Digraphs ( 12.4) Directed Graphs. Digraph Application. Digraph Properties. A digraph is a graph whose edges are all directed.

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1 igraphs ( 12.4) irected Graphs OR OS digraph is a graph whose edges are all directed Short for directed graph pplications one- way streets flights task scheduling irected Graphs 1 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 inedges and out-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 irected Graphs 3 irected Graphs 4

2 irected FS 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 a a vertex s determines the vertices reachable from s irected Graphs Reachability FS tree rooted at v: vertices reachable from v via directed paths F F 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 irected Graphs 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* irected Graphs irected Graphs 10 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 irected Graphs 11 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 (add this edge if it s not already in) i Uses only vertices numbered 1,,k- 1 k irected Graphs 12 j Uses only vertices numbered 1,,k- 1

4 Floyd-Warshall s lgorithm Floyd-Warshall s algorithm numbers the vertices of G as,, v n and computes a series of digraphs G 0,, G n G 0 =G G k has a directed edge (v i ) if G has a directed path from v i to v j with intermediate vertices in the set {,, 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) Floyd-Warshall xample return v lgorithm FloydWarshall(G) OS Input digraph G Output transitive closure G* of G OR 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 ) if G k.aredjacent(v i ) G k.insertirecteddge(v i, k) G n irected Graphs 13 irected Graphs 14 Floyd-Warshall, Iteration 1 OS v Floyd-Warshall, Iteration 2 OS v OR OR irected Graphs 1 irected Graphs 1

5 Floyd-Warshall, Iteration 3 OS v Floyd-Warshall, Iteration 4 OS v OR OR irected Graphs 1 irected Graphs 1 Floyd-Warshall, Iteration OS v Floyd-Warshall, Iteration OS v OR OR irected Graphs 1 irected Graphs 20

6 Floyd-Warshall, onclusion OR 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 ), 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 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 make cookies for professors 10 sleep 11 dream about graphs irected Graphs 23 lgorithm for Topological Sorting Note: This algorithm is different than the one in Goodrich-Tamassia Running time: O(n + m). How? Method 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 irected Graphs 24

7 Topological Sorting lgorithm using FS Simulate the algorithm by using depth-first search lgorithm topologicalfs(g) Input dag G Output topological ordering of G n G.numVertices() for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR topologicalfs(g, v) O(n+m) time. 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.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) topologicalfs(g, w) else {e is a forward or cross edge} Label v with topological number n n n - 1 Topological Sorting xample irected Graphs 2 irected Graphs 2 Topological Sorting xample Topological Sorting xample irected Graphs 2 irected Graphs 2

8 Topological Sorting xample Topological Sorting xample irected Graphs 2 irected Graphs 30 Topological Sorting xample Topological Sorting xample 4 irected Graphs 31 irected Graphs 32

9 Topological Sorting xample Topological Sorting xample irected Graphs 33 irected Graphs 34 Topological Sorting xample irected Graphs 3

Directed Graphs BOS Goodrich, Tamassia Directed Graphs 1 ORD JFK SFO DFW LAX MIA

Directed Graphs BOS Goodrich, Tamassia Directed Graphs 1 ORD JFK SFO DFW LAX MIA Directed Graphs BOS ORD JFK SFO LAX DFW MIA 2010 Goodrich, Tamassia Directed Graphs 1 Digraphs A digraph is a graph whose edges are all directed Short for directed graph Applications one-way streets flights