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 $d\leq d_{\star}$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_{\star} m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. 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 w.h.p. 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 $U\subseteq V$ 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 $s$

Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning

During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size d; after the update we have to answer queries. In this paper, we consider the dynamic subgraph connectivity problem with sensitivity d: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to $d$ vertices are changed. Then we get a sequence of connectivity queries in the subgraph of activated vertices. We present the first fully dynamic algorithm for this problem which has an update and query time only slightly worse than the best decremental algorithm. In addition, we present the first incremental algorithm which is tight with respect to the best known conditional lower bound; moreover, the algorithm is simple and we believe it is implementable and efficient in practice

Path-Fault-Tolerant Approximate Shortest-Path Trees

Let $G=(V,E)$ be an $n$-nodes non-negatively real-weighted undirected graph. In this paper we show how to enrich a {\em single-source shortest-path tree} (SPT) of $G$ with a \emph{sparse} set of \emph{auxiliary} edges selected from $E$, in order to create a structure which tolerates effectively a \emph{path failure} in the SPT. This consists of a simultaneous fault of a set $F$ of at most $f$ adjacent edges along a shortest path emanating from the source, and it is recognized as one of the most frequent disruption in an SPT. We show that, for any integer parameter $k \geq 1$, it is possible to provide a very sparse (i.e., of size $O(kn\cdot f^{1+1/k})$) auxiliary structure that carefully approximates (i.e., within a stretch factor of $(2k-1)(2|F|+1)$) the true shortest paths from the source during the lifetime of the failure. Moreover, we show that our construction can be further refined to get a stretch factor of $3$ and a size of $O(n \log n)$ for the special case $f=2$, and that it can be converted into a very efficient \emph{approximate-distance sensitivity oracle}, that allows to quickly (even in optimal time, if $k=1$) reconstruct the shortest paths (w.r.t. our structure) from the source after a path failure, thus permitting to perform promptly the needed rerouting operations. Our structure compares favorably with previous known solutions, as we discuss in the paper, and moreover it is also very effective in practice, as we assess through a large set of experiments.Comment: 21 pages, 3 figures, SIROCCO 201

Fault-Tolerant ST-Diameter Oracles

We study the problem of estimating the ST-diameter of a graph that is subject to a bounded number of edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a given graph G, two sets of vertices S,T, and positive integer f. When queried with a set F of at most f edges, the oracle returns an estimate D? of the ST-diameter diam(G-F,S,T), the maximum distance between vertices in S and T in G-F. The oracle has stretch ? ? 1 if diam(G-F,S,T) ? D? ? ? diam(G-F,S,T). If S and T both contain all vertices, the data structure is called an f-edge fault-tolerant diameter oracle (f-FDO). An f-edge fault-tolerant distance sensitivity oracles (f-DSO) estimates the pairwise graph distances under up to f failures. We design new f-FDOs and f-FDO-STs by reducing their construction to that of all-pairs and single-source f-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature. We also provide an information-theoretic lower bound on the space requirement of approximate f-FDOs. We show that there exists a family of graphs for which any f-FDO with sensitivity f ? 2 and stretch less than 5/3 requires ?(n^{3/2}) bits of space, regardless of the query time