For fixed @math , we consider the task of adding to a graph @math a set of weighted shortcut edges on the same vertex set, such that the length of a shortest @math -hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in @math . A set of shortcut edges with this property is called an exact @math -hopset and may be applied in processing distance queries on graph @math . In particular, a @math -hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on @math -hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that @math -hopsets require exponentially fewer shortcuts per node than any previously described distance oracle while incurring only a quadratic increase in the query decoding time, and actually offer a speedup when compared to simple oracles based on a direct application of @math -hopsets. Finally, we consider the problem of computing minimum-size @math -hopset (for any @math ) for a given graph @math , showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When @math , for a given bound on the space used by the distance oracle, we provide a construction of hopsets achieving polylog approximation both for space and query time compared to the optimal @math -hopset oracle given the space bound.
A long line of research studies the interplay between data structure space and query decoding time. A lot of attention has been given to distance oracles for planar graphs @cite_36 @cite_33 @cite_31 @cite_32 @cite_2 @cite_0 @cite_25 , and it has recently been shown that a distance oracle with @math space and @math query-time is possible @cite_25 . In the context of weighted directed graphs with treewidth @math , Chaudhuri and Zaroliagis @cite_38 propose a distance oracle using @math space and @math query time for integral @math where @math is the @math th-row inverse Ackermann function (as defined in Subsection ). In the context of unweighted graphs with treewidth @math , Farzan and Kamali @cite_13 obtain distance oracles with @math query time using optimal space (within low order terms). This construction heavily relies on the unweighted setting as exhaustive look-up tables are constructed for handling graphs with polylogarithmic size.
@cite_38 @cite_33 @cite_36 @cite_32 @cite_0 @cite_2 @cite_31 @cite_13 @cite_25
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A (1+e)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Ω(e-1n lg n) space for n-node graphs. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of e and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(nlg2 n). However, our constructions have slower query times. For planar graphs, the query time is O(e-2 lg2 n). For all our linear-space results, we can in fact ensure, for any δ > 0, that the space required is only 1 + δ times the space required just to represent the graph itself. We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation @math , we show how to construct in @math time a data structure of size @math that answers distance queries in @math time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever @math . * We provide a linear-space exact distance oracle for planar graphs with query time @math for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space @math such that for any pair of nodes at distance D the query time is @math . Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with @math nodes, we can preprocess G in @math time to produce a data structure of size @math that can answer the following queries in @math time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA'05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For @math and space @math , we essentially improve the query time from @math to @math . We present an O(n1.5)-space distance oracle for directed planar graphs that answers distance queries in O(log n) time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses O(n5 3)-space and answers queries in O(log n) time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any S ∈ [n, n2], we show an oracle of size S that answers queries in O(max 1, n1.5 S ) time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of S and improves by polynomial factors over all previously known tradeoffs for the range S ∈ [n, n5 3]. For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n 5 3)-space distance oracle which answers exact distance queries in O(log n) time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space O(n 2- ) for some constant 0) either required query time polynomial in n or could only answer approximate distance queries.Furthermore, we show how to trade-off time and space: for any S &#x2265; n 3 2, we show how to obtain an S-space distance 5 2 oracle that answers queries in time O(S n 3 2 log n). This is a polynomial improvement over the previous planar distance oracles with o(n 1 4) query time.