35 Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] marked[] isit done Phase. Compute reerse in G R. Phase. Compute reerse in G R. marked[] check reerse digraph G R
36 Kosaraju's algorithm Simple (but mysterious) algorithm for computing strong components. Run DS on G R to compute reerse. Run DS on G, considering ertices in order gien by first DS. DS in reerse graph DS in original graph KOSARAJU'S ALGORIHM DS in reerse digraph G R check unmarked ertices in the order reerse 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... Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] original digraph G isit
37 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] done strong component: Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit isit
38 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit isit Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit isit
39 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit done Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] done isit
40 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] done done Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit done
41 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] strong component: check Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] check check
42 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] check isit Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit isit
43 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit isit Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit done
44 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] done done Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] done strong component:
45 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] check check Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] check isit
46 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit isit Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit done
47 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit done Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] strong component: isit
48 Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] isit done Phase. Run DS in G, isiting unmarked ertices in reerse of G R. Phase. Run DS in G, isiting unmarked ertices in reerse of G R. scc[] scc[] strong component: check
49 Kosaraju's algorithm Phase. Run DS in G, isiting unmarked ertices in reerse of G R. done scc[] Simple (but mysterious) algorithm for computing strong components. Run DS on G R to compute reerse. 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 dfs() check dfs() dfs() check dfs() check done done done done check check check check check check check Proposition. Second DS gies strong components. () dfs() check check dfs() check done check done dfs() check check done check Connected components in an undirected graph (with DS) Strong components in a digraph (with two DSs) 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]; 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];
50 Digraph-processing summary: algorithms of the day single-source reachability DS topological sort (DAG) DS strong components Kosaraju DS (twice)
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