Chapter 22. Elementary Graph Algorithms

Transcription

1 Graph Algorithms - Spring 2011 Set 7. Lecturer: Huilan Chang Reference: (1) Cormen, Leiserson, Rivest, and Stein, Introduction to Algorithms, 2nd Edition, The MIT Press. (2) Lecture notes from C. Y. Chen in NCTU. Chapter 22. Elementary Graph Algorithms Chapter overview: Sec. 1: computational representations of graphs: adjacency lists and adjacency matrices. Sec. 2: graph-searching algorithm breadth-first search Sec. 3: depth-first search. Sec. 4&5: real application of depth-first search: topologically sorting and finding the strongly connected components. 1. Representations of graphs Two standard ways to represent a graph G = (V, E) : adjacency-list and adjacency-matrix. Simple graph: array Adj Figure 1. (a) An undirected graph. (b) An adjacency-list representation of G. (c) The adjacency-matrix representation of G. Directed graph: Example: G = (V, E) where E={(1, 2), (1, 4), (2, 5), (3, 5), (3, 6), (4, 2), (5, 4), (6, 6)}. Figure 2. (a) A directed graph. (b) An adjacency-list representation of G. (c) The adjacency-matrix representation of G. 1

2 The adjacency-list representation Use an array Adj of V lists, one for each vertex in V. 1. undirected graph If {u, v} E, then u appears in Adj[v] and v appears in Adj[u]. 2. directed graph If (u, v) E, then v appears in Adj[u]. 3. weighted graph The weight w(u, v) of the edge (u, v) E is simply stored with vertex v in u s adjacency list. The adjacency-matrix representation Use a V V matrix A = (a ij ). 1. undirected graph 1 if { i, j} E a ij =. 0 otherwise Since G is undirected, A = A T. 2. directed graph 1 if ( i, j) E a ij =. 0 otherwise 3. weighted graph The weight w(u, v) of the edge (u, v) E is stored as a uv. If an edge es not exist, store a NIL value (or 0 or ) at the entry. The memory needed adjacency-list representation: θ ( V + E) adjacency-matrix representation: 2 θ ( V ) (independent of then number of edges) Which representation is usually preferred? use adjacency-list representation: when graph is sparse (i.e., E is much less 2 than V ). use adjacency-matrix representation: when graph is dense or small (i.e., V is small), or when we need to answer quickly if there is an edge connecting two given vertices. 2

3 2. Breadth-first search(bfs, 寬先搜尋 ) Breadth-first search is one of the simplest algorithms for searching a graph and the archetype for many important graph algorithms: Prim's minimum-spanning-tree algorithm (Sec 23.2) and Dijkstra's single-source shortest-paths algorithm (Sec 24.3) Given a graph G = (V, E) and a source vertex s, breadth-first search: - explores the edges of G to "discover" every vertex that is reachable from s. - computes the distance from s. - produces a "breadth-first tree" with root s that contains all reachable vertices. - the path in the breadth-first tree from s to v corresponds to a "shortest path" from s to v in G. - The algorithm works on both directed and undirected graphs. Queue (first-in, first-out) ENQUEUE(Q, w, r) ENQUEUE(Q, s) DEQUEUE(Q, w) 3

4 Figure 3. Tree edges are shown shaded as they are produced by BFS. Within each vertex u is shown d[u]. color[u] (white, gray, or black) denotes the state of vertex u: - All vertices start out white (not discovered). - u is discovered the first time, at which time color[u]=gray. - If all vertices adjacent to u have been discovered, color[u]=black. d[u]: distance from the source s to u (initial= ) π[u] denotes the predecessor of u in the breadth-first tree (π[u] is NIL if u has no predecessor in the breadth-first tree (u is not reachable from s)) (see the shaded edges in Figure 3). BFS(G, s) 1 for each vertex u V[ G] { s} 2 { color[u] WHITE 3 d[u] 4 π[u] NIL } 5 color[s] GRAY 6 d[s] 0 7 π[s] NIL 8 Q Ø 9 ENQUEUE(Q, s) 10 while Q Ø 11 { u DEQUEUE(Q) 12 for each v Adj[ u] 13 if color[v] = WHITE 14 then { color[v] GRAY 15 d[v] d[u] π[v] u 17 ENQUEUE(Q, v) } 18 color[u] BLACK } Give Initial values Perform BFS Analysis If use the adjacency-list representation, then: Steps 1 to 9 take O(V) time and Steps 10 to 18 take O(E) time. Thus total running time of BFS is O(V + E). 4

5 Let δ(s, v) = minimum number of edges in any path from s to v. We can prove: (i) When the above algorithm terminates, d[v] = δ(s, v) for all v V. (ii) For any vertex v s that is reachable from s, one of the shortest paths from s to v is a shortest path from s to π[v] followed by the edge (π[v], v). Exercises : The diameter of a graph G =(V, E) is given by maxδ ( u, v) u, v V that is, the diameter is the largest of all shortest-path distances G. 1. Give an efficient algorithm (O(V)) to compute the diameter of a tree, 2. use an example to depict your algorithm, and 3. analyze the running time of your algorithm. 3. Depth-first search(dfs, 深先搜尋 ) To search "deeper" in the graph whenever possible! - edges are explored out of the most recently discovered vertex v that still has unexplored edges leaving it. - when all of v's edges have been explored, the search "backtracks" to explore edges leaving the vertex from which v was discovered. - this process continues until we have discovered all the vertices that are reachable from the original source vertex. - if any undiscovered vertices remain, then one of them is selected as a new source and the search is repeated from that source. source new source 5

6 Each vertex v has two timestamps( 時間戳 ): 1 st timestamp: discovery time d[v] (v will be colored GRAY). 2 nd timestamp: finishing time f[v] (when the search finishes examining v's adjacency list (v will be colored BLACK). (For every vertex u, d[u] < f[u].) source backtrack new source DFS(G) 1 for each vertex u V[ G] 2 { color[u] WHITE 3 π[u] NIL } 4 time 0 5 for each vertex u V[ G] 6 if color[u] = WHITE 7 then DFS-VISIT(u) DFS-VISIT(u) 1 color[u] GRAY 2 time time d[u] time 4 for each v Adj [u] 5 if color[v] = WHITE 6 then { π[v] u 7 DFS-VISIT(v) } 8 color[u] = BLACK 9 f[u] time time + 1 6

7 Analysis The procedure DFS-VISIT is called exactly once for each vertex v V. During an execution of DFS-VISIT(v), the loop on lines 4-7 is executed Adj[v] times. The total cost of executing lines 4-7 of DFS-VISIT is Θ(E). The running time of DFS is therefore Θ(V + E). Properties of depth-first search Discovery and finishing times have parenthesis structure Theorem 22.7: (Parenthesis theorem) For any two vertices u and v, exactly one of the following three conditions holds: [d[u], f[u]] I [d[v], f[v]] =, and neither u nor v is a descendant ( 子孫 ) of the other in the depth-first forest, [d[u], f[u]] [d[v], f[v]], and u is a descendant of v in a depth-first tree, [d[v], f[v]] [d[u], f[u]], and v is a descendant of u in a depth-first tree. Corollary 22.8: (Nesting of Descendants' Intervals) Vertex v is a proper descendant of vertex u in the depth-first forest d[u] < d[v] < f[v] < f[u]. Theorem 22.9: (White-path theorem) In a depth-first forest, vertex v is a descendant of vertex u at the time d[u], vertex v can be reached from u along a path consisting entirely of white vertices. 7

8 Classification of edges (will be used in Section 4) The DFS forest of a directed graph may have four types of edges: 1. Tree edges Edges in the DFS forest. 2. Back edges Nontree edges (u, v) connecting a vertex u to an ancestor v in a depth-first tree. 3. Forward edges Nontree edges (u, v) connecting a vertex u to a descendant v in a depth-first tree. 4. Cross edges All other edges: (i) between vertices in the same depth-first tree, as long as one vertex is not ancestor of the other, or (ii) between vertices in different depth-first trees. The DFS forest of an undirected graph only have two types of edges: 1. Tree edges 2. Back edges 4. Topological sort In the following, dag means directed acyclic graph. A topological sort of a dag G = (V, E) is a linear ordering of all its vertices such that if G contain an edge (u, v), then u appears before v in the ordering. If the graph is not acyclic, then no linear ordering is possible Example: 8

9 The following simple algorithm topologically sorts a dag. TOPOLOGICAL-SORT(G) 1 call DFS(G) to compute finishing times f[v] for each vertex v and as each vertex is finished, insert it to the front of a linked list 2 return the linked list of vertices Vertices appear in reverse order of their finishing times in the linked list. Analysis If linked-list representation is used, then the algorithm takes θ ( V + E ) time. The correctness of this algorithm uses the following key lemma characterizing directed acyclic graphs. Lemma A directed graph G is acyclic a depth-first search of G yields no back edges. Theorem TOPOLOGICAL-SORT (G) produces a topological sort of a directed acyclic graph G. Proof It suffices to show that if G contains an edge (u, v), then f[u] > f[v]. Consider any edge (u, v). When this edge is explored, (1) v cannot be gray, since then v would be an ancestor of u and (u, v) would be a back edge, contradicting Lemma (2) If v is white, it becomes a descendant of u, and so f[v] < f[u]. (3) If v is black, it has already been finished, so that f[v] has already been set. Hence, f[v] < f[u]. 9