Exact Distance Oracles for Planar Graphs

Abstract

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: 1• Given a desired space allocation S ∈ [n lg lg n, n 2 ], we show how to construct inÕ(S) time a data structure of size O(S) that answers distance queries inÕ(n/ √ S) time per query. The best distance oracles for planar graphs until the current work are due to Cabello (SODA 2006), Chen and Xu (STOC 2000), Djidjev (WG 1996), and Fakcharoenphol and Rao (FOCS 2001). For σ ∈ (1, 4/3) and space S = n σ , we essentially improve the query time from n 2 /S to p n 2 /S.• As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever k ∈ [ √ n, n).• We provide a linear-space exact distance oracle for planar graphs with query time O(n 1/2+ ) for any constant > 0. This is the first such data structure with provable sublinear query time.• For edge lengths ≥ 1, we provide an exact distance oracle of spaceÕ(n) such that for any pair of nodes at distance the query time isÕ(min{ , √ n}). Comparable query performance had been observed experimentally but could not be explained theoretically.Our data structures with superlinear space are based on the following new tool: given a non-self-crossing cycle C with c = O( √ n) nodes, we can preprocess G inÕ(n) time to produce a data structure of size O(n lg lg c) that can answer the following queries inÕ(c) time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and provides an alternative for a related data structure of Klein (SODA 2005), which reports distances to the boundary of a face, rather than a cycle.

Near-Optimal Compression for the Planar Graph Metric

“…Many exact distance oracles for planar graphs have been proposed [13,20,31,35,65,78], mostly focusing on the tradeoff between space and query time, showing that with space s the query time can be madeÕ(n/ √ s) [62].…”

Decrementai Transitive Closure and Shortest Paths for Planar Digraphs and Beyond

“…For the special case of planar graphs, we prove that there exists a decremental transitive closure algorithm with nearly-linear total update time and O( √ n) query time (Theorem 5.18). Consequently, using known techniques [28], we show a trade-off data structure with O(n 2 /t) total update time and O( √ t) query time (Theorem 6.6). Setting t = n 2/3 , we obtain a data structure with all operations running in amortized O(n 1/3 ) time, provided that all edges are eventually deleted.…”

“…Although our graphs are in fact of the form G * In this section we sketch how to obtain a decremental transitive closure algorithm for planar graphs with a faster query time, at a cost of a slower update time and larger space consumption. The method we use is due to Mozes and Sommer [28] who used it to develop the so-called cycle-MSSP data structure, which they subsequently leveraged to show a space-time trade-off for exact distance oracles for planar graphs. In the following, let t be a parameter.…”

Proxies for Shortest Path and Distance Queries

“…Both of them utilize the property of spatial coherence, i.e., spatial positions of both source and destination nodes and the shortest paths between them that facilitate the aggregation of source and destination nodes into groups sharing common nodes or edges on the shortest paths between them. (2) Road networks are also often assumed to be planar graphs with non-negetive weights, and the properties of planar graphs are further utilized to simplify the search process [12], [16], [20], [28]. Moreover, (3) a (spatial) hierarchical index structure is used by several techniques [14], [30], [41].…”