Connectivity Oracles for Graphs Subject to Vertex Failures

Abstract

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. Our deterministic structure processes a batch of d ≤ d failed vertices inÕ(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 timeÕ(d 2 ), query time O(d), and spaceÕ(m) for any d . This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures.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 ⊆ V we can remove a set B of |U |/(s − 2) vertices such that the remaining graph contains a Steiner forest for U − B with maximum degree s. IntroductionThe dynamic subgraph model [19,21,36,39,37,42,65] is a constrained dynamic graph model. Rather than allow the graph to evolve in completely arbitrary ways (via an unbounded sequence of edge insertions and deletions), there is assumed to be a fixed ideal graph G = (V, E) that can be preprocessed in advance. The ideal graph is susceptible only to the failure of edges/vertices and their subsequent recovery, possibly with a bound d on the number of failures at one time. Queries naturally answer questions about the current failure-free subgraph. This model is useful because it more accurately represents the behavior of many real-world networks: changes to the underlying topology are relatively rare but transient failures very common. More importantly, this model offers the algorithm designer the freedom to explore exotic graph rep-

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

“…Recently, [20] provided an efficient construction of distance sensitivity oracles that support f = O(log n/ log log n) many faults with polylogarithmic query time. In an another breakthrough, [25] showed a connectivity sensitivity oracle that supports f ∈ [1, n] vertex failures with O(f m log n) space, update time O(g 2 ) and query time O(g) where g ≤ f is the number of actual faults.…”

Pairwise Reachability Oracles and Preservers under Failures

“…For small values of k, Duan and Pettie [24] improved the update time of [36] to O(k 2 log log n) by presenting a data structure of O(m) size. For handling vertex failures, Duan and Pettie [25] provided a data structure of O(mk log n) size with O(k 3 log 3 n) update time and O(k) query time.…”

Planar Reachability Under Single Vertex or Edge Failures

“…For the case of undirected graphs, there exist nearly optimal f -sensitivity oracles for answering connectivity queries under edge and vertex failures. Duan and Pettie [23] presented a near-optimal preprocessing O(n log n)-space f -sensitivity oracle that, for any set F of up to f edge-failures their oracle, spends O(f log f log log n) time to process the failed edges and then can answer connectivity queries in G \ F in time O(log log n) per query. This result is nearly optimal also for the case of planar undirected graphs.…”

“…We remark that previous data structures handling failures in O(polylog n) time either work only for the single-source version of the problem (see dominator trees, or [19] for two failures), or work only on undirected graphs (see, e.g., [22,23] for oracles for general graphs, and [2,12] for planar graphs), or achieve nearly linear space only for dense graphs [51]. It is worth not-ing that for planar digraphs vertex failures are generally more challenging than edge failures, since, whereas one can easily reduce edge failures to vertex failures, the standard opposite reduction of splitting a vertex into an in-and an out-vertex does not preserve planarity.…”