Path-based depth-first search for strong and biconnected components

Abstract

Depth-first search, as developed by Tarjan and coauthors, is a fundamental technique of efficient algorithm design for graphs [23]. This note presents depth-first search algorithms for two basic problems, strong and biconnected components. Previous algorithms either compute auxiliary quantities based on the depth-first search tree (e.g., LOWPOINT values) or require two passes. We present one-pass algorithms that only maintain a representation of the depth-first search path. This gives a simplified view of depth-first search without sacrificing efficiency. In greater detail, most depth-first search algorithms (e.g., [23,10,11]) compute so-called LOWPOINT values that are defined in terms of the depth-first search tree. Because of the success of this method LOWPOINT values have become almost synonymous with depth-first search. LOWPOINT values are regarded as crucial in the strong and biconnected component algorithms, e.g., [14, pp. 94, 514]. Tarjan’s LOWPOINT method for strong components is presented in texts [1, 7,14,16,17,21]. The strong component algorithm of Kosaraju and Sharir [22] is often viewed as conceptu-