Depth first search and linear graph algorithms-英文文献.pdfVIP

SIAM J. COMPUT. Vol. 1, No. 2, June 1972 DEPTH-FIRST SEARCH AND LINEAR GRAPH ALGORITHMS* ROBERT TARJAN Abstract. The value of depth-first search or bacltracking as a technique for solving problems is illustrated by two examples. An improved version of an algorithm for finding the strongly connected components of a directed graph and ar algorithm for finding the biconnected components of an un- direct graph are presented. The space and time requirements of both algorithms are bounded by k 1V + k2E d- k for someconstants kl, k2, and ka, where Vis the number ofvertices and E is the number of edges of the graph being examined. Key words. Algorithm, backtracking, biconnectivity, connectivity, depth-first, graph, search, spanning tree, strong-connectivity. 1. Introduction. Consider a graph G, consisting of a set of vertices U and a set of edges g. The graph may either be directed (the edges are ordered pairs (v, w) of vertices; v is the tail and w is the head of the edge) or undirected (the edges are unordered pairs of vertices, also represented as (v, w)). Graphs form a suitable abstraction for problems in many areas; chemistry, electrical engineering, and sociology, for example. Thus it is important to have the most economical algo- rithms for answering graph-theoretical questions. In studying graph algorithms we cannot avoid at least a few definitions. These definitions are more-or-less standard in the literature. (See Harary [3], for instance.) If G (,g) is a graph, a path p’v w in G is a sequence of vertices and edges leading from v to w. A path is simple if all its vertices are distinct. A path p’v v is called a closed path. A closed path p’v v is a cycle if all its edges are distinct and the only vertex to occur twice in p is v, which occurs exactly twice. Two cycles which are cyclic permutations of each other are considered to be the