1. CS2420: Lecture 36 Vladimir Kulyukin Computer Science Department Utah State University

2. Outline Graph Algorithms (Chapter 9)

3. Weighted Graphs Weighted graphs and weighted digraphs (networks) have weights assigned to their edges. Weighted graphs can be represented as adjacency matrices or adjacency lists.

4. Weighted Graphs: Example AB CD 1 2 4 7 5

5. Paths A path from vertex u to vertex v in a graph G is a sequence of adjacent (connected by an edge) vertices that starts with u and ends in v. If all vertices in a path are distinct (except possibly for the first and last one), the path is simple. The length of the path is the number of edges in the path.

6. Cycles A cycle is a simple path of positive length that starts and ends in the same vertex. A graph with cycles is cyclic. A graph without cycles is acyclic. DAG stands for directed acyclic graph.

7. Graphs: Connectivity An undirected graph G is connected if, for every pair of vertices u and v, there is a path from u to v. A directed graph with the same property is called strongly connected. If every vertex is a ball and every edge is a string, one can pick up any ball and lift the entire graph without losing any other balls.

8. Topological Sorting Let G = (V, E) be a directed acyclic graph (dag). Find a list of vertices such that for every edge (u, v) in E, u is listed before v.

9. Topological Sorting: Motivation There are five required courses for a student to take. The courses are: C 1, C 2, C 3, C 4, and C 5. C 1 and C 2 have no prerequisites. C 3 ’s prerequisites are C 1 and C 2. C 4 has C 3 as its prerequisite. C 5 has C 3 and C 4 as its prerequisites.

10. Topological Sorting: Motivation C1 C3 C2 C4 C5

11. Topological Sorting Edges: –(C 1, C 3 ), (C 2, C 3 ), (C 3, C 4 ), (C 3, C 5 ), (C 4, C 5 ) This graph has two possible topological sortings: –C 1, C 2, C 3, C 4, C 5 –C 2, C 1, C 3, C 4, C 5

12. Topological Sorting: Algorithm A source is a vertex with no incoming edges, i.e. the indegree of the source is 0. –Identify a source. –Delete it with all its outgoing edges from the graph. –Keep going until no more sources are left. –If, at any point, a source cannot be found, stop and return false. –Make sure that all vertices are included in the topological sorting.

13. Topological Sorting: Pseudocode TopologicalSort(G) { Initialiaze Q; int counter = 0; for each vertex V in G if ( V.indegree == 0 ) Q.Push(V); while ( !Q.IsEmpty() ) { V = Q.Pop(); V.TopNum = ++counter; for each W adjacent to V { W.indegree--; if ( W.indegree == 0 ) { Q.Pop(W); } if ( counter != NUM_VERTICES ) return an error, because G has a cycle; }

14. Topological Sorting: Example C1 C3 C2 C4 C5 C1 is a source; Delete C1 and its outgoing edges.

15. Topological Sorting: Example C3 C2 C4 C5 C2 is a source; Delete C2 and its outgoing edges.

16. Topological Sorting: Example C3 C4 C5 C3 is a source; Delete C3 and its outgoing edges.

17. Topological Sorting: Example C4 C5 C4 is a source; Delete C4 and its outgoing edges.

18. Topological Sorting: Example C5 C5 is a source, and we are done.

19. Topological Sort: Run Time O(|V| + |E|)