Elementary Graph Algorithms

Transcription

1 Elementary Graph Algorithms Representations Breadth-First Search Depth-First Search Topological Sort Strongly Connected Components CS 5633 Analysis of Algorithms Chapter 22: Slide 1

2 Graph Representations 2 example analysis G.V = vertices of graph G G.E = edges of graph G G.Adj[u] = all vertices v where (u,v) G.E A graph can be represented as an adjacency list or adjacency matrix. CS 5633 Analysis of Algorithms Chapter 22: Slide 2

3 Representations Continued 2 example analysis How efficient are these? Space? Determining if (u,v) is an edge in the graph? Visiting each edge once? CS 5633 Analysis of Algorithms Chapter 22: Slide 3

4 Breadth-First Search 2 example analysis BFS is O(V +E). BFS(G, s) finds shortest paths from s. Algorithm notes: Search starts from vertex s. white, gray, black respectively means undiscovered, discovered, finished. v.d = distance from s v.π = previous vertex on path from s to v CS 5633 Analysis of Algorithms Chapter 22: Slide 4

5 Breadth-First Search Algorithm 2 example analysis BFS(G, s) for each vertex v G.V v.color white, v.d, v.π nil s.color gray, s.d 0, Q {s} while Q u Dequeue(Q) for each v G.Adj[u] if v.color = white then Enqueue(Q, v) v.color gray v.d u.d+1 v.π u u.color black CS 5633 Analysis of Algorithms Chapter 22: Slide 5

6 Breadth-First Search Example 2 example analysis CS 5633 Analysis of Algorithms Chapter 22: Slide 6

7 Depth-First Search 2 example analysis DFS is Θ(V +E). v.d = discovery time v.f = finishing time v.d and v.f form a parenthesis structure. DFS(G) for each vertex u G.V u.color white u.π nil time 0 for each vertex u G.V if u.color = white then time DFS-Visit(u,time+1) CS 5633 Analysis of Algorithms Chapter 22: Slide 7

8 DFS Algorithm Continued 2 example analysis DFS-Visit(u, time) u.color gray u.d time for each vertex v G.Adj[u] if v.color = white then v.π u time DFS-Visit(v,time+1) u.color black return u.f time+1 CS 5633 Analysis of Algorithms Chapter 22: Slide 8

9 Depth-First Search Example 2 example analysis CS 5633 Analysis of Algorithms Chapter 22: Slide 9

10 Topological Sort 2 example analysis A of a directed acyclic graph (dag) orders the vertices. u v if (u,v) is an edge. Topological-Sort(G) Call DFS(G). As each vertex is finished, insert the vertex onto the front a linked list. a b c d e f g h i CS 5633 Analysis of Algorithms Chapter 22: Slide 10

11 Topological Sort Analysis 2 example analysis Topological-Sort is Θ(V + E). Correct because: If there is a path from u to v, then either: Case 1: DFS-Visit(u) before DFS-Visit(v). u.d < v.d < v.f < u.f because DFS-Visit(u) discovers v. Case 2: DFS-Visit(v) before DFS-Visit(u). v.d < v.f < u.d < u.f because DFS-Visit(v) doesn t discover u. CS 5633 Analysis of Algorithms Chapter 22: Slide 11

12 Strongly Connected Components 2 example analysis A strongly connected component of a directed graph is a maximal set of vertices where every vertex can reach the others. Strongly-Connected-Components(G) Call DFS(G) saving finishing times u.f. Compute G T, i.e., reverse all the edges. Call DFS(G T ), loop order is decreasing u.f. Each DFS(G T ) tree is a SCC. CS 5633 Analysis of Algorithms Chapter 22: Slide 12

13 SCC Example 2 example analysis G a e b c d f g h G T a e b c d f g h CS 5633 Analysis of Algorithms Chapter 22: Slide 13

14 SCC Analysis 2 example analysis Let C 1 and C 2 be two SCCs in a graph G. Suppose path from C 1 to C 2. Suppose u is the first vertex discovered in C 1. u s finish time will be after other vertices in C 1. Case 1: C 1 is discovered before C 2. DFS will explore C 2 before u is finished. Case 2: C 2 is discovered before C 1. DFS will explore C 2 before u is discovered. In both cases, u s finish time will be after all vertices in C 2. When G T is explored, u (and C 1 ) will be discovered and finished before C 2. CS 5633 Analysis of Algorithms Chapter 22: Slide 14