CONNECTIVITY ORACLES FOR GRAPHS SUBJECT TO VERTEX FAILURES

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of d <= d(*) failed vertices in (O) over tilde (d(3)) time and thereafter answers connectivity queries in O(d) time. It occupies space O(d(*)m log n). We develop a randomized Monte Carlo version of our data structure with update time (O) over tilde (d(2)), query time O(d), and space (O) over tilde (m) for any failure bound d <= n. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. We also develop a more efficient Monte Carlo edge failure connectivity oracle. Using space O(n log(2) n), d edge failures are processed in O(d log d log log n) time, and thereafter, connectivity queries are answered in O(log log n) time, which are correct with high probability. Our data structures are based on a new decomposition theorem for an undirected graph G = (V, E), which is of independent interest. It states that for any terminal set U subset of V we can remove a set B of vertical bar U vertical bar/(s - 2) vertices such that the remaining graph contains a Steiner forest for U - B with maximum degree s.