1,529 research outputs found

    Preserving Distances in Very Faulty Graphs

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    Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain replacement paths around the faulted set. Unfortunately, the complexity of the interaction between replacement paths blows up significantly, even from 1 to 2 faults, and the structure of optimal preservers and spanners is poorly understood. In particular, no nontrivial bounds for preservers and additive spanners are known when the number of faults is bigger than 2. Even the answer to the following innocent question is completely unknown: what is the worst-case size of a preserver for a single pair of nodes in the presence of f edge faults? There are no super-linear lower bounds, nor subquadratic upper bounds for f>2. In this paper we make substantial progress on this and other fundamental questions: - We present the first truly sub-quadratic size fault-tolerant single-pair preserver in unweighted (possibly directed) graphs: for any n node graph and any fixed number f of faults, O~(fn^{2-1/2^f}) size suffices. Our result also generalizes to the single-source (all targets) case, and can be used to build new fault-tolerant additive spanners (for all pairs). - The size of the above single-pair preserver grows to O(n^2) for increasing f. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size O(n^{2-eps}) for eps>0, then the additive error has to be Omega(eps f). This surprisingly matches known upper bounds in the literature. - For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is Theta(n^2) for f > 1 in both directed and undirected graphs, while for f=1 the size is Theta(n) in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound. Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure

    Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

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    A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node vGv \in G stores its distance to the so-called hubs SvVS_v \subseteq V, chosen so that for any u,vVu,v \in V there is wSuSvw \in S_u \cap S_v belonging to some shortest uvuv path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with E(G)=O(n)|E(G)| = O(n), for which we show a lowerbound of n2O(logn)\frac{n}{2^{O(\sqrt{\log n})}} for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size O(nRS(n)c)O(\frac{n}{RS(n)^{c}}) for some 0<c<10 < c < 1, where RS(n)RS(n) is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to n2(logn)o(1)\frac{n}{2^{(\log n)^{o(1)}}} would require a breakthrough in the study of lower bounds on RS(n)RS(n), which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of 12O(logn)SumIndex(n)\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n), where SumIndex(n)SumIndex(n) is the communication complexity of the Sum-Index problem over ZnZ_n. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be Θ(n2(logn)c)\Theta(\frac{n}{2^{(\log n)^c}}) for some 0<c<10<c < 1

    Path-Reporting Distance Oracles with Near-Logarithmic Stretch and Linear Size

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    Given an nn-vertex undirected graph G=(V,E,w)G=(V,E,w), and a parameter k1k\geq1, a path-reporting distance oracle (or PRDO) is a data structure of size S(n,k)S(n,k), that given a query (u,v)V2(u,v)\in V^2, returns an f(k)f(k)-approximate shortest uvu-v path PP in GG within time q(k)+O(P)q(k)+O(|P|). Here S(n,k)S(n,k), f(k)f(k) and q(k)q(k) are arbitrary functions. A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen, has S(n,k)=O(kn1+1k)S(n,k)=O(k\cdot n^{1+\frac{1}{k}}), f(k)=2k1f(k)=2k-1 and q(k)=O(logk)q(k)=O(\log k). The size of this oracle is Ω(nlogn)\Omega(n\log n) for all kk. Elkin and Pettie and Neiman and Shabat devised much sparser PRDOs, but their stretch was polynomially larger than the optimal 2k12k-1. On the other hand, for non-path-reporting distance oracles, Chechik devised a result with S(n,k)=O(n1+1k)S(n,k)=O(n^{1+\frac{1}{k}}), f(k)=2k1f(k)=2k-1 and q(k)=O(1)q(k)=O(1). In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. We devise a PRDO with size S(n,k)=O(kloglognlognn1+1k)S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}}), stretch f(k)=O(k)f(k)=O(k) and query time q(k)=O(logkloglognlogn)q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil). We can also have size O(n1+1k)O(n^{1+\frac{1}{k}}), stretch O(kkloglognlogn)O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil) and query time q(k)=O(logkloglognlogn)q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil). Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any ϵ>0\epsilon>0, any k=1,2,...k=1,2,..., and any graph GG and a collection P\mathcal{P} of pp vertex pairs, there exists a (1+ϵ)(1+\epsilon)-approximate preserver with O(γ(ϵ,k)p+nlogk+n1+1k)O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}}) edges, where γ(ϵ,k)=(logkϵ)O(logk)\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.Comment: 61 pages, 3 figure

    New Fault Tolerant Subset Preservers

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    New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures

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    In this paper, we consider the question of computing sparse subgraphs for any input directed graph G=(V,E)G=(V,E) on nn vertices and mm edges, that preserves reachability and/or strong connectivity structures. We show O(n+min{Pn,nP})O(n+\min\{|{\cal P}|\sqrt{n},n\sqrt{|{\cal P}|}\}) bound on a subgraph that is an 11-fault-tolerant reachability preserver for a given vertex-pair set PV×V{\cal P}\subseteq V\times V, i.e., it preserves reachability between any pair of vertices in P{\cal P} under single edge (or vertex) failure. Our result is a significant improvement over the previous best O(nP)O(n |{\cal P}|) bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set PV×V{\cal P}\subseteq V\times V of size Ω(nϵ)\Omega(n^\epsilon), for even any arbitrarily small ϵ\epsilon, requires at least Ω(n1+ϵ/8)\Omega(n^{1+\epsilon/8}) edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. We also present the first sub-quadratic bound of at most O~(k2kn21/k)\tilde{O}(k 2^k n^{2-1/k}) size, for strong-connectivity preservers of directed graphs under kk failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general kk. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95]
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