Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Abstract

We present an O(n 1.5 )-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(n 5/3 )-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, n 2 ], we show an oracle of size S that answers queries inÕ(max{1, n 1.5 /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, n 5/3 ].

Near-Optimal Compression for the Planar Graph Metric

“…Indeed, in SODA'18 [41] the oracle was improved to O(n 3/2 ) space andÕ(1) query time and the tradeoff was improved to s space andÕ(n 3/2 /s) query time. To get even better tradeoffs we might allow a (1+ε) error [52,56,71]: we can achieve very small (1 + δ)n space and fastÕ(1) query time.…”

Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n5/3) Time

“…They show an oracle of sizeÕ(n 5/3 ), query time O(log n), and preprocessing time O(n 2 ). They also improve the stateof-the-art tradeoff between space and query time (see also [33]). Using Theorem 1.1 yields an optimal preprocessing time for the oracles in [23], matching their space requirements up to polylogarithmic factors.…”

Almost optimal distance oracles for planar graphs

“…Finally, we define VD * to be the graph obtained from VD * 1 after replacing the node h * by multiple copies, one for each occurrence of h as an endpoint of an edge in VD * 1 . It was shown in [22] that VD * is a forest, and that, if all vertices of h are sites and if the additive weights are such that each site is in its own nonempty Voronoi cell, then VD * is a ternary tree. See Fig.…”

“…Note that this includes the two trivial approaches mentioned above. Over the past three decades, many other works presented exact distance oracles for planar graphs with increasingly better space to query-time tradeoffs [1,6,11,12,15,20,22,37,39]. Figure 1 illustrates the advancements in the space/query-time tradeoffs over the years.…”

“…Cohen-Addad et al [12], inspired by Cabello's use of Voronoi diagrams for the diameter problem in planar graphs [7], provided the first such oracle. The currently best known tradeoff [22] is an oracle withÕ(n 3/2 /Q) space andÕ(Q) query-time for any Q ∈ [1, n 1/2 ]. Note that all known oracles with nearly linear (i.e.Õ(n)) space require Ω( √ n) time to answer queries.…”