DFS on Directed Graphs BOS. Outline and Reading ( 6.4) Digraphs. Reachability ( 6.4.1) Directed Acyclic Graphs (DAG s) ( 6.4.4)

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1 S on irected Graphs OS OR JK SO LX W MI irected Graphs S Outline and Reading ( 6.4) Reachability ( 6.4.1) irected S Strong connectivity irected cyclic Graphs (G s) ( 6.4.4) Topological Sorting irected Graphs S igraphs digraph is a graph whose edges are all directed Short for directed graph pplications one-way streets flights task scheduling irected Graphs S 1.3 3

2 igraph pplication Scheduling: edge (a,b) means task a must be completed before b can be started ics21 ics22 ics23 ics51 ics53 ics52 ics161 ics131 ics141 ics121 ics11 ics151 The good life irected Graphs S irected S S on digraphs traverses edges only along their proper direction In the directed S algorithm, we have four types of edges discovery edges back edges forward edges cross edges directed S starting at a vertex s determines the vertices reachable from s irected Graphs S irected S example S_Sweep starts at, then, tree, back, cross irected Graphs S 1.3 6

3 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 1,, 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 v 2 precedence constraints Theorem digraph admits a topological v ordering if and only if it is a G 1 irected Graphs S 1.3 G G v 4 v 5 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. play make cookies for professors 4 5 nap more c.s. write c.s. program 10 sleep 6 work out 11 dream about graphs irected Graphs S 1.3 lgorithm for Topological Sorting Note: This algorithm is different than the one in Goodrich-Tamassia 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 Running time: O(n + m) [with smart implementation] How? irected Graphs S 1.3

4 Topological Sorting lgorithm using S Simulate the algorithm by using depth-first search lgorithm topos_sweep(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 topologicals(g, v) O(n+m) time. lgorithm topologicals(g, v) Input graph G and a start vertex v of G Output a labeling of the vertices of G in the connected component of v in topological order setlabel(v, VISIT) for all e G.outIncidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) topologicals(g, w) else {e is a forward or cross edge} setlabel(e, VISIT) Label v with topological number n n n - 1 irected Graphs S Topological Sorting xample irected Graphs S Topological Sorting xample irected Graphs S

5 Topological Sorting xample irected Graphs S Topological Sorting xample irected Graphs S Topological Sorting xample 6 irected Graphs S

6 Topological Sorting xample 6 5 irected Graphs S Topological Sorting xample irected Graphs S Topological Sorting xample irected Graphs S 1.3 1

7 Topological Sorting xample irected Graphs S Topological Sorting xample irected Graphs S Strong onnectivity ach vertex can reach all other vertices a c g d e f b irected Graphs S

8 Strong onnectivity lgorithm Pick a vertex v in G. Perform a S from v in G. If there s a w not visited, print no. Let G be G with edges reversed. Perform a S from v in G. If there s a w not visited, print no. G: a f d c e g b lse, print yes. G : a c g Running time: O(n+m). d e f b irected Graphs S Strongly onnected omponents Maximal subgraphs such that each vertex can reach all other vertices in the subgraph an be computed in O(n+m) time using S a c g { a, c, g } f d e b { f, d, e, b } irected Graphs S S algorithm Using S_Sweep for directed graphs, construct list L of reverse finish order of the vertices in the traversal. Node is finished when traversal leaves it permanently. o another S_Sweep on G R, (G with edges reversed), with the following modification: in S_Sweep outer loop, start S calls on vertices according to the order in list L. ach spanning tree produced by S_Sweep on G R will contain all nodes from exactly one S of G irected Graphs S

9 Strongly onnected omponents irected Graphs S Strongly onnected omponents irected Graphs S S lgorithm, more detail // Phase 1 Run S_Sweep on G, returning a list L of nodes in reverse finish order. one by adding vertex v to the front of L after traversal on v is finished in S_Sweep. // Phase 2a onstruct G R from G by copying the vertices, and then adding the reverse of every edge from G to G R. // Phase 2b o a modified S_Sweep traversal on G R, where list L is used to order the S calls. ach S call labels vertices traversed with a different S number. // inal Phase: Label vertices and edges of G. irected Graphs S 1.3 2

10 S Phase 1 onstruct list L Similar to topological sort lgorithm S1S_Sweep(G) Input dag G Output list L of vertices of G in reverse finish order. L empty list 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 S1S(G, v) O(n+m) time. lgorithm S1S(G, v) Input graph G and a start vertex v of G Output vertices of G in the connected component of v added to L, according to reverse finish order setlabel(v, VISIT) for all e G.outIncidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) S1S(G, w) else {e is a forward or cross edge} L.insertirst(v) irected Graphs S S Phase 2b Similar to onnected lgorithm S2bS(G R, v, sccnum) omponents Input graph G R and a start vertex v lgorithm S2bS_Sweep(G R,L) Output vertices of G R in the connected component of v labeled by Input dag G R, list L sccnum Output Labeling of vertices in setlabel(v, VISIT) G R by scc component number Label v with sccnum sccnum 1 for all e G.outIncidentdges(v) for all u G.vertices() if getlabel(e) = UNXPLOR setlabel(u, UNXPLOR) w opposite(v,e) for all e G.edges() if getlabel(w) = UNXPLOR setlabel(e, UNXPLOR) setlabel(e, ISOVRY) for all v L, {traverse L in order} S2bS(G, w) if getlabel(v) = UNXPLOR else S2bS(G, v, sccnum) {e is a forward or cross edge} sccnum++ O(n+m) time. irected Graphs S orrectness of S algorithm Lemma 1: In terms of vertices, S s of G are the same as the S s of G R. Lemma 2: or graph G, let be a forest generated by S_Sweep on G. Let S be a tree of. Then S contains one or more complete S s of G. (No partial S s). Lemma 3a: Let be the forest generated by S phase 2b, and S be a spanning tree in. Let x be the root of S, and v be a descendent of x. Then there is a path from v to x in G R. Lemma 3b: Let S be as in Lemma 3a. S combined with other edges in G R form a strongly connected subgraph of G R. irected Graphs S

11 readth-irst Search irected Graphs S Outline and Reading readth-first search ( 6.3.3) lgorithm xample Properties nalysis pplications S vs. S ( 6.3.3) omparison of applications omparison of edge labels irected Graphs S readth-irst Search readth-first search (S) is general graph traversal technique Visits all the vertices and edges with n vertices and m edges takes O(n + m ) time Like searching a binary tree level by level S can etermine whether G is connected ompute the connected components of G ompute a spanning forest of G ind and report a path with the minimum number of edges between two given vertices ind a simple cycle, if there is one irected Graphs S

12 xample unexplored vertex visited vertex unexplored edge discovery edge cross edge irected Graphs S xample (cont.) irected Graphs S xample (cont.) irected Graphs S

13 S lgorithm The algorithm uses a queue to keep track of vertices lgorithm S_Sweep(G) Input graph G Output labeling of the edges and the vertices of G 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 S(G, v) lgorithm S(G, s) Q new empty queue Q.enqueue(s) setlabel(s, VISIT) while Q.ismpty() v Q.dequeue() for all e G.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) setlabel(w, VISIT) Q.enqueue(w) else setlabel(e, ROSS) irected Graphs S Properties Notation G s : connected component of s L i : nodes at depth i in S tree. Property 1 S(G, s) visits all the vertices and edges of G s Property 2 The discovery edges labeled by S(G, s) form a spanning tree T s of G s ; T s called S tree Property 3 or each vertex v in L i The path of T s from s to v has i edges very path from s to v in G s has at least i edges irected Graphs S nalysis Setting/getting a vertex/edge label takes O(1) time ach vertex is labeled twice once as UNXPLOR once as VISIT ach edge is labeled twice once as UNXPLOR once as ISOVRY or ROSS ach vertex is inserted once into Q Inner loop of S runs in O(deg(v)) time S runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that Σ v deg(v) = 2m irected Graphs S 1.3 3

14 pplications an specialize the S traversal of a graph G to solve the following problems in O(n + m) time ompute the connected components of G ompute a spanning forest of G ind a simple cycle in G, or report that G is a forest Given two vertices of G, find a minimum length path in G (if it exists) irected Graphs S S vs. S pplications Spanning forest, connected components, paths, cycles Shortest paths Topological sort, iconnected components, S S S S S irected Graphs S S vs. S (cont.) On undirected graphs ack edge (v,w) ross edge (v,w) w is an ancestor of v in the tree of discovery edges w is in the same level as v or in the next level in the tree of discovery edges S S irected Graphs S