Directed Graphs. digraph API digraph search transitive closure topological sort strong components. Directed graphs

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1 irected graphs igraph. Set of vertices connected pairwise by oriented edges. irected raphs digraph AP digraph search transitive closure topological sort strong components References: Algorithms in Java, Chapter oogle - Map data Sanborn, NAVTQ - Terms of Use Algorithms in Java, th dition Robert Sedgewick and Kevin Wayne Copyright November, :: AM Web graph WordNet graph Vertex = web page. Vertex = synset. dge = hyperlink. dge = hypernym relationship.

2 igraph applications Some digraph problems graph vertex edge transportation street intersection one-way street web web page hyperlink WordNet synset hypernym scheduling task precedence constraint financial stock, currency transaction food web species predator-prey relationship cell phone person placed call infectious disease person infection game board position legal move citation journal article citation object graph object pointer Path. s there a directed path from s to t? Shortest path. What is the shortest directed path from s and t? Strong connectivity. Are all vertices mutually reachable? Transitive closure. or which vertices v and w is there a path from v to w? Topological sort. Can you draw the digraph so that all edges point from left to right? PRT/CPM. iven a set of tasks with precedence constraints, how can we best complete them all? PageRank. What is the importance of a web page? inheritance hierarchy class inherits from control flow code block jump igraph representations Vertices. This lecture: use integers between and V-. Real world: convert between names and integers with symbol table. digraph AP digraph search transitive closure topological sort strong components dges: four options. [same as undirected graph, but orientation matters] List of vertex pairs. Adjacency matrix. Adjacency lists. Adjacency sets.

3 igraph AP Set of edges representation public class igraph igraph(int V) igraph(n in) graph data type create an empty digraph with V vertices create a digraph from input stream Store a list of the edges (linked list or array). void adddge(int v, int w) add an edge from v to w terable<nteger> adj(int v) return an iterator over the neighbors of v int V() return number of vertices String tostring() return a string representation n in = new n(); raph = new igraph(in); StdOut.println(); for (int v = ; v <.V(); v++) for (int w :.adj(v)) /* process edge v w */ Adjacency-matrix representation Adjacency-list representation Maintain a two-dimensional V-by-V boolean array; for each edge v w in the digraph: adj[v][w] = true. Maintain vertex-indexed array of lists. from to : : : : : : same as undirected graph, but one entry for each edge : : : : : : :

4 Adjacency-set representation Adjacency-set representation: Java implementation Maintain vertex-indexed array of sets. Same as raph, but only insert one copy of each edge. : public class igraph private final int V; private final ST<nteger>[] adj; adjacency sets : : : : : : : :, same as undirected graph, but one entry for each edge public igraph(int V) this.v = V; adj = (ST<nteger>[]) new ST[V]; for (int v = ; v < V; v++) adj[v] = new ST<nteger>(); public void adddge(int v, int w) adj[v].add(w); create empty graph with V vertices add edge from v to w (no parallel edges) : : : :,, public terable<nteger> adj(int v) return adj[v]; iterator for v's neighbors igraph representations Typical digraph application: oogle's PageRank algorithm n practice. use adjacency-set (or adjacency-list) representation. Real-world digraphs tend to be sparse. Algorithms all based on iterating over edges incident to v. representation space edge between v and w? iterate over edges incident to v? list of edges adjacency matrix V V adjacency list + V degree(v) degree(v) adjacency set + V log (degree(v)) degree(v) oal. etermine which pages on web are important. Solution. gnore keywords and content, focus on hyperlink structure. Random surfer model. Start at random page. With probability., randomly select a hyperlink to visit next; with probability., randomly select any page. PageRank = proportion of time random surfer spends on each page. Solution. Simulate random surfer for a long time. Solution. Compute ranks directly until they converge. Solution. Compute eigenvalues of adjacency matrix! None feasible without sparse digraph representation.

5 Reachability Problem. ind all vertices reachable from s along a directed path. s digraph AP digraph search transitive closure topological sort strong components epth-first search in digraphs epth-first search (single-source reachability) Same method as for undirected graphs. dentical to undirected version (substitute igraph for raph). very undirected graph is a digraph. appens to have edges in both directions. S is a digraph algorithm. S (to visit a vertex v) Mark v as visited. Recursively visit all unmarked vertices w adjacent to v. public class Searcher private boolean[] marked; public Searcher(igraph, int s) marked = new boolean[.v()]; dfs(, s); private void dfs(igraph, int v) marked[v] = true; for (int w :.adj(v)) if (!marked[w]) dfs(, w); true if connected to s constructor marks vertices connected to s recursive S does the work public boolean isreachable(int v) return marked[v]; client can ask whether any vertex is reachable from s

6 Reachability application: program control-flow analysis Reachability application: mark-sweep garbage collector very program is a digraph. Vertex = basic block of instructions (straight-line program). dge = jump. ead code elimination. ind (and remove) unreachable code. nfinite loop detection. etermine whether exit is unreachable. very data structure is a digraph. Vertex = object. dge = 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). Reachability application: mark-sweep garbage collector epth-first search (S) 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 S stack. S enables direct solution of simple digraph problems. Reachability. Cycle detection. Topological sort. Transitive closure. s there a path from s to t? Basis for solving difficult digraph problems. irected uler path. Strong connected components.

7 Breadth-first search in digraphs igraph BS application: web crawler very undirected graph is a digraph. appens to have edges in both directions. BS is a digraph algorithm. BS (from source vertex s) Put s onto a O queue. Repeat until the queue is empty: remove the least recently added vertex v add each of v's unvisited neighbors to the queue and mark them as visited. oal. Crawl web, starting from some root web page, say Solution. BS with implicit graph. BS. Start at some root web page. Maintain a Queue of websites to explore. Maintain a ST of discovered websites. equeue the next website and enqueue websites to which it links (provided you haven't done so before). Property. Visits vertices in increasing distance from s. Q. Why not use S? Web crawler: BS-based Java implementation Queue<String> q = new Queue<String>(); ST<String> visited = new ST<String>(); queue of websites to crawl set of visited websites String s = " q.enqueue(s); visited.add(s); start crawling from website s while (!q.ismpty()) String v = q.dequeue(); StdOut.println(v); n in = new n(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 (!visited.contains(w)) visited.add(w); q.enqueue(w); read in raw html for next website in queue use regular expression to find all URLs in website of form if unvisited, mark as visited and put on queue digraph AP digraph search transitive closure topological sort strong components

8 raph-processing challenge (revisited) igraph-processing challenge Problem. s there an undirected path between v and w? Problem. s there a directed path from v to w? oals. Linear preprocessing time, constant query time. oals. Linear preprocessing time, constant query time. ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible. can't do better than V (reduction from boolean matrix multiplication) Transitive closure igraph-processing challenge (revised) ef. The transitive closure of a digraph is another digraph with a directed edge from v to w if there is a directed path from v to w in. Problem. s there a directed path from v to w? oals. ~ V preprocessing time, constant query time. digraph transitive closure TC() digraph is usually sparse TC() is usually dense ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible. open research problem

9 igraph-processing challenge (revised again) Transitive closure: Java implementation Problem. s there a directed path from v to w? oals. ~ V preprocessing time, ~ V space, constant query time. Use an array of Searcher objects, one for each row of transitive closure. ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible. Use S once for each vertex to compute rows of transitive closure public class TransitiveClosure private Searcher[] tc; public TransitiveClosure(igraph ) tc = new Searcher[.V()]; for (int v = ; v <.V(); v++) tc[v] = new Searcher(, v); public boolean reachable(int v, int w) return tc[v].isreachable(w); array of Searcher objects initialize array is there a directed path from v to w? igraph application: scheduling Scheduling. iven a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks? raph model. Create a vertex v for each task. Create an edge v w if task v must precede task w. tasks digraph AP digraph search transitive closure topological sort strong components precedence constraint graph. read programming assignment. download files. write code. attend precept. sleep feasible schedule

10 Topological sort igraph-processing challenge A. irected acyclic graph. Problem. Check that a digraph is a A; if so, find a topological order. oal. Linear time. Topological sort. Redraw A so all edges point left to right. act. igraph is a A iff no directed cycle. ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible. use S Topological sort in a A: Java implementation Topological sort in a A: trace public class TopologicalSorter private boolean[] marked; private Stack<nteger> sorted; public TopologicalSorter(igraph ) marked = new boolean[.v()]; sorted = new Stack<nteger>(); for (int v = ; v <.V(); v++) if (!marked[v]) tsort(, v); private void tsort(igraph, int v) marked[v] = true; for (int w :.adj(v)) if (!marked[w]) tsort(, w); sorted.push(v); public terable<nteger> order() return sorted; vertices in topological order reverse S postorder Visit means call tsort() and leave means return from tsort(). marked[] sorted visit : - visit : - visit : - leave : leave : visit : leave : visit : check : leave : leave : check : check : visit : check : check : check : visit : leave : leave : check : check : check :

11 Topological sort in a A: correctness proof igraph-processing challenge Proposition. f digraph is a A, algorithm yields a topological order. Pf. Algorithm terminates in O( + V) time since it's just a version of S. Consider any edge v w. When tsort(, v) is called, - Case : tsort(, w) has already returned. Thus, w will appear after v in topological order. - Case : tsort(, w) has not yet been called, so it will get called directly or indirectly by tsort(, v) and it will finish before tsort(, v). Thus, w will appear after v in topological order. - Case : tsort(, w) has already been called, but not returned. Then the function call stack contains a directed path from w to v. Combining this path with the edge v w yields a directed cycle, contradicting A. Problem. iven a digraph, is there a directed cycle? oal. Linear time. ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible. run S-based topological sort algorithm; if it yields a topological sort, no directed cycle (can modify code to find cycle) Cyclic inheritance Spreadsheet recalculation The Java compiler does cycle detection. Microsoft xcel does cycle checking (and has a circular reference toolbar!) public class A extends B... % javac A.java A.java:: cyclic inheritance involving A public class A extends B ^ error public class B extends C... public class C extends A...

12 Symbolic links Topological sort and cycle detection applications The Linux file system does not do cycle detection. % ln -s a.txt b.txt % ln -s b.txt c.txt % ln -s c.txt a.txt % more a.txt a.txt: Too many levels of symbolic links Causalities. mail loops. Compilation units. Class inheritance. Course prerequisites. eadlocking detection. Temporal dependencies. Pipeline of computing jobs. Check for symbolic link loop. valuate formula in spreadsheet.. Topological sort application (weighted A) Precedence scheduling. Task v takes time[v] units of time. Can work on jobs in parallel. Precedence constraints: must finish task v before beginning task w. oal: finish each task as soon as possible. x. A B C index task time prereqs A begin - B framing A C roofing B siding B windows plumbing electricity C, paint C, finish, PRT/CPM algorithm. nitialize fin[v] = for all vertices v. A B C

13 PRT/CPM algorithm. nitialize fin[v] = for all vertices v. PRT/CPM algorithm. nitialize fin[v] = for all vertices v. A B C A B C PRT/CPM algorithm. nitialize fin[v] = for all vertices v. PRT/CPM algorithm. nitialize fin[v] = for all vertices v. A B C A B C

14 PRT/CPM algorithm. nitialize fin[v] = for all vertices v. PRT/CPM algorithm. nitialize fin[v] = for all vertices v. A B C A B C PRT/CPM algorithm. nitialize fin[v] = for all vertices v. PRT/CPM algorithm. nitialize fin[v] = for all vertices v. A B C A B C

15 PRT/CPM: Java implementation Critical path. Longest path from source to sink. To compute: Remember vertex that set value (parent-link). Work backwards from sink. A B C index time prereqs finish A - B A C B B C, C,, Assume is digraph of precedence constraints. double[] fin = new double[.v()]; for (int v = ; v <.V(); v++) fin[v] = time[v]; TopologicalSorter ts = new TopologicalSorter(); for (int v : ts.order()) for (int w :.adj(v)) fin[w] = Math.max(fin[w], fin[v] + time[w]); initialize finish times apply updates to vertices in topological order igraph-processing challenge ef. Vertices v and w are strongly connected if there is a directed path from v to w and from w to v. Problem. Are v and w strongly connected? oal. Linear preprocessing time, constant query time. digraph AP digraph search transitive closure topological sort strong components ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert. ntractable. No one knows. mpossible.

16 igraph-processing challenge cological food web graph ef. Vertices v and w are strongly connected if there is a directed path from v to w and from w to v. Vertex = species. dge: from producer to consumer. Problem. Are v and w strongly connected? oal. Linear preprocessing time, constant query time. ow difficult? Any COS student could do it. Need to be a typical diligent COS student. ire an expert (or a COS student). ntractable. No one knows. correctness proof mpossible. implementation: use S twice (see textbook) strongly connected components Strong component. Subset of species with common energy flow. Software module dependency graph Strong components algorithms: brief history Vertex = software module. dge: from module to dependency. irefox nternet explorer s: Core OR problem. Widely studied; some practical algorithms. Complexity not understood. : linear-time S algorithm (Tarjan). Classic algorithm. level of difficulty: CS++. demonstrated broad applicability and importance of S. s: easy two-pass linear-time algorithm (Kosaraju). orgot notes for teaching algorithms class; developed alg in order to teach it! Later found in Russian scientific literature (). Strong component. Subset of mutually interacting modules. Approach. Package strong components together. Approach. Use to improve design! s: more easy linear-time algorithms (abow, Mehlhorn). abow: fixed old OR algorithm. Mehlhorn: needed one-pass algorithm for LA.

17 Kosaraju's algorithm igraph-processing summary: algorithms of the day Simple (but mysterious) algorithm for computing strong components Run S on R and compute postorder. Run S on, considering vertices in reverse postorder. single-source reachability S transitive closure S (from each vertex) R topological sort (A) S Proposition. Trees in second S are strong components. (!) Pf. [see COS ] strong components Kosaraju S (twice)