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LinearSpace Approximate Distance Oracles for Planar, BoundedGenus, and MinorFree Graphs
"... Abstract. A (1 + ɛ)approximate distance oracle for a graph is a data structure that supports approximate pointtopoint shortestpathdistance queries. The relevant measures for a distanceoracle construction are: space, query time, and preprocessing time. There are strong distanceoracle construct ..."
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Cited by 11 (5 self)
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Abstract. A (1 + ɛ)approximate distance oracle for a graph is a data structure that supports approximate pointtopoint shortestpathdistance queries. The relevant measures for a distanceoracle construction are: space, query time, and preprocessing time. There are strong distanceoracle constructions known for planar graphs (Thorup) and, subsequently, minorexcluded graphs (Abraham and Gavoille). However, these require Ω(ɛ −1 n lg n) space for nnode graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, boundedgenus graphs, and minorexcluded graphs we give distanceoracle constructions that require only
Submatrix maximum queries in Monge matrices and partial Monge matrices, and their applications
, 2012
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Exact Distance Oracles for Planar Graphs
, 2010
"... We provide the first linearspace data structure with provable sublinear query time for exact pointtopoint shortest path queries in planar graphs. We prove that for any planar graph G with nonnegative arc lengths and for any ɛ> 0 there is a data structure that supports exact shortest path and d ..."
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Cited by 8 (4 self)
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We provide the first linearspace data structure with provable sublinear query time for exact pointtopoint shortest path queries in planar graphs. We prove that for any planar graph G with nonnegative arc lengths and for any ɛ> 0 there is a data structure that supports exact shortest path and distance queries in G with the following properties: the data structure can be created in time O(n lg(n) lg(1/ɛ)), the space required is O(n lg(1/ɛ)), and the query time is O(n 1/2+ɛ). Previous data structures by Fakcharoenphol and Rao (JCSS’06), Klein, Mozes, and Weimann (TransAlg’10), and Mozes and WulffNilsen (ESA’10) with query time O(n 1/2 lg 2 n) use space at least Ω(n lg n / lg lg n). We also give a construction with a more general tradeoff. We prove that for any integer S ∈ [n lg n, n 2], we can construct in time Õ(S) a data structure of size O(S) that answers distance queries in O(nS −1/2 lg 2.5 n) time per query. Cabello (SODA’06) gave a comparable construction for the smaller range S ∈ [n 4/3 lg 1/3 n, n 2]. For the range S ∈ (n lg n, n 4/3 lg 1/3 n), only data structures of size O(S) with query time O(n 2 /S) had been known (Djidjev, WG’96). Combined, our results give the best query times for any shortestpath data structure for planar graphs with space S = o(n 4/3 lg 1/3 n). As a consequence, we also obtain an algorithm that computes k–many distances in planar graphs in time O((kn) 2/3 (lg n) 2 (lg lg n) −1/3 + n(lg n) 2 / lg lg n). 1
More compact oracles for approximate distances in undirected planar graphs
 In SODA ’13
, 2013
"... Distance oracles are data structures that provide fast (possibly approximate) answers to shortestpath and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS‘01, Thorup int ..."
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Cited by 4 (2 self)
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Distance oracles are data structures that provide fast (possibly approximate) answers to shortestpath and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS‘01, Thorup introduced approximate distance oracles for planar graphs. He proved that, for any > 0 and for any planar graph on n nodes, there exists a (1 + )–approximate distance oracle using space O(n−1 logn) such that approximate distance queries can be answered in time O(−1). Ten years later, we give the first improvements on the space–query time tradeoff for planar graphs. • We give the first oracle having a space–time product with subquadratic dependency on 1/. For space Õ(n logn) we obtain query time Õ(−1) (assuming polynomial edge weights). We believe that the dependency on may be almost optimal. • For the case of moderate edge weights (average bounded by poly(logn), which appears to be the case for many realworld road networks), we hit a “sweet spot, ” improving upon Thorup’s oracle both in terms of and n. Our oracle uses space Õ(n log log n) and it has query time Õ(−1 + log log log n). (Notation: Õ(·) hides lowdegree polynomials in log(1/) and log∗(n).) ar X iv
Faster Shortest Paths in Dense Distance Graphs, with Applications
, 2014
"... We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the lineartime shortestpath algorithm of Henzinger, Klein, Subramanian, and Rao [STOC’94]. The second is Fakcharoenphol and Rao’s algorithm [FOCS’01] for emulating Dijkstra’s alg ..."
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We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the lineartime shortestpath algorithm of Henzinger, Klein, Subramanian, and Rao [STOC’94]. The second is Fakcharoenphol and Rao’s algorithm [FOCS’01] for emulating Dijkstra’s algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph G into regions of at most r vertices each, for some parameter r < n. The vertex set of the DDG is the set of Θ(n/ r) vertices of G that belong to more than one region (boundary vertices). The DDG has Θ(n) arcs, such that distances in the DDG are equal to the distances in G. Fakcharoenphol and Rao’s implementation of Dijkstra’s algorithm on the DDG (nicknamed FRDijkstra) runs in O(n log(n)r−1/2 log r) time, and is a key component in many stateoftheart planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the logn dependency in the running time of the shortestpath algorithm, making it O(nr−1/2 log2 r). This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly lineartime, algorithms for problems such as minimumcut, maximumflow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in O(n log p) time, where p is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with r vertices and k boundary vertices in O(r log k) time, which is faster than has been previously known for small values of k.