6,512 research outputs found

    Approximating Approximate Distance Oracles

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    Given a finite metric space (V,d), an approximate distance oracle is a data structure which, when queried on two points u,v in V, returns an approximation to the the actual distance between u and v which is within some bounded stretch factor of the true distance. There has been significant work on the tradeoff between the important parameters of approximate distance oracles (and in particular between the size, stretch, and query time), but in this paper we take a different point of view, that of per-instance optimization. If we are given an particular input metric space and stretch bound, can we find the smallest possible approximate distance oracle for that particular input? Since this question is not even well-defined, we restrict our attention to well-known classes of approximate distance oracles, and study whether we can optimize over those classes. In particular, we give an O(log n)-approximation to the problem of finding the smallest stretch 3 Thorup-Zwick distance oracle, as well as the problem of finding the smallest Pv{a}trac{s}cu-Roditty distance oracle. We also prove a matching Omega(log n) lower bound for both problems, and an Omega(n^{frac{1}{k}-frac{1}{2^{k-1}}}) integrality gap for the more general stretch (2k-1) Thorup-Zwick distance oracle. We also consider the problem of approximating the best TZ or PR approximate distance oracle with outliers, and show that more advanced techniques (SDP relaxations in particular) allow us to optimize even in the presence of outliers

    Approximate Distance Oracles for Planar Graphs with Improved Query Time-Space Tradeoff

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    We consider approximate distance oracles for edge-weighted n-vertex undirected planar graphs. Given fixed epsilon > 0, we present a (1+epsilon)-approximate distance oracle with O(n(loglog n)^2) space and O((loglog n)^3) query time. This improves the previous best product of query time and space of the oracles of Thorup (FOCS 2001, J. ACM 2004) and Klein (SODA 2002) from O(n log n) to O(n(loglog n)^5).Comment: 20 pages, 9 figures of which 2 illustrate pseudo-code. This is the SODA 2016 version but with the definition of C_i in Phase I fixed and the analysis slightly modified accordingly. The main change is in the subsection bounding query time and stretch for Phase

    Fast and Compact Exact Distance Oracle for Planar Graphs

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    For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n5/3)O(n^{5/3})-space distance oracle which answers exact distance queries in O(logn)O(\log n) time for nn-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space O(n2ϵ)O(n^{2 - \epsilon}) for some constant ϵ>0\epsilon > 0) either required query time polynomial in nn or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any Sn3/2S \ge n^{3/2}, we show how to obtain an SS-space distance oracle that answers queries in time O((n5/2/S3/2)logn)O((n^{5/2}/ S^{3/2}) \log n). This is a polynomial improvement over the previous planar distance oracles with o(n1/4)o(n^{1/4}) query time

    Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs

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    Let GG be a graph where each vertex is associated with a label. A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex vv and a label λ\lambda, returns a (1+ε)(1+\varepsilon)-approximation of the distance from vv to the closest vertex with label λ\lambda in GG. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements

    Approximate Distance Sensitivity Oracles in Subquadratic Space

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    An ff-edge fault-tolerant distance sensitive oracle (ff-DSO) with stretch σ1\sigma \ge 1 is a data structure that preprocesses a given undirected, unweighted graph GG with nn vertices and mm edges, and a positive integer ff. When queried with a pair of vertices s,ts, t and a set FF of at most ff edges, it returns a σ\sigma-approximation of the ss-tt-distance in GFG-F. We study ff-DSOs that take subquadratic space. Thorup and Zwick [JACM 2015] showed that this is only possible for σ3\sigma \ge 3. We present, for any constant f1f \ge 1 and α(0,12)\alpha \in (0, \frac{1}{2}), and any ε>0\varepsilon > 0, an ff-DSO with stretch 3+ε 3 + \varepsilon that takes O~(n2αf+1/ε)O(logn/ε)f+1\widetilde{O}(n^{2-\frac{\alpha}{f+1}}/\varepsilon) \cdot O(\log n/\varepsilon)^{f+1} space and has an O(nα/ε2)O(n^\alpha/\varepsilon^2) query time. We also give an improved construction for graphs with diameter at most DD. For any constant kk, we devise an ff-DSO with stretch 2k12k-1 that takes O(Df+o(1)n1+1/k)O(D^{f+o(1)} n^{1+1/k}) space and has O~(Do(1))\widetilde{O}(D^{o(1)}) query time, with a preprocessing time of O(Df+o(1)mn1/k)O(D^{f+o(1)} mn^{1/k}). Chechik, Cohen, Fiat, and Kaplan [SODA 2017] presented an ff-DSO with stretch 1+ε1{+}\varepsilon and preprocessing time Oε(n5+o(1))O_\varepsilon(n^{5+o(1)}), albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to Oε(mn2+o(1))O_{\varepsilon}(mn^{2+o(1)}).Comment: accepted at STOC 202
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