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).
For a general undirected @math -vertex graph @math , Thorup and Zwick showed that for any parameter @math , there is a @math -approximate distance oracle for @math with @math space and @math query time which is believed to be essentially optimal due to a girth conjecture of Erd o s @cite_14 . Variations and slight improvements have since been presented; see, e.g., @cite_12 @cite_11 @cite_10 @cite_15 @cite_0 .
@cite_14 @cite_0 @cite_15 @cite_10 @cite_12 @cite_11
1581792577 1601859867 2951814478 2158860088 2951493689 2069572385
Given an undirected graph @math with @math edges, @math vertices, and non-negative edge weights, and given an integer @math , we show that for some universal constant @math , a @math -approximate distance oracle for @math of size @math can be constructed in @math time and can answer queries in @math time. We also give an oracle which is faster for smaller @math . Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all @math and improve the @math time bound of Thorup and Zwick except for very sparse graphs and small @math . When @math and @math , our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erd o s. We give the first improvement to the space approximation trade-off of distance oracles since the seminal result of Thorup and Zwick. For unweighted undirected graphs, our distance oracle has size @math and, when queried about vertices at distance @math , returns a path of length at most 2d+1. For weighted undirected graphs with @math edges, our distance oracle has size @math and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space @math . For unweighted graphs, this implies a @math space lower bound to achieve approximation 2d+1. We give the first improvement to the space approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size @math and, when queried about vertices at distance @math , returns a path of length @math . For weighted graphs with @math edges, our distance oracle has size @math and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space @math . For unweighted graphs, this implies a @math space lower bound to achieve approximation @math . We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Thorup and Zwick showed that for any integer k 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn @math ) expected time and construct a data structure (a (2k1)-approximate distance oracle) of size O(kn @math ) capable of returning in O(k) time an approximation @math of the distance (u,v) from u to v in G that satisfies @math , for any two vertices u,v V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn @math log n). We present here a deterministic O(kmn @math ) time algorithm for constructing oracles of size O(kn @math ). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP03) for constructing (2k1)-spanners of size O(kn @math ) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced. We consider distance queries in vertex-labeled planar graphs. For any fixed @math we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex @math and a label @math return a @math -approximation of the distance from @math to its closest vertex with label @math . For a directed planar graph with @math vertices, such that the ratio of the largest to smallest arc length is bounded by @math , the preprocessing time is @math , the data structure size is @math , and the query time is @math . We also point out that a vertex label distance oracle for undirected planar graphs suggested in an earlier version of this paper is incorrect.