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

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1 Algorithms ROBERT SEDGEWICK KEVIN WAYNE. DIRECTED GRAPHS. DIRECTED GRAPHS introduction introduction digraph API digraph API Algorithms F O U R T H E D I T I O N digraph search topological sort Algorithms 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. outdegree = indegree = 8 directed path from to directed cycle 8 Google - Map data 8 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.!"#$%& '(!&)&%*+,$-). -&/%,% &/&&% '8 :; <=>>? #"*-/ &*+@ A--&A/&) A-&-/8 The Topology of? the Federal Funds Market, A-&-/8 Bech and Atalay, 8 B>? )".A--&A/&) A-&-/8 <>> #"*-/./%-#+@ A--&A/&) <CD? #"*-/ "-EA-&-/8 <FGH? #"*-/ $/E A-&-/; F- /I". )*@ /I&%& &%& JK -)&. "- /I& <>>8 LL -)&. "- /I& <CD8 :K -)&. "- <FGH8 J -)&. 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"- /I& <FGH ", "/ I*. * */I,% /I& <>> $/ -/ / /I& <>>; C- A-/%*./8 * -)& ". "- /I& <CD ", "/ I*. * */I / /I& <>> $/ -/,% "/; R -)& ". "- * /&-)%"+ ", "/ )&. -/ %&.")& - * )"%&A/&) */I / %,% /I& <>>;S!%/%'$( C- /I& -&/%, *@&-/..&-/ N&%!&)"%& %*U" "& *- O:P8 /I& <>> ". /I& +*%#&./ A-&-/; F- *N&%*#&8 *+./ %&', /I& -)&. "- /I*/ -&/% &+-# / /I& <>>; CA-/%*./8 /I& <>> ". $AI.*++&%,% /I&,&)&%*+,$-). -&/%; C- :8 -+@ (&' ) (', /I& -)&. &+-# / /I". A-&-/; Q@,*% /I& +*%#&./ A-&-/ ". /I& <CD; C- :8 )%' ) )', /I& -)&. &%& "- /I". A-&-/; HI& <FGH A-/*"-&) (*' ) +', *++ -)&. &% )*@8 I"+& /I&%& &%& (+' ),', /I& -)&. +A*/&) "- /I& /&-)%"+.;SS V&.. /I*- -' ) (', /I& -)&. &%& "- /I& %&*"-"-# )".A--&A/&) A-&-/. O.&& H*+& JP; Uber taxi graph longer existed, or had moved 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). This had the disadvantage of not differentiating between blogs that were Vertex = taxi pickup; edge = taxi ride. actively mentioned in a post on that day, from blogroll links that remain static over many weeks []. 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. 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 page. We constructed a citation network by identifying whether a URL present on the page of one blog references another political blog. We called a link found anywhere 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 that community. An effect that may not be as apparent from the visualization is that even though we started with a balanced set of blogs, conservative blogs show a greater tendency to link. 8% of conservative blogs link to at least one other blog, and 8% receive a link. In contrast, % of liberal blogs link to another blog, while only % are linked to by another blog. So overall, we see a slightly higher tendency for conservative blogs to link. Liberal blogs linked to. blogs 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 8 and links from our single-day snapshot S HI& /&-)%"+. *@ *+. & )"W&%&-/"*/&) "-/ /I%&&.$A-&-/.( *.&/, -)&. /I*/ *%& - * */I &*-*/"-#,% <CD8 *.&/, -)&. /I*/ *%& - * */I +&*)"-# / <FGH8 *-) *.&/, -)&. /I*/ *%& - * */I /I*/ &#"-. "- <CD *-) &-). "- <FGH; S S!!"#, -)&. &%& "- X,%E<CDY /&-)%"+.8 $!%#, -)&. &%& "- /I& X/E<FGHY /&-)%"+. *-) "!&#, -)&. &%& "X/$&.Y,% <CD / <FGH; S 8

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 happeningoccurrence occurrentnatural_event change alteraon modicaon damage harm.. impairment transion miracle increase miracle act human_acon human_acvity forfeit forfeiture acon group_acon food web species predator-prey relationship WordNet synset hypernym scheduling task precedence constraint 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 descent object graph object pointer run running dash sprint jump parachung inheritance hierarchy class inherits from control flow code block jump Some digraph problems problem description s t path Is there a path from s to t? shortest s t path What is the shortest path from s to t? directed cycle Is there a directed cycle in the graph? topological sort Can the digraph be drawn 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 directed path from v to w? PageRank What is the importance of a web page? Algorithms ROBERT SEDGEWICK KEVIN WAYNE DIRECTED GRAPHS introduction digraph API digraph search topological sort strong components

4 Digraph API Digraph API Almost identical to Graph 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 adjacent from v int V() number of vertices int E() number of edges V tinydg.txt E 8 8 % java Digraph tinydg.txt -> -> -> -> -> -> -> -> -> -> -> -> Digraph reverse() reverse of this digraph String tostring() string representation In in = new In(args[]); Digraph G = new Digraph(in); for (int v = ; v < G.V(); v++) for (int w : G.adj(v)) StdOut.println(v + "->" + w); read digraph from input stream print out each edge (once) Digraph representation: adjacency lists Directed graphs: quiz Maintain vertex-indexed array of lists. Which is order of growth of running time to iterate over all vertices nydg.txt 8 8 E V tinydg.txt 8 8 E adj[] adj[] adjacent from v in a digraph using the adjacency-lists representation? A. indegree(v) B. outdegree(v) C. degree(v) D. V E. I don't know.

5 Digraph representations Adjacency-lists graph representation (review): Java implementation In practice. Use adjacency-lists representation. Algorithms based on iterating over vertices adjacent from v. Real-world digraphs tend to be sparse. representation space insert edge from v to w huge number of vertices, small average vertex outdegree edge from v to w? iterate over vertices adjacent from v? list of edges E E E adjacency matrix V V adjacency lists E + V outdegree(v) outdegree(v) 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); adjacency lists create empty graph with V vertices add edge vw disallows parallel edges public Iterable<Integer> adj(int v) return adj[v]; iterator for vertices adjacent to v 8 Adjacency-lists digraph representation: Java implementation public class Digraph private final int V; private final Bag<Integer>[] adj; adjacency lists 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); create empty digraph with V vertices add edge v w Algorithms ROBERT SEDGEWICK KEVIN WAYNE. DIRECTED GRAPHS introduction digraph API digraph search topological sort strong components public Iterable<Integer> adj(int v) return adj[v]; iterator for vertices adjacent from v

6 Reachability Depth-first search in digraphs Problem. Find all vertices reachable from s along a directed path. s 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 vertex v. Recursively visit all unmarked vertices w adjacent from v. Depth-first search demo Depth-first search demo To visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices adjacent from v To visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices adjacent from v. 8 reachable from vertex v 8 marked[] T T T T T T F F F F F F F edgeto[] a directed graph reachable from

7 Depth-first search (in undirected graphs) Depth-first search (in directed graphs) Recall code for undirected graphs. Code for directed graphs identical to undirected one. [substitute Digraph for Graph] public class DepthFirstSearch private boolean[] marked; true if connected to s public class DirectedDFS private boolean[] marked; true if path from s public DepthFirstSearch(Graph G, int s) marked = new boolean[g.v()]; dfs(g, s); constructor marks vertices connected to s public DirectedDFS(Digraph G, int s) marked = new boolean[g.v()]; dfs(g, s); constructor marks vertices reachable from s private void dfs(graph G, int v) marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); recursive DFS does the work private void dfs(digraph G, int v) marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); recursive DFS does the work public boolean visited(int v) return marked[v]; client can ask whether any vertex is connected to s public boolean visited(int v) return marked[v]; client can ask whether any vertex is reachable from s Reachability application: program control-flow analysis Reachability application: mark-sweep garbage collector 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. Every 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 8

8 Reachability application: mark-sweep garbage collector Depth-first search in digraphs summary 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). DFS enables direct solution of simple digraph problems. Reachability. Path finding. Topological sort. Directed cycle detection. roots Basis for solving difficult digraph problems. -satisfiability. Directed Euler path. Strongly-connected components. SIAM J. COMPUT. Vol., No., June 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 ka, where Vis the number of vertices ande is thenumber of edges of the graph being examined. Breadth-first search in digraphs Directed breadth-first search demo Same method as for undirected graphs. Every undirected graph is a digraph (with edges in both directions). BFS is a digraph algorithm. Repeat until queue is empty: Remove vertex v from queue. Add to queue all unmarked vertices adjacent from v and mark them. 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 adjacent from v: add to queue and mark as visited. V tinydg.txt 8 E Proposition. BFS computes shortest paths (fewest number of edges) from s to all other vertices in a digraph in time proportional to E + V. graph G

9 Directed breadth-first search demo Repeat until queue is empty: Remove vertex v from queue. Add to queue all unmarked vertices adjacent from v and mark them. 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. v edgeto[] distto[] Ex. S =,,. Shortest path to is. Shortest path to is. Shortest path to is. Q. How to implement multi-source shortest paths algorithm? done Breadth-first search in digraphs application: web crawler Bare-bones web crawler: Java implementation Goal. Crawl web, starting from some root web page, say Queue<String> queue = new Queue<String>(); SET<String> marked = new SET<String>(); queue of websites to crawl set of marked websites 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. Why not use DFS? String root = " queue.enqueue(root); marked.add(root); 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); 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 relative URLs] if unmarked, mark it and put on the queue How many strong components are there in this digraph?

10 Web crawler output BFS crawl DFS crawl Algorithms ROBERT SEDGEWICK KEVIN WAYNE DIRECTED GRAPHS introduction digraph API digraph search topological sort strong components 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 topological order

11 Topological sort demo Run depth-first search. Return vertices in reverse postorder. Topological sort demo Run depth-first search. Return vertices in reverse postorder. tinydag.txt postorder topological order a directed acyclic graph done Depth-first search order Topological sort in a DAG: intuition public class DepthFirstOrder private boolean[] marked; private Stack<Integer> reversepostorder; public DepthFirstOrder(Digraph G) reversepostorder = new Stack<Integer>(); marked = new boolean[g.v()]; for (int v = ; v < G.V(); v++) if (!marked[v]) dfs(g, v); Why does topological sort algorithm work? First vertex in postorder has outdegree. Second-to-last vertex in postorder can only point to last vertex.... postorder private void dfs(digraph G, int v) marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); reversepostorder.push(v); topological order public Iterable<Integer> reversepostorder() return reversepostorder; returns all vertices in reverse DFS postorder

12 Topological sort in a DAG: correctness proof Directed cycle detection 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 appears before v in postorder Case : dfs(w) has not yet been called. dfs(w) will get called directly or indirectly by dfs(v) so, dfs(w) will finish before dfs(v) thus, w appears before v in postorder Case : dfs(w) has already been called, but has not yet returned. function-call stack contains path from w to v so v w would complete a cycle (contradiction) v = case (w =,, ) case (w = ) dfs() dfs() dfs() done done dfs() done dfs() check done done check check dfs() check check check dfs() check check done done check check check done 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. a digraph with a directed cycle Goal. Given a digraph, find a directed cycle. Solution. DFS. What else? See textbook. Directed cycle detection application: precedence scheduling Directed cycle detection application: cyclic inheritance Scheduling. Given a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks? The Java compiler does cycle detection. 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... Remark. A directed cycle implies scheduling problem is infeasible. 8

13 Directed cycle detection application: spreadsheet recalculation Depth-first search orders Microsoft Excel does cycle detection (and has a circular reference toolbar!) Observation. DFS visits each vertex exactly once. The order in which it does so can be important. Orderings. Preorder: order in which dfs() is called. Postorder: order in which dfs() returns. Reverse postorder: reverse order in which dfs() returns. private void dfs(graph G, int v) marked[v] = true; preorder.enqueue(v); for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); postorder.enqueue(v); reversepostorder.push(v); 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. strongly-connected components

14 Directed graphs: quiz Connected components vs. strongly-connected components How many strong components are in a DAG with V vertices and E edges? A. B. C. V D. E E. I don't know. v and w are connected if there is a path between v and w A graph and its connected components connected components 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 strongly-connected components connected component id (easy to compute with DFS) 8 id[] strongly-connected component id (how to compute?) 8 id[] public boolean connected(int v, int w) return id[v] == id[w]; public boolean stronglyconnected(int v, int w) return id[v] == id[w]; constant-time client connectivity query constant-time client strong-connectivity query Strong component application: ecological food webs Strong component application: software modules Food web graph. Vertex = species; edge = from producer to consumer. Software module dependency graph. Vertex = software module. Edge: from module to dependency. Firefox Internet Explorer Strong component. Subset of species with common energy flow. Strong component. Subset of mutually interacting modules. Approach. Package strong components together. Approach. Use to improve design!

15 Strong components algorithms: brief history Kosaraju-Sharir algorithm: intuition s: Core OR problem. Widely studied; some practical algorithms. Complexity not understood. : linear-time DFS algorithm (Tarjan). Classic algorithm. Level of difficulty: Algs++. Demonstrated broad applicability and importance of DFS. Reverse graph. Strong components in G are same as in G R. Kernel DAG. Contract each strong component into a single vertex. Idea. how to compute? Compute topological order (reverse postorder) in kernel DAG. Run DFS, considering vertices in reverse topological order. 8s: 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 (). first vertex is a sink (has no edges pointing from it) s: more easy linear-time algorithms. Gabow: fixed old OR algorithm. Cheriyan-Mehlhorn: needed one-pass algorithm for LEDA. digraph G and its strong components E D Kernel DAG in reverse Ctopological order kernel DAG of G (topological order: A B C D E) B A 8 Kosaraju-Sharir algorithm demo Kosaraju-Sharir algorithm demo Phase. Compute reverse postorder in G R. Phase. Run DFS in G, visiting unmarked vertices in reverse postorder of G R. Phase. Compute reverse postorder in G R digraph G reverse digraph G R

16 Kosaraju-Sharir algorithm demo Kosaraju-Sharir algorithm Phase. Run DFS in G, visiting unmarked vertices in reverse postorder of G R. 8 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. v id[] DFS in reverse digraph G R done 8 8 check unmarked vertices in the order 8 reverse postorder for use in second dfs() 8 dfs() dfs() dfs(8) check 8 done dfs() done done dfs() dfs() dfs() dfs() dfs() check dfs() check done done check check done... 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 original digraph G 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! check unmarked vertices in the order 8 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(8) check 8 done check done dfs() check check done check 8

17 Connected components in an undirected graph (with DFS) Strong components in a digraph (with two DFSs) 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 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.reversepostorder()) if (!marked[v]) dfs(g, v); count++; 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); public boolean connected(int v, int w) return id[v] == id[w]; public boolean stronglyconnected(int v, int w) return id[v] == id[w]; Digraph-processing summary: algorithms of the day single-source reachability in a digraph DFS topological sort in a DAG DFS strong components in a digraph 8 Kosaraju-Sharir DFS (twice)