Lecture 16: Directed Graphs

Transcription

1 Lecture 16: Directed Graphs ORD JFK BOS SFO LAX DFW MIA Courtesy of Goodrich, Tamassia and Olga Veksler Instructor: Yuzhen Xie

2 Outline Directed Graphs Properties Algorithms for Directed Graphs DFS and BFS Strong Connectivity Transitive Closure DAG and Topological Ordering 2

3 Digraphs A digraph is a graph whose edges are all directed Short for directed graph C E D Applications one-way streets flights task scheduling A B 3

4 Digraph Properties E D A graph G=(V,E) such that Each edge goes in one direction: Edge (A,B) goes from A to B, Edge (B,A) goes from B to A, If G is simple, m < n*(n-1). A If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of in-edges and out-edges in time proportional to their size OutgoingEdges(A): (A,C), (A,D), (A,B) IngoingEdges(A): (E,A),(B,A) C B We say that vertex w is reachable from vertex v if there is a directed path from v to w Eis reachable hablefoma from E is not reachable from D 4

5 Directed DFS We can specialize the traversal algorithms (DFS and BFS) to digraphs by traversing edges only along their direction In the directed DFS algorithm, we have four types of edges discovery edges back edges forward edges cross edges A directed DFS starting at a vertex s visits all the vertices reachable from s Unlike undirected graph, if u is reachable from s, it does not mean that s is reachable from u Algorithm DFS(G, v) Input digraph G and a start vertex v of G Output traverses vertices in G reachable from v setlabel(v, VISITED) for all e G.outgoingEdges(v) w opposite(v,e) if getlabel(w) = UNEXPLORED DFS(G, w) C E A 1 D B 5

6 Graph G Reachability E D C F A B DFS tree rooted at v: vertices reachable from v via directed d paths BFS tree rooted at v: vertices reachable from v via directed paths E D E D A C DFS(G,C) A B BFS (G,B) C F

7 Strong Connectivity A graph is strongly connected if we can reach all other vertices from any fixed vertex a g a g c c d e d e f strongly connected graph b f b not strongly connected graph cannot reach a from g

8 Inefficient Algorithm to Check for Strong Connectivity stronglyconnected gy = true for every vertex v in G DFS(G,v) if some vertex w is not visited stronglyconnected = false f a d G: c e g b DFS(G,v) visits every vertex reachable from v Simple, yet very inefficient, O[(n+m)n] ) ] 8

9 Efficient Algorithm to Check for Strong Connectivity 1) Pick a vertex v in G 2) Perform a DFS from v in G If there s a vertex w not visited, print no and terminate the algorithm else (all vertices are visited) from v we can reach any other graph vertex 3) Let G be G with edges reversed G 4) Perform a DFS from v in G. If there is path from v to w in G, then there is path from w to v in the original graph G If every vertex is reachable from v in G, then there is a path from any vertex w to v in the original graph G Then G is strongly connected To find path between any nodes b and d: first go from b to v, then go from v to d Running time is O(n + m), perform DFS 2 times 9 w w v v d d c c G e e g b g b

10 Transitive Closure D E Given a digraph G, the transitive closure of G is B the digraph G* such that G C G* has the same vertices as G A if G has a directed path from u to v (u v), G* has a directed edge from u to v The transitive closure D E summarizes reachability B information about a digraph C A G* 10

11 Computing Transitive Closure We can perform DFS(G,v) for each vertex v for all vertices v in graph G DFS(G,v) for all vertices w in graph G if w is reachable from v put edge (v,w) in G* Running time is O(n(n+m)) This is O(n 2 ) for sparse graphs A graph is sparse if it has O(n) edges This is O(n 3 ) for dense graphs A graph is dense if it has O(n 2 ) edges 11

12 DAGs and Topological Ordering A directed acyclic graph (DAG) is a digraph that has no directed cyclescles A topological ordering of a digraph is a numbering of vertices: 1, 2,, n such that for every edge (v,, w), number(v) < number(w) Example: in a task scheduling digraph, a topological l ordering of a task sequence satisfies the precedence constraints 2 1 B A B A D E DAG G C 4 5 D E 3 C Topological l ordering of G

13 Topological Ordering Scheduling problem: edge (a,b) means task a must be completed before b can be started Number vertices, so that u.number < v.number for any edge (u,v) wake up eat study computer sci. 7 play 9 make cookies for professors nap 8 write c.s. program 10 sleep A typical student day 4 5 more c.s work out 11 dream about graphs

14 Topological Ordering Theorem Theorem: Digraph admits topological ordering if and only if it is a DAG Proof: Suppose graph is not a DAG, that is it has a cycle v 1, v 2,, v n-1, v n, v 1. If topological ordering existed, it would imply that v 1.num < v 2.num < < v n-1.num < v n.num < v 1.num If graph is a DAG, there is a vertex v with no outgoing edges if every vertex has an outgoing edge, then performing DFS we will encounter a back edge which means cycle This vertex can be the last one in topological ordering Removing this vertex from the graph with incoming and outgoing edges leaves the graph acyclic counter = n Repeat until counter = 0 1. Find vertex v with no outgoing edges 2. Set topological label of v to counter 3. Remove v from the graph together with ihall llincoming i and outgoing edges. 4. counter = counter -1

15 Topological Sorting Algorithm Using DFS Simulate the algorithm by using depth-first search counter is an instance (global) variable Algorithm topologicaldfs(g) Input dag G Output topological ordering of G counter G.numVertices() for all u G.vertices() setlabel(u, UNEXPLORED) for all v G.vertices() if getlabel(v) = UNEXPLORED topologicaldfs(g, v) O(n + m) time. Algorithm topologicaldfs(g, v) Input graph G and a start vertex v Output labeling of the vertices of G in the connected component of v setlabel(v, VISITED) for all e G.outgoingEdges(v) w opposite(v,e) if getlabel(w) = UNEXPLORED topologicaldfs(g, w) v.topologicalnumber = counter counter counter

16 Topological Sorting Example 16

17 Topological Sorting Example 9 17

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