Topological Sort

1. Topological Sort Directed Acyclic Graph (dag): a directed graph which contains no directed cycles. A topological order for a dag G: a sequential listing of all the vertices of G such that if (u,v)  E(G) then u precedes v in the listing. Topological Sort: an algorithm that takes as input a dag G and produces a topological order of the vertices of G Example Applications: ·Prerequisites dag for courses at a University ·Glossary of technical terms: (t1,t2)  E(G) iff t1 is used in the definition of t2.

2. In general, topological order of a dag is not unique.

3. We will now apply dfs and bfs to the topological sort problem. For dfs, the idea is to find vertices with no successors; these vertices may be placed at the end of the list. Such a vertex may be found by probing as deep as possible along edges, which is what dfs does. Then when all successors of a vertex have been placed in the list, that vertex may be placed. This corresponds to backing out of a vertex when you have completed the exploration from that vertex in dfs. Thus when we are ready to back out from a vertex, we add it to the list in the back-to-front order. Thus dfs will fill the list from the end of the list back toward the first entry.

25. Code for DFS Topological Sort • Input parameter: adj Output parameter: ts • top_sort( adj, ts ) { n = adj_last k = n // placement index for values into the array ts for I = 1 to n visit[i] = false for I = 1 to n if ( !visit[i] ) top_sort_recurs(adj, i, ts)}

26. Code for DFS Topological Sort • top_sort_recurs( adj, start, ts ) { visit[start] = true trav = adj[start] while ( trav != null ) { v = trav.ver if ( !visit[v] ) top_sort_recurs(adj, v, ts) trav = trav.next } ts[k] = start k = k-1} • By an analysis similar to that for regular dfs, the running time is (n + m)

27. Homework • Page 193: # 2, 3, 6