! Vertex = species. ! Edge = from prey to predator.

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irected Graphs irected Graphs igraph. Set of objects with oriented pairwise connections. x. One-way street, hyperlink. Reference: Chapter 19, Algorithms in Java, 3 rd dition, Robert Sedgewick.

1 irected Graphs irected Graphs igraph. Set of objects with oriented pairwise connections. x. One-way street, hyperlink. Reference: Chapter 19, Algorithms in Java, 3 rd dition, Robert Sedgewick. Robert Sedgewick and Kevin Wayne Copyright igraph Applications cological ood Web igraph Vertex dge financial stock, currency transaction ood web graph.! Vertex = species.! dge = from prey to predator. transportation street intersection, airport highway, airway route scheduling task precedence constraint control flow code block jump nternet web page hyperlink game board position legal move telephone person placed call food web species predator-prey relation infectious disease person infection citation journal article citation object graph object pointer inheritance hierarchy class inherits from Reference: 3 4

2 Some igraph Problems igraph Representation Transitive closure. s there a directed path from v to w? Strong connectivity. Are all vertices mutually reachable? Vertex names.! This lecture: use integers between 0 and V-1.! Real world: convert between names and integers with symbol table. Topological sort. Can you draw the graph so that all of the edges point from left to right? Orientation of edge matters. PRT/CPM. Given a set of tasks with precedence constraints, what is the earliest that we can complete each task? A H B C G J K Shortest path. Given a weighted graph, find best route from v to w? PageRank. What is the importance of a web page? L M 5 6 Adjacency Matrix Representation Adjacency List Representation Adjacency matrix representation.! Two-dimensional V! V boolean array.! dge v"w in graph: adj[v][w] = true. Vertex indexed array of lists.! Space proportional to number of edges.! One representation of each edge. H A B C J L G K M A B C G H J K L M 0 A B C G H J K L M H A B C J L G K M A: C B G B: C: : : : G: H: : J: K L M K: L: M M: 7 8

3 Adjacency List: Java mplementation igraph Representations mplementation. Same as Graph, but only insert one copy of each edge. public class igraph { private int V; private Sequence<nteger>[] adj; public igraph(int V) { this.v = V; adj = (Sequence<nteger>[]) new Sequence[V]; for (int v = 0; v < V; v++) adj[v] = new Sequence<nteger>(); igraphs are abstract mathematical objects.! AT implementation requires specific representation.! fficiency depends on matching algorithms to representations. Representation Space Adjacency matrix #(V 2 ) Adjacency list #( + V) dge from v to w? #(1) O(outdeg(v)) numerate edges leaving v? List of edges #( + V ) O() #() #(V) #(outdeg(v)) public void insert(int v, int w) { adj[v].add(w); public terable<nteger> adj(int v) { return adj[v]; igraphs in practice. [use adjacency list representation]! Real world digraphs are sparse.! Bottleneck is iterating over edges leaving v Reachability igraph Search Goal. ind all vertices reachable from s along a directed path. epth first search. To visit a vertex v:! Mark v as visited.! Recursively visit all unmarked vertices w adjacent to v. Running time. O() since each edge examined at most once. s Robert Sedgewick and Kevin Wayne Copyright

4 epth irst Search Control low Graph Remark. Same as undirected version, except igraph instead of Graph. public class Searcher { private boolean[] marked; public Searcher(igraph G, int s) { marked = new boolean[g.v()]; dfs(g, s); private void dfs(igraph G, int v) { marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(g, w); public boolean isreachable(int v) { return marked[v]; Control-flow graph.! Vertex = basic block (straight-line program).! dge = jump. ead code elimination. ind (and remove) code blocks that are unreachable during execution. dead code might arise from compiler optimizations (or careless programmer) nfinite loop detection. xit block is unreachable from entry block. Caveat. Not all infinite loops are detectable Mark-Sweep Garbage Collector epth irst Search Roots. Objects known to be accessible by program (e.g., stack). Live objects. Objects that the program could get to by starting at a root and following a chain of pointers. Mark-sweep algorithm. [McCarthy 1960]! Mark: run S from roots to mark live objects. easy to identify pointers in type-safe language! Sweep: if object is unmarked, it is garbage, so add to free list. S enables direct solution of simple digraph problems.! Reachability.! Cycle detection.! Topological sort.! Transitive closure.! ind path from s to t. Basis for solving difficult digraph problems.! irected uler path.! Strong connected components. xtra memory. Uses 1 extra mark bit per object, plus S stack

5 Breadth irst Search Application: Web Crawler Shortest path. ind the shortest directed path from s to t. BS. Analogous to BS in undirected graphs. Web graph. Vertex = website, edge = hyperlink. Goal. Crawl nternet, starting from some root website. Solution. BS with implicit graph. BS.! Start at some root website, say 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). t Q. Why not use S? s Web Crawler: Java mplementation Queue<String> q = new Queue<String>(); ST<String> visited = new ST<String>(); String s = " q.enqueue(s); visited.add(s); while (!q.ismpty()) { String v = q.dequeue(); System.out.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); start crawling from s read in raw html if unvisited, mark as visited and put on queue queue of sites to crawl set of visited sites search using regular expression Transitive Closure 19 Robert Sedgewick and Kevin Wayne Copyright

6 Transitive Closure Transitive Closure Transitive closure. s there a directed path from v to w? Transitive closure. s there a directed path from v to w? Lazy. Run separate S for each query. ager. Run S from every vertex v. G Method Preprocess Query Space S (lazy) O(1) O( + V) O( + V) S (eager) O( V) O(1) O(V 2 ) Transitive closure Remark. irected problem is harder than undirected one. Open research problem. O(1) query, O(V 2 ) preprocessing time. tc[v][w] = 1 iff path from v to w Transitive Closure: Java mplementation public class TransitiveClosure { private boolean[][] tc; Topological Sort public TransitiveClosure(igraph G) { tc = new boolean[g.v()][g.v()]; for (int v = 0; v < G.V(); v++) dfs(g, v, v); run dfs from every vertex // reachability from s, made it to v private void dfs(igraph G, int s, int v) { tc[s][v] = true; for (int w : G.adj(v)) if (!tc[s][w]) dfs(g, s, w); public boolean reachable(int v, int w) { return tc[v][w]; is w reachable from v? 23 Robert Sedgewick and Kevin Wayne Copyright

7 Topological Sort Application: Scheduling AG. irected acyclic graph. Scheduling. Given a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks? Topological sort. Redraw AG so all edges point left to right. Graph model.! Create a vertex v for each task.! Create an edge v"w if task v must precede task w.! Schedule tasks in topological order. 0. read programming assignment 1. download files 2. write code 12. sleep Observation. Not possible if graph has a directed cycle Topological Sort: S Topological Sort: Java mplementation Topologically sort a AG.! Run S.! Reverse postorder numbering yields a topological sort. public class TopologicalSorter { private int cnt; private boolean[] visited; private int[] ts; Pf of correctness. When S backtracks from a vertex v, all vertices reachable from v have already been explored. Running time. O( + V). Q. f not a AG, how would you identify a cycle? no back edges in AG A H S tree public TopologicalSorter(igraph G) { visited = new boolean[g.v()]; ts = new int[g.v()]; cnt = G.V(); for (int v = 0; v < G.V(); v++) if (!visited[v]) tsort(g, v); private void tsort(igraph G, int v) { visited[v] = true; for (int w : G.adj(v)) if (!visited[w]) tsort(g, w); ts[--cnt] = v; assign numbers in reverse S postorder 27 28

8 Topological Sort: Applications Program valuation and Review Technique / Critical Path Method Topological sort applications.! Causalities.! Compilation units.! Class inheritance.! Course prerequisites.! eadlocking detection.! Temporal dependencies.! Pipeline of computing jobs.! Check for symbol link loop.! valuate formula in spreadsheet. PRT/CPM.! Task v takes time[v] units of time.! Can work on jobs in parallel.! Precedence constraints: must finish task v before beginning task w.! What's earliest we can finish each task? ndex Task Time Prereq A Begin 0 - B raming 4 A C Roofing 2 B Siding 6 B Windows 5 Plumbing 3 G lectricity 4 C, H Paint 6 C, inish 0, H A B C G H time[v] Program valuation and Review Technique / Critical Path Method PRT/CPM algorithm.! Compute topological order of vertices.! nitialize fin[v] = 0 for all vertices v.! Consider vertices v in topological order. for each edge v"w, set fin[w]= max(fin[w], fin[v] + time[w]) critical path fin[v] time[v] A B C G H

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

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