Distance Oracles for Sparse Graphs

Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn^1+1/k) that can answer queries in time O(k) with a distance estimate that is at most ¿ = 2k-1 times larger than the actual shortest distance (this ratio is called the stretch).They proved that, under a combinatorial conjecture, their data structure is optimal in terms of space: if a stretch of at most 2k-1 is desired, then the space complexity is at least n^1+1/k. Their proof holds even if infinite query time is allowed: it is essentially an "incompressibility" result. Also, the proof only holds for dense graphs, and the best bound it can prove only implies that the size of the data structure is lower bounded by the number of edges of the graph. Naturally, the following question arises: what happens for sparse graphs? In this paper we give a new lower bound for approximate distance oracles in the cell-probe model. This lower bound holds even for sparse (polylog(n)-degree) graphs, and it is not an "incompressibility" bound: we prove a three-way tradeoff between space, stretch, and query time. We show that when the query time is t and the stretch is ¿, then the space S must be S ¿ n^1+¿(1/t¿)/lg n. This lower bound follows by a reduction from lopsided set disjointness to distance oracles, based on and motivated by recent work of Patrascu. Our results in fact show that for any high-girth regular graph, an approximate distance oracle that supports efficient queries for all subgraphs of G must obey this tradeoff. We also prove some lemmas that count sets of paths in high-girth regular graphs and high-girth regular expanders, which might be of independent interest.

1. Introduction

An approximate distance oracle is a data structure that answers distance queries for a (connected) graph . If the reported distance satisfies for all , the distance oracle is said to have multiplicative stretch . In all our lower bounds, the graphs are unweighted, ¹

As usual, denotes the number of nodes, denotes the number of edges of the graph . and is the big-Oh notation hiding poly-logarithmic factors. As usual, we abbreviate . All logarithms are base-2 unless explicitly stated otherwise. ln denotes lg_e, The distance between and in is the length, in edges, of the shortest path between and Three other definitions we use in the introduction are: the girth of , denoted by , is the length of the shortest cycle in is called -regular if each vertex has exactly neighbors. Two (or more) paths are called vertex-disjoint if they do not have any vertices in common.

.