Algorithms. Algorithms. Algorithms 4.2 DIRECTED GRAPHS. introduction. introduction. digraph API digraph search topological sort

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1 Algorithms ROBERT SEDGEWICK KEVIN WAYNE. DIRECTED GRAPHS. DIRECTED GRAPHS introduction introduction Algorithms F O U R T H E D I T I O N digraph API digraph search topological sort Algorithms digraph API digraph search topological sort ROBERT SEDGEWICK KEVIN WAYNE strong components ROBERT SEDGEWICK KEVIN WAYNE strong components Directed graphs Road network Digraph. Set of vertices connected pairwise by directed edges. Vertex = intersection; edge = one-way street. vertex of outdegree and indegree directed path from to directed cycle Google - Map data Sanborn, NAVTEQ - Terms of Use

2 Political blogosphere graph Overnight interbank loan graph Vertex = political blog; edge = link. Vertex = bank; edge = overnight loan. GSCC GWCC GIN GOUT DC Tendril The Political Blogosphere and the U.S. Election: Divided They Blog, Adamic and Glance, Figure : Community structure of political blogs (expanded set), shown using utilizing a GEM layout [] in the GUESS[] visualization and analysis tool. The colors reflect political orientation, red for conservative, and blue for liberal. Orange links go from liberal to conservative, and purple ones from conservative to liberal. The size of each blog reflects the number of other blogs that link to it. not make a distinction between blog references made in blogrolls (blogroll links) from those made in posts (post citations). This had the disadvantage of not differentiating between blogs that were actively mentioned in a post on that day, from implication. blogroll links that remain static over many weeks []. Vertex = variable; edge = logical Since posts usually contain sparse references to other blogs, and blogrolls usually contain dozens of blogs, we assumed that the network obtained by crawling the front page of each blog would strongly reflect blogroll links. 7 blogs had blogrolls through blogrolling.com, while many others simply maintained a list of links to their favorite blogs. We did not include blogrolls placed on a secondary if x is true, page. then x is true We constructed a citation network by identifying whether a URL present on the page of one blog ~x found anywhere x references another political blog. We called a link on a blog s page, a page link to distinguish it from a post citation, a link to another blog that occurs strictly within a post. Figure shows the unmistakable division between the liberal and conservative political (blogo)spheres. In fact, % of the links originating within either the conservative or liberal communities stay within x not be as ~xapparent x the visualization is that even though that community. An effect that may from we started with a balanced set of blogs, conservative blogs show a greater tendency to link. % of conservative blogs link to at least one other blog, and % receive a link. In contrast, 7% of liberal blogs link to another blog, while only 7% are linked to by another blog. So overall, we see a slightly higher tendency to link. Liberal ~x for conservative blogs~x x blogs linked to. xblogs on average, while conservative blogs linked to an average of., and this difference is almost entirely due to the higher proportion of liberal blogs with no links at all. Although liberal blogs may not link as generously on average, the most popular liberal blogs, Daily Kos and Eschaton (atrios.blogspot.com), had and links from our single-day snapshot x S HI& /&-)%"+. *@ *+. & )"W&%&-/"*/&) "-/ /I%&&.$A-&-/.( *.&/, -)&. /I*/ *%& - * */I &*-*/"-#,% <CD *.&/, -)&. /I*/ *%& - * */I +&*)"-# / <FGH *-) *.&/, -)&. /I*/ *%& - * */I /I*/ &#"-. "- <CD *-) &-). "- <FGH; S S!!"#, -)&. &%& "- X,%E<CDY /&-)%"+. $!%#, -)&. &%& "- /I& X/E<FGHY /&-)%"+. *-) "!&#, -)&. &%& "X/$&.Y,% <CD / <FGH; ~x S7 ~x!"#$%&%'$( HI& -)&., * -&/% A*- & *%/"/"-&) "-/ * A++&A/"-, )".M"-/.&/. A*++&) )".A--&A/&) A-&-/.!!! " # "!!circuit!!! "; HI& -)&. "/I"- &*AI )".A--&A/&) A-&-/ ) -/ I*N& +"-. / %,% Combinational -)&. "- *-@ /I&% A-&-/ ";&; #!"# $"#!$# "" $ " $!!!! " % $ $!!!!! "& # ' ", % (# %! ; HI& A-&-/ "/I /I& +*%#&./ -$&%, -)&. ". %&,&%%&) / *. /I& )%*$& +"*,-. /'$$"/&" /''$"$& O<=>>P; C- /I&% %). /I& <=>> ". /I& +*%#&./ A-&-/, /I& -&/% "- I"AI *++ -)&. A--&A/ / &*AI /I&% N"* Vertex = logical gate; edge = wire. $-)"%&A/&) */I.; HI& %&*"-"-# )".A--&A/&) A-&-/. OB>.P *%&.*++&% A-&-/.,% I"AI /I&.*& ". /%$&; C- &"%"A*+./$)"&. /I& <=>> ".,/&-,$-) / &.&N&%*+ %)&%., *#-"/$)& +*%#&% /I**-@, /I& B>. O.&& Q%)&% "& *- O7PP; HI& <=>> A-."./., * )%*$& &'$)-. /'$$"/&" /''$"$& O<>>P * )%*$& '&7/''$"$& O<FGHP * )%*$& %$7/''$"$& O<CDP *-) &"$%- O.&&!"#$%& 'P; HI& <>> A%".&. *++ -)&. /I*/ A*- %&*AI &N&%@ /I&% -)& "- /I& <>> /I%$#I * )"%&A/&) */I; R -)& ". "- /I& <FGH ", "/ I*. * */I,% /I& <>> $/ -/ / /I& <>>; C- A-/%*./ * -)& ". "- /I& <CD ", "/ I*. * */I / /I& <>> $/ -/,% "/; R -)& ". "- * /&-)%"+ ", "/ )&. -/ %&.")& - * )"%&A/&) */I / %,% /I& <>>;S!%/%'$( C- /I& -&/%, *@&-/..&-/ N&%!&)"%& %*U" "& *- O7:P /I& <>> ". /I& +*%#&./ A-&-/; F- *N&%*#& *+./ %&', /I& -)&. "- /I*/ -&/% &+-# / /I& <>>; CA-/%*./ /I& <>> ". $AI.*++&%,% /I&,&)&%*+,$-). -&/%; C- 7: -+@ (&' ) (', /I& -)&. &+-# / /I". A-&-/; Q@,*% /I& +*%#&./ A-&-/ ". /I& <CD; C- 7: )%' ) )', /I& -)&. &%& "- /I". A-&-/; HI& <FGH A-/*"-&) (*' ) +', *++ -)&. &% )*@ I"+& /I&%& &%& (+' ),', /I& -)&. +A*/&) "- /I& /&-)%"+.;SS V&.. /I*- -' ) (', /I& -)&. &%& "- /I& %&*"-"-# )".A--&A/&) A-&-/. O.&& H*+& JP; Implication longer existed, or graph had moved to a different location. When looking at the front page of a blog we did ~x!"#$%& '(!&)&%*+,$-). -&/%,% &/&&% 7' 7:; <=>>? #"*-/ &*+@ A--&A/&) A-&-/ The Topology of? the Federal Funds Market, A-&-/ Bech and Atalay, B>? )".A--&A/&) A-&-/ <>> #"*-/./%-#+@ A--&A/&) <CD? #"*-/ "-EA-&-/ <FGH? #"*-/ $/E A-&-/; F- /I". )*@ /I&%& &%& JK -)&. "- /I& <>> LL -)&. "- /I& <CD :K -)&. "- <FGH J -)&. "- /I& /&-)%"+. *-) 7 -)&. "- * )".A--&A/&) A-&-/; x 7

3 WordNet graph Digraph applications Vertex = synset; edge = hypernym relationship. event digraph vertex directed edge transportation street intersection one-way street web web page hyperlink happening occurrence occurrent natural_event change alteraon modicaon miracle miracle act human_acon human_acvity food web species predator-prey relationship WordNet synset hypernym scheduling task precedence constraint damage harm.. impairment transion increase forfeit forfeiture acon group_acon financial bank transaction leap jump saltaon jump leap resistance opposion change transgression cell phone person placed call infectious disease person infection demoon variaon moon movement move game board position legal move citation journal article citation locomoon travel run running descent jump parachung object graph object pointer inheritance hierarchy class inherits from dash sprint control flow code block jump Some digraph problems Path. Is there a directed path from s to t? s Shortest path. What is the shortest directed path from s to t? Topological sort. Can you draw a digraph so that all edges point upwards? Strong connectivity. Is there a directed path between all pairs of vertices? Transitive closure. For which vertices v and w is there a path from v to w? t Algorithms ROBERT SEDGEWICK KEVIN WAYNE DIRECTED GRAPHS introduction digraph API digraph search topological sort strong components PageRank. What is the importance of a web page?

4 Digraph API Digraph API public class Digraph Digraph(int V) create an empty digraph with V vertices Digraph(In in) create a digraph from input stream void addedge(int v, int w) add a directed edge v w Iterable<Integer> adj(int v) vertices pointing from v int V() number of vertices int E() number of edges Digraph reverse() reverse of this digraph String tostring() string representation V tinydg.txt E 7 % java Digraph tinydg.txt -> -> -> -> -> -> -> -> -> -> -> - In in = new In(args[]); Digraph G = new Digraph(in); read digraph from input stream In in = new In(args[]); Digraph G = new Digraph(in); read digraph from input stream for (int v = ; v < G.V(); v++) for (int w : G.adj(v)) StdOut.println(v + "->" + w); print out each edge (once) for (int v = ; v < G.V(); v++) for (int w : G.adj(v)) StdOut.println(v + "->" + w); print out each edge (once) Adjacency-lists digraph representation Adjacency-lists graph representation (review): Java implementation.txt E Maintain vertex-indexed array of lists. V tinydg.txt 7 7 E adj[] adj[] 7 7 public class Graph private final int V; private final Bag<Integer>[] adj; public Graph(int V) this.v = V; adj = (Bag<Integer>[]) new Bag[V]; for (int v = ; v < V; v++) adj[v] = new Bag<Integer>(); public void addedge(int v, int w) adj[v].add(w); adj[w].add(v); public Iterable<Integer> adj(int v) return adj[v]; adjacency lists create empty graph with V vertices add edge vw iterator for vertices adjacent to v

5 Adjacency-lists digraph representation: Java implementation Digraph representations public class Digraph private final int V; private final Bag<Integer>[] adj; public Digraph(int V) this.v = V; adj = (Bag<Integer>[]) new Bag[V]; for (int v = ; v < V; v++) adj[v] = new Bag<Integer>(); public void addedge(int v, int w) adj[v].add(w); adjacency lists create empty digraph with V vertices add edge v w In practice. Use adjacency-lists representation. Algorithms based on iterating over vertices pointing from v. Real-world digraphs tend to be sparse. representation space insert edge from v to w huge number of vertices, small average vertex degree edge from v to w? iterate over vertices pointing from v? list of edges E E E adjacency matrix V V adjacency lists E + V outdegree(v) outdegree(v) public Iterable<Integer> adj(int v) return adj[v]; iterator for vertices pointing from v 7 disallows parallel edges Reachability Problem. Find all vertices reachable from s along a directed path. s. DIRECTED GRAPHS Algorithms ROBERT SEDGEWICK KEVIN WAYNE introduction digraph API digraph search topological sort strong components

6 Depth-first search in digraphs Depth-first search demo Same method as for undirected graphs. Every undirected graph is a digraph (with edges in both directions). DFS is a digraph algorithm. DFS (to visit a vertex v) Mark v as visited. Recursively visit all unmarked vertices w pointing from v. To visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. a directed graph 7 7 Depth-first search demo Depth-first search (in undirected graphs) To visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeto[] Recall code for undirected graphs. public class DepthFirstSearch private boolean[] marked; true if path to s reachable from 7 reachable from vertex 7 T T T T T T F F F F F F F public DepthFirstSearch(Graph G, int s) marked = new boolean[g.v()]; dfs(g, s); private void dfs(graph G, int v) marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); public boolean visited(int v) return marked[v]; constructor marks vertices connected to s recursive DFS does the work client can ask whether any vertex is connected to s

7 Depth-first search (in directed graphs) Reachability application: program control-flow analysis Code for directed graphs identical to undirected one. [substitute Digraph for Graph] public class DirectedDFS private boolean[] marked; public DirectedDFS(Digraph G, int s) marked = new boolean[g.v()]; dfs(g, s); private void dfs(digraph G, int v) marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); true if path from s constructor marks vertices reachable from s recursive DFS does the work Every program is a digraph. Vertex = basic block of instructions (straight-line program). Edge = jump. Dead-code elimination. Find (and remove) unreachable code. Infinite-loop detection. Determine whether exit is unreachable. public boolean visited(int v) return marked[v]; client can ask whether any vertex is reachable from s Reachability application: mark-sweep garbage collector Reachability application: mark-sweep garbage collector Every data structure is a digraph. Vertex = object. Edge = reference. Roots. Objects known to be directly accessible by program (e.g., stack). 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 DFS stack). Reachable objects. Objects indirectly accessible by program (starting at a root and following a chain of pointers). roots roots 7

8 Depth-first search in digraphs summary Breadth-first search in digraphs DFS enables direct solution of simple digraph problems. Reachability. Path finding. Topological sort. Directed cycle detection. Basis for solving difficult digraph problems. -satisfiability. Directed Euler path. Strongly-connected components. Same method as for undirected graphs. Every undirected graph is a digraph (with edges in both directions). BFS is a digraph algorithm. BFS (from source vertex s) Put s onto a FIFO queue, and mark s as visited. Repeat until the queue is empty: - remove the least recently added vertex v - for each unmarked vertex pointing from v: add to queue and mark as visited. SIAM J. COMPUT. Vol., No., June 7 DEPTH-FIRST SEARCH AND LINEAR GRAPH ALGORITHMS* ROBERT TARJAN" Abstract. The value of depth-first search or "bacltracking" as a technique for solving problems is illustrated by two examples. An improved version 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. The 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 vertices and E is the number of edges of the graph being examined. Proposition. BFS computes shortest paths (fewest number of edges) from s to all other vertices in a digraph in time proportional to E + V. Directed breadth-first search demo Directed breadth-first search demo Repeat until queue is empty: Remove vertex v from queue. Add to queue all unmarked vertices pointing from v and mark them. Repeat until queue is empty: Remove vertex v from queue. Add to queue all unmarked vertices pointing from v and mark them. V tinydg.txt E v edgeto[] distto[] graph G done

9 Multiple-source shortest paths Breadth-first search in digraphs application: web crawler Multiple-source shortest paths. Given a digraph and a set of source vertices, find shortest path from any vertex in the set to each other vertex. Goal. Crawl web, starting from some root web page, say Ex. S =, 7,. Shortest path to is 7. Shortest path to is 7. Shortest path to is. Solution. [BFS with implicit digraph] Choose root web page as source s. Maintain a Queue of websites to explore. Maintain a SET of discovered websites. Dequeue the next website and enqueue websites to which it links (provided you haven't done so before) Q. How to implement multi-source shortest paths algorithm? A. Use BFS, but initialize by enqueuing all source vertices. Q. Why not use DFS? 7 How many strong components are there in this digraph? Bare-bones web crawler: Java implementation Queue<String> queue = new Queue<String>(); SET<String> marked = new SET<String>(); queue of websites to crawl set of marked websites String root = " queue.enqueue(root); marked.add(root); start crawling from root website while (!queue.isempty()) String v = queue.dequeue(); StdOut.println(v); In in = new In(v); String input = in.readall(); String regexp = " Pattern pattern = Pattern.compile(regexp); Matcher matcher = pattern.matcher(input); while (matcher.find()) String w = matcher.group(); if (!marked.contains(w)) marked.add(w); queue.enqueue(w); read in raw html from next website in queue use regular expression to find all URLs in website of form [crude pattern misses relative URLs] if unmarked, mark it and put on the queue Algorithms ROBERT SEDGEWICK KEVIN WAYNE DIRECTED GRAPHS introduction digraph API digraph search topological sort strong components

10 Precedence scheduling Topological sort Goal. Given a set of tasks to be completed with precedence constraints, in which order should we schedule the tasks? Digraph model. vertex = task; edge = precedence constraint. DAG. Directed acyclic graph. Topological sort. Redraw DAG so all edges point upwards.. Algorithms. Complexity Theory. Artificial Intelligence. Intro to CS. Cryptography. Scientific Computing. Advanced Programming tasks precedence constraint graph directed edges DAG Solution. DFS. What else? feasible schedule 7 topological order Topological sort demo Run depth-first search. Return vertices in reverse postorder. Topological sort demo Run depth-first search. Return vertices in reverse postorder. postorder topological order a directed acyclic graph done

11 Depth-first search order Topological sort in a DAG: correctness proof public class DepthFirstOrder private boolean[] marked; private Stack<Integer> reversepost; public DepthFirstOrder(Digraph G) reversepost = new Stack<Integer>(); marked = new boolean[g.v()]; for (int v = ; v < G.V(); v++) if (!marked[v]) dfs(g, v); private void dfs(digraph G, int v) marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); reversepost.push(v); public Iterable<Integer> reversepost() return reversepost; returns all vertices in reverse DFS postorder Proposition. Reverse DFS postorder of a DAG is a topological order. Pf. Consider any edge v w. When dfs(v) is called: Case : dfs(w) has already been called and returned. Thus, w was done before v. Case : dfs(w) has not yet been called. dfs(w) will get called directly or indirectly by dfs(v) and will finish before dfs(v). Thus, w will be done before v. 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 v, so v w would complete a cycle. v = case case all vertices pointing from are done before is done, so they appear after in topological order dfs() dfs() dfs() done done dfs() done dfs() check done done check check dfs() check check check dfs() done done check check check done Directed cycle detection Directed cycle detection application: precedence scheduling Proposition. A digraph has a topological order iff no directed cycle. Pf. If directed cycle, topological order impossible. If no directed cycle, DFS-based algorithm finds a topological order. Scheduling. Given a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks? a digraph with a directed cycle Goal. Given a digraph, find a directed cycle. Solution. DFS. What else? See textbook. Remark. A directed cycle implies scheduling problem is infeasible.

12 Directed cycle detection application: cyclic inheritance Directed cycle detection application: spreadsheet recalculation The Java compiler does cycle detection. Microsoft Excel does cycle detection (and has a circular reference toolbar!) public class A extends B... public class B extends C... % javac A.java A.java:: cyclic inheritance involving A public class A extends B ^ error public class C extends A... Strongly-connected components Def. Vertices v and w are strongly connected if there is both a directed path from v to w and a directed path from w to v. Algorithms ROBERT SEDGEWICK KEVIN WAYNE. DIRECTED GRAPHS introduction digraph API digraph search topological sort strong components Key property. Strong connectivity is an equivalence relation: v is strongly connected to v. If v is strongly connected to w, then w is strongly connected to v. If v is strongly connected to w and w to x, then v is strongly connected to x. Def. A strong component is a maximal subset of strongly-connected vertices.

13 Connected components vs. strongly-connected components Strong component application: ecological food webs v and w are connected if there is a path between v and w v and w are strongly connected if there is both a directed path from v to w and a directed path from w to v Food web graph. Vertex = species; edge = from producer to consumer. A graph and connected its connected components components strongly-connected components connected component id (easy to compute with DFS) 7 cc[] strongly-connected component id (how to compute?) 7 scc[] public int connected(int v, int w) return cc[v] == cc[w]; constant-time client connectivity query public int stronglyconnected(int v, int w) return scc[v] == scc[w]; constant-time client strong-connectivity query Strong component. Subset of species with common energy flow. Strong component application: software modules Strong components algorithms: brief history Software module dependency graph. Vertex = software module. Edge: from module to dependency. s: Core OR problem. Widely studied; some practical algorithms. Complexity not understood. 7: linear-time DFS algorithm (Tarjan). Classic algorithm. Level of difficulty: Algs++. Demonstrated broad applicability and importance of DFS. Firefox Internet Explorer s: easy two-pass linear-time algorithm (Kosaraju-Sharir). Forgot notes for lecture; developed algorithm in order to teach it! Later found in Russian scientific literature (7). Strong component. Subset of mutually interacting modules. Approach. Package strong components together. Approach. Use to improve design! s: more easy linear-time algorithms. Gabow: fixed old OR algorithm. Cheriyan-Mehlhorn: needed one-pass algorithm for LEDA.

14 Kosaraju-Sharir algorithm: intuition Kosaraju-Sharir algorithm demo Reverse graph. Strong components in G are same as in G R. Kernel DAG. Contract each strong component into a single vertex. Phase. Compute reverse postorder in G R. Phase. Run DFS in G, visiting unmarked vertices in reverse postorder of G R. Idea. how to compute? Compute topological order (reverse postorder) in kernel DAG. Run DFS, considering vertices in reverse topological order. 7 first vertex is a sink (has no edges pointing from it) Kernel DAG in reverse topological order digraph G and its strong components kernel DAG of G (in reverse topological order) digraph G Kosaraju-Sharir algorithm demo Kosaraju-Sharir algorithm demo Phase. Compute reverse postorder in G R. 7 Phase. Run DFS in G, visiting unmarked vertices in reverse postorder of G R. 7 v scc[] reverse digraph G R done

15 Kosaraju-Sharir algorithm Kosaraju-Sharir algorithm Simple (but mysterious) algorithm for computing strong components. Phase : run DFS on GR to compute reverse postorder. Phase : run DFS on G, considering vertices in order given by first DFS. DFS in reverse digraph G R Simple (but mysterious) algorithm for computing strong components. Phase : run DFS on GR to compute reverse postorder. Phase : run DFS on G, considering vertices in order given by first DFS. DFS in original digraph G check unmarked vertices in the order 7 reverse postorder for use in second dfs() 7 dfs() dfs() dfs() check done dfs(7) 7 done done dfs() dfs() dfs() dfs() dfs() check dfs() check done done check 7 check done... 7 check unmarked vertices in the order 7 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 dfs() check check dfs() check done check done dfs(7) check check 7 done check Kosaraju-Sharir algorithm Connected components in an undirected graph (with DFS) Proposition. Kosaraju-Sharir algorithm computes the strong components of a digraph in time proportional to E + V. Pf. Running time: bottleneck is running DFS twice (and computing GR ). Correctness: tricky, see textbook (nd printing). Implementation: easy! public class CC private boolean marked[]; private int[] id; private int count; public CC(Graph G) marked = new boolean[g.v()]; id = new int[g.v()]; for (int v = ; v < G.V(); v++) if (!marked[v]) dfs(g, v); count++; private void dfs(graph G, int v) marked[v] = true; id[v] = count; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); public boolean connected(int v, int w) return id[v] == id[w];

16 Strong components in a digraph (with two DFSs) Digraph-processing summary: algorithms of the day public class KosarajuSharirSCC private boolean marked[]; private int[] id; private int count; public KosarajuSharirSCC(Digraph G) marked = new boolean[g.v()]; id = new int[g.v()]; DepthFirstOrder dfs = new DepthFirstOrder(G.reverse()); for (int v : dfs.reversepost()) if (!marked[v]) dfs(g, v); count++; single-source reachability in a digraph topological sort in a DAG DFS DFS private void dfs(digraph G, int v) marked[v] = true; id[v] = count; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); strong components in a digraph 7 Kosaraju-Sharir DFS (twice) public boolean stronglyconnected(int v, int w) return id[v] == id[w];