Chapter 13 Lafores Book Part 2 Graphs Hwajung

Chapter 13 (Lafore’s Book) – Part 2 Graphs Hwajung Lee ITEC 324 Principle of CS III

Directed Acyclic Graph (DAG) Example C I E F A B J H G D DAG can have one or more topological orders. (ex) I B D G C E F J A H or C E G A I B F J D H

Topological Sorting �Definition Topological sort or topological ordering of a directed acyclic graph (DAG) is a linear ordering of its nodes in which each node comes before all nodes to which it has outbound edges. Every DAG has one or more topological sorts.

Topological Sort – Version 1 void topsort( ) throws Cycle. Found { Queue q; int counter = 0; Vertex v, w; q = new Queue(); for each vertex v if (indegree == 0) q. enqueue(v); while (!q. empty()) { v = q. dequeue(); v. top. Num = ++counter; for each w adjacent to v if (--w. indegree==0) q. enqueue(w); } if (counter != NUM_VERTICES) throw new Cycle. Found(); } Initialize Q with vertices of indegree 0 Decrement indegrees of vertices w, if 0 then enqueue Vertices will remain unenumerated if a cycle exists

Top. Sort Example Queue I C G E B D A F J H C I Load Q with initial v with indegree == 0 E F A B J H G D Output I C G B E D A F J H

Topological Sort – Version 2 �Source Code: topo. java

Connectivity in Directed Graph (1) �Cf. Connectivity in a non-directed graph can be easily checked by using a depth-first or bread-first search. That is, if you can reach all the vertices from a randomly selected vertex using one of the search, the non-directed graph is connected.

Connectivity in Directed Graph (2) � However, in a directed graph, you can not check a connectivity with a dfs or a bfs by starting from a randomly selected vertex. It is a reachability test from a vertex to another vertex. B A D C E � Shall we start the search on each vertex in turn and see if we can reach all the vertices? Not efficient. Whrshall’s algorithm

Warshall’s Algorithm (1) �Warshall’s algorithm systematically modifies a graph’s adjacency matrix. �The graph represented by the revised adjacency matrix is called the transitive closure of the original graph. �Idea: If you can get from vertex L to vertex M, and you can get from M to N, then you can get from L to N.

Warshall’s Algorithm (2) A A B C D E 1 B C 1 1 D E 1 1 1 A A B C D E 1 B C 1 D E 1 1 1 B Row A: 1 is added at (B, C) A Row B, C, D: No update D Row E: 1 is added at (D, C) E After update it, a dsf or bsf can be used. C