Recognition of DFS trees: sequential and parallel algorithms with refined verifications

AbstractThe depth-first search (DFS) algorithm is one of the basic techniques used in a very large variety of graph algorithms. Every application of the DFS involves, besides traversing the graph, constructing a special structured tree, called a DFS tree, that may be used subsequently.In a previous work we have shown that the family of graphs in which every spanning tree is a DFS tree is quite limited. Therefore, the question: Given an undirected graph G=(V, E) and an undirected spanning tree T, is T a DFS tree (T-DFS) in G? was naturally raised and answered by sequential linear-time algorithms. Here we present a parallel algorithm which solves this problem in O(t) time complexity and uses O(|E|/t) processors, where t⩾log|V|, on a CREW PRAM. We also study the problem for directed graphs. A linear (O(|E|)) time algorithm for solving it in the sequential case and a parallel Kimplemetation of it, which has O(log2|V|) time complexity and uses O(|V|2.376) processors on a CREW PRAM, are presented.An important feature of our algorithms, that we call refined verification, is that some of their decisions are endowed with proofs that can be verified with a better complexity than that of the algorithms themselves: In the undirected case, if the answer of the algorithm is negative then it outputs a proof for the fact that can be verified in O(t) time complexity with O(|V|/t) processors, where t⩾log|V|, on a CREW PRAM. In the directed case, if T is not a DFS tree in G then the sequential algorithm supplies an O(|V|) time proof for that fact and the parallel implementation supplies a proof for the fact that can be verified in O(t) time complexity with O(|V|/t) processors, where t⩾log|V|, on a CREW PRAM. If T is a DFS tree in G then the parallel implementation of the algorithm outputs a proof that can be verified in O(t) time complexity with O(|E|/t) processors, where t⩾log|V|, on a CREW PRAM. Hence, all the verification have an optimal speed-up