Decremental Strongly Connected Components and Single-Source Reachability in Near-Linear Time

Computing the strongly connected Components (SCCs) in a graph G = (V, E) is known to take only O(m+n) time using an algorithm by Tarjan [SIAM J. Comput., 1 (1972), pp. 146-160] where m = |E|, n = |V |. For fully dynamic graphs, conditional lower bounds provide evidence that the update time cannot be improved by polynomial factors over recomputing the SCCs from scratch after every update. Nevertheless, substantial progress has been made to find algorithms with fast update time for decremental graphs, i.e., graphs that undergo edge deletions. In this paper, we present the first algorithm for general decremental graphs that maintains the SCCs in total update time O(m), thus only a polylogarithmic factor from the optimal running time. (We use O(f(n)) notation to suppress logarithmic factors, i.e., g(n) = O(f(n)) if g(n) = O(f(n)polylog(n)).) Our result also yields the fastest algorithm for the decremental single-source reachability (SSR) problem which can be reduced to decrementally maintaining SCCs. Using a well-known reduction, we use our decremental result to achieve new update/query-time trade-offs in the fully dynamic setting. We can maintain the reachability of pairs S × V , S ⊆ V in fully dynamic graphs with update time O(^|S|m_t) and query time O(t) for all t ∊ [1, |S|]; this matches to polylogarithmic factors the best all-pairs reachability algorithm for S = V .