An Improved Algorithm for Finding the Strongly Connected Components of a Directed Graph

For a directed graph D = (V,E), a Strongly Connected Component (SCC) is a maximal induced subgraph S = (VS , ES) where, for every x, y ∈ VS , there is a path from x to y (and vice-versa). Tarjan presented a now well-established algorithm for computing the strongly connected components of a digraph in time Θ(v+e) [8]. In the worst case, this needs v(2 + 5w) bits of storage, where w is the machine’s word size. Nuutila and Soisalon-Soininen reduced this to v(1 + 4w) [6]. In this paper, we present for the first time an algorithm requiring only 3vw bits in the worst case. Tarjan’s algorithm has found numerous uses in the literature, often as a subcomponent of larger algorithms, such as those for transitive closure [5], compiler optimisation [3] and program analysis [1, 7] to name but a few. Of particular relevance is its use in model checking, where the algorithm’s storage requirements are a critical factor limiting the number of states which can be explored [4].