Digraphs. Directed Graphs. Digraph Application. Digraph Properties. A digraph is a graph whose edges are all directed.

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1 igraphs irected Graphs LX FW OR MI OS digraph is a graph whose edges are all directed Short for directed graph pplications one-way streets flights task scheduling 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 2 igraph Properties graph G=(V,) such that ach edge goes in one direction: dge (a,b) goes from a to b, but not b to a If G is simple, m < n(n - 1) If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of incoming edges and outgoing edges in time proportional to their size igraph pplication Scheduling: edge (a,b) means task a must be completed before b can be started ics21 ics1 ics131 ics11 ics22 ics3 ics141 ics23 ics2 ics121 ics11 ics11 The good life 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 4

2 irected FS Reachability We can specialize the traversal algorithms (FS and FS) to digraphs by traversing edges only along their direction In the directed FS algorithm, we have four types of edges discovery edges back edges forward edges cross edges directed FS starting at a vertex s determines the vertices reachable from s FS tree rooted at v: vertices reachable from v via directed paths F F 2010 Goodrich, Tamassia irected Graphs 2010 Goodrich, Tamassia irected Graphs Strong onnectivity Strong onnectivity lgorithm ach vertex can reach all other vertices a g c d e b f Pick a vertex v in G Perform a FS from v in G If there s a w not visited, print no Let G be G with edges reversed Perform a FS from v in G If there s a w not visited, print no lse, print yes Running time: O(n+m) G: G : a f a f d d c c e e g b g b 2010 Goodrich, Tamassia irected Graphs 2010 Goodrich, Tamassia irected Graphs

3 Strongly onnected omponents Transitive losure Maximal subgraphs such that each vertex can reach all other vertices in the subgraph an also be done in O(n+m) time using FS, but is more complicated (similar to biconnectivity). f a d c e g b { a, c, g } { f, d, e, b } Given a digraph G, the transitive closure of G is the digraph G* such that G* has the same vertices as G if G has a directed path from u to v (u v), G* has a directed edge from u to v The transitive closure provides reachability information about a digraph G G* 2010 Goodrich, Tamassia irected Graphs omputing the Transitive losure We can perform FS starting at each vertex O(n(n+m)) If there's a way to get from to and from to, then there's a way to get from to. lternatively... Use dynamic programming: The Floyd-Warshall lgorithm 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 10 Floyd-Warshall Transitive losure Idea #1: Number the vertices 1, 2,, n. Idea #2: onsider paths that use only vertices numbered 1, 2,, k, as intermediate vertices: Uses only vertices numbered 1,,k-1 i Uses only vertices numbered 1,,k (add this edge if it s not already in) 2010 Goodrich, Tamassia irected Graphs 12 k j Uses only vertices numbered 1,,k-1

4 Floyd-Warshall s lgorithm Number vertices,, v n ompute digraphs G 0,, G n G 0 =G G k has directed edge (v i, v j ) if G has a directed path from v i to v j with intermediate vertices in {,, v k } We have that G n = G* In phase k, digraph G k is computed from G k - 1 Running time: O(n 3 ), assuming aredjacent is O(1) (e.g., adjacency matrix) lgorithm FloydWarshall(G) Input digraph G Output transitive closure G* of G i 1 for all v G.vertices() denote v as v i i i + 1 G 0 G for k 1 to n do G k G k - 1 for i 1 to n (i k) do for j 1 to n (j i, k) do if G k - 1.aredjacent(v i, v k ) G k - 1.aredjacent(v k, v j ) if G k.aredjacent(v i, v j ) G k.insertirecteddge(v i, v j, k) return G n 2010 Goodrich, Tamassia irected Graphs 13 Floyd-Warshall xample LX OR FW OS MI 2010 Goodrich, Tamassia irected Graphs 14 v Floyd-Warshall, Iteration 1 OS v Floyd-Warshall, Iteration 2 OS v OR OR LX FW MI LX FW MI 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 1

5 Floyd-Warshall, Iteration 3 OS v Floyd-Warshall, Iteration 4 OS v OR OR LX FW MI LX FW MI 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 1 Floyd-Warshall, Iteration OS v Floyd-Warshall, Iteration OS v OR OR LX FW MI LX FW MI 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 20

6 Floyd-Warshall, onclusion LX OR FW MI 2010 Goodrich, Tamassia irected Graphs 21 OS v Gs and Topological Ordering directed acyclic graph (G) is a digraph that has no directed cycles topological ordering of a digraph is a numbering,, v n of the vertices such that for every edge (v i, v j ), we have i < j xample: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints Theorem digraph admits a topological ordering if and only if it is a G 2010 Goodrich, Tamassia irected Graphs 22 G G v v 3 Topological ordering of G Topological Sorting Number vertices, so that (u,v) in implies u < v 1 wake up typical student day 2 3 eat study computer sci. 4 nap more c.s. play write c.s. program work out bake cookies 10 sleep 11 dream about graphs 2010 Goodrich, Tamassia irected Graphs 23 lgorithm for Topological Sorting Note: This algorithm is different than the one in the book lgorithm TopologicalSort(G) H G // Temporary copy of G n G.numVertices() while H is not empty do Let v be a vertex with no outgoing edges Label v n n n - 1 Remove v from H Running time: O(n + m) 2010 Goodrich, Tamassia irected Graphs 24

7 Implementation with FS Simulate the algorithm by using depth-first search O(n+m) time. lgorithm topologicalfs(g) Input dag G Output topological ordering of G n G.numVertices() for all u G.vertices() setlabel(u, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR topologicalfs(g, v) lgorithm topologicalfs(g, v) Input graph G and a start vertex v of G Output labeling of the vertices of G in the connected component of v setlabel(v, VISIT) for all e G.outdges(v) { outgoing edges } w opposite(v,e) if getlabel(w) = UNXPLOR { e is a discovery edge } topologicalfs(g, w) else { e is a forward or cross edge } Label v with topological number n n n Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 2

8 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs 32

9 2010 Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs Goodrich, Tamassia irected Graphs