Exact Distance Oracles for Planar Graphs with Failing Vertices

TL;DR

This paper develops exact distance oracles for directed weighted planar graphs that efficiently handle multiple vertex failures, offering new tradeoffs between space and query time, and improving upon previous results especially for single failures.

Contribution

It introduces novel exact distance oracles for planar graphs capable of handling any number of vertex failures with improved space-query tradeoffs, including for multiple failures.

Findings

Achieves $ ilde{O}(n)$ space and $ ilde{O}( oot n)$ query time for constant $k$

Provides a space vs. query time tradeoff for any $q ange [1, oot n]$

Improves previous results for single vertex failures with polynomial factor improvements

Abstract

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex $u$, a target vertex $v$ and a set $X$ of $k$ failed vertices, such an oracle returns the length of a shortest $u$-to-$v$ path that avoids all vertices in $X$. We propose oracles that can handle any number $k$ of failures. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with $n$ vertices and any constant $k$, we show an $\tilde{\mathcal{O}}(n)$-size, $\tilde{\mathcal{O}}(\sqrt{n})$-query-time oracle. We then present a space vs. query time tradeoff: for any $q \in \lbrack 1,\sqrt n \rbrack$, we propose an oracle of size $n^{k+1+o(1)}/q^{2k}$ that answers queries in $\tilde{\mathcal{O}}(q)$ time. For single vertex failures ($k=1$), our $n^{2+o(1)}/q^2$-size, $\tilde{\mathcal{O}}(q)$-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for $q \geq n^t$, for any $t \in (0,1/2]$. For multiple failures, no planarity exploiting results were previously known. A preliminary version of this work was presented in SODA 2019. In this version, we show improved space vs. query time tradeoffs relying on the recently proposed almost optimal distance oracles for planar graphs [Charalampopoulos et al., STOC 2019; Long and Pettie, SODA 2021].