Algorithms ROBERT SEDGEWICK | KEVIN WAYNE 4.2 DIRECTED GRAPHS ‣ introduction ‣ digraph API Algorithms ‣ digraph search FOURTH EDITION ‣ topological sort ‣ strong components ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu 4.2 DIRECTED GRAPHS ‣ introduction ‣ digraph API Algorithms ‣ digraph search ‣ topological sort ‣ strong components ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu Directed graphs Digraph. Set of vertices connected pairwise by directed edges. outdegree = 4 indegree = 2 0 6 8 7 directed path 2 from 0 to 2 1 3 9 10 4 directed cycle 5 11 12 3 Road network To see all the details that are visible on the screen,use the Address Holland Tunnel "Print" link next to the map. New York, NY 10013 Vertex = intersection; edge = one-way street. ©2008 Google - Map data ©2008 Sanborn, NAVTEQ™ - Terms of Use 4 Political blogosphere graph Vertex = political blog; edge = link. The Political Blogosphere and the 2004 U.S. Election: Divided They Blog, Adamic and Glance, 2005 Figure 1: Community structure of political blogs (expanded set), shown using utilizing a GEM layout [11] in the GUESS[3] visualization and analysis tool. The colors reflect political orientation, 5 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. 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 actively mentioned in a post on that day, from blogroll links that remain static over many weeks [10]. 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. 479 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 1 shows the unmistakable division between the liberal and conservative political (blogo)spheres. In fact, 91% 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. 84% of conservative blogs link to at least one other blog, and 82% receive a link. In contrast, 74% of liberal blogs link to another blog, while only 67% are linked to by another blog. So overall, we see a slightly higher tendency for conservative blogs to link. Liberal blogs linked to 13.6 blogs on average, while conservative blogs linked to an average of 15.1, 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 338 and 264 links from our single-day snapshot 4 Overnight interbank loan graph Vertex = bank; edge = overnight loan. 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PageRank What is the importance of a web page ? 12 4.2 DIRECTED GRAPHS ‣ introduction ‣ digraph API Algorithms ‣ digraph search ‣ topological sort ‣ strong components ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu 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 pointing from v int V() number of vertices int E() number of edges Digraph reverse() reverse of this digraph String toString() string representation 14 Digraph API % java Digraph tinyDG.txt tinyDG.txt V 0->5 5 1 13 E 22 0->1 4 2 2->0 adj[] 2 3 2->3 0 3 3 2 0 6 0 3->5 1 5 2 0 1 3->2 2 0 2 4->3 11 12 3 2 3 12 9 4->2 9 10 4 5->4 4 9 11 5 7 9 ⋮ 9 4 8 0 10 12 6 11->4 11 4 7 11->12 4 3 6 9 8 3 5 12->9 6 8 9 6 8 6 10 ⋮5 4 0 5 11 11 10 6 4 12 6 9 12 In in = new In(args[0]); read digraph from 7 6 input stream Digraph G = new Digraph(in); 4 12 for (int v = 0; v < G.V(); v++) 9 print out each for (int w : G.adj(v)) edge (once) StdOut.println(v + "->" + w); 15 Digraph representation: adjacency lists Maintain vertex-indexed array of lists.