Graph Algorithms Overview for Elementary Level

1. Elementary Graph Algorithms Peter Schröder

2. Representations • For graphs G=(V,E) • adjacency list • for every vertex v list Adj[v] • better for sparse graphs: size of O(max(E,V)) • adjacency matrix A: aij=1 if (vi,vj) in E • single bit entry for each edge in |V|x|V| matrix • better for dense graphs: size is O(|V|2) • meaning of A2?

3. Breadth First Search • Given a graph G=(V,E) and a distinguished vertex s in V • discover every vertex reachable from s • find shortest path from s • the name reflects the fact that all vertices at distance k-1 are discovered before any at distance k are discovered • 3 colors • white: undiscovered; gray: discovered; black: all neighbors have been discovered

4. Breadth First Tree • Incremental buildup of a tree with a predecessor relationship BFS(G,s) for( u in V[G]-{s} ) color[u] = white; d[u] = infinity; p[u] = nil; color[s] = gray; d[s] = 0; p[s] = nil; Q = {s} while( Q != {} ) u = head[Q] for( v in Adj[u] ) if( color[v] == white ) color[v] = gray; d[v] = d[u]+1; p[v] = u; Enqueue(Q,v) Dequeue(Q) color[u] = black

5. Analysis • Running time? • Shortest paths • Lemma 1: Let G=(V,E) be (un-)directed, s in V arbitrary. Then for any (u,v) in E: ∂(s,v)≤ ∂(s,u)+1 • Lemma 2: Let G=(V,E) be (un-)directed, suppose BFS(G,s) is run. Then upon termination d[v]≥ ∂(s,v) for all v in V

6. Analysis • Shortest paths • Lemma 3: Suppose during execution of BFS(G,s) Q contains (v1,v2,…vr). Then d[vr] ≤d[v1]+1 and d[vi] ≤d[vi+1] for i=1,2,…r-1 • Theorem: Correctness of BFS. Suppose BFS(G,s) is run then every v in V reachable from s is discovered and d[v]= ∂(s,v) upon termination. One of the shortest paths from s to v is given by the shortest path from s to p[v] and the edge (p[v],v)

7. Breadth First Trees • BFS builds a breadth first tree in the p[] field • predecessor subgraph of G=(V,E) • Gp(Vp,Ep) where Vp contains all v in V with non-nil p[] field plus {s} and Ep contains all predecessor edges for all v in Vp • this is a tree. Why? • Lemma 5: BFS(G,s) produces a predecessor subgraph which is a breadth first tree

8. Depth First Search • Different strategy • always go as deep as possible before going on • not just a single tree, but a forest results • Gp=(V,Ep) where Ep consists of all edges (p[v],v) for any v in V with predecessor field not empty • contrast with BFS tree: Gp(Vp,Ep), where Ep depends on Vp not V! • there will be no distinguished start vertex

9. Depth First Search DFS(G) for( u in V[G] ) color[u] = white; p[u] = nil; time = 0 for( u in V[G] ) if( color[u] == white ) DFS-visit(u) DFS-visit(u) color[u] = gray; d[u] = time = time+1; for( v in adj[u] ) if( color[v] == white ) p[v] = u; DFS-visit(v); color[u] = black; f[u] = time = time + 1;

10. Analysis • Running time • Correctness • Theorem 6: (Parenthesis theorem) In any DFS of a (un-)directed graph G=(V,E) for any pair u,v one of the following holds • [d[u],f[u]] and [d[v],f[v]] are entirely disjoint • [d[u],f[u]] is contained in [d[v],f[v]] and u is a descencent of v • vice versa • Corollary 7: descendent intervals are nested

11. Analysis • Correctness • Theorem 8: (White path theorem) In a DFF of a (un-)directed G=(V,E) vertex v is a descendent of u iff at the time d[u] v can be reached from u along a path of white vertices

12. Classification • DFS admits classification of edges • Tree edges: members of DFF of Gp. (u,v) is a tree edge if v was discovered exploring u • Back edges: edges (u,v) connecting a vertex u to an ancestor v (self loops are back edges) • Forward edges: non-tree edges (u,v) connecting u to a descendent v • Cross edges: all others (c0uld be same tree, could be different trees)

13. Edge Types • Undirected graphs • Theorem 9: in a DFS of an undirected graph G=(V,E) every edge of G is either a tree edge or a back edge

14. Topological Sort • For a directed acyclic graph (DAG) • a topological sort is a linear ordering of all its vertices such that if G contains (u,v) then u appears before v in the ordering Topological-Sort(G) call DFS(G) to compute finish times enter vertices onto front of list according to their finish times return linked list of vertices

15. Analysis • Running time • Correctness • Lemma 10: a directed graph is acyclic iff DFS(G) yields no back edges • Theorem 11: Topological-Sort(G) produces a topological sort of a DAG G

16. A Classic Problem • Strongly connected components • a strongly connected component of a directed graph G=(V,E) is a maximal subset U of V such that for any u,v in U there is a path from u to v and a path from v to u • form the transpose GT (runtime?) • u is reachable from v in G iff v is reachable from u in GT

17. Algorithm Strongly-Connected-Components(G) call DFS(G) to compute finish times compute GT call DFS(GT) and consider vertices decreasing finish time order output vertices of each tree in DFS(GT) as separate strongly connected component

18. Analysis • Running time • Correctness • Lemma 12: if two vertices are in the same SCC then no path between them ever leaves the SCC • Theorem 13: in any DFS all vertices in the same SCC are placed in the same DFT

19. Analysis • Correctness • Theorem 14: in a directed graph G=(V,E) the forefather phi(u) of any u in V in any DFS of G is an ancestor of u • Corollary 15: in any DFS of a directed graph G=(V,E) vertices u and phi(u) for all u in V lie in the same SCC • Theorem 16: in a directed graph G=(V,E) two vertices u,v in V lies in the same SCC iff they have the same forefather in DFS(G)