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Distance oracles for sparse graphs
| Venue: | IN PROCEEDINGS OF THE 50TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS |
| Citations: | 26 - 4 self |
BibTeX
@INPROCEEDINGS{Sommer_distanceoracles,
author = {Christian Sommer and Elad Verbin and Wei Yu},
title = {Distance oracles for sparse graphs},
booktitle = {IN PROCEEDINGS OF THE 50TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS},
year = {},
pages = {703--712},
publisher = {}
}
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Abstract
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 α = 2k − 1 times larger than the actual shortest distance (α 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 α, then the space S must be S ≥ n 1+Ω(1/tα) / lg n. (1) This lower bound follows by a reduction from lopsided set disjointness to distance oracles, based on and motivated by recent work of Pǎtras¸cu. 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 Eq. (1). 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.
Keyphrases
data structure approximate distance oracle sparse graph distance oracle query time high-girth regular graph cellprobe model independent interest following question arises seminal work keywords-distance oracle lopsided set disjointness recent work dense graph efficient algorithm actual shortest distance cell-probe model count set answer distance query combinatorial conjecture distance estimate incompressibility result abstract thorup three-way tradeoff efficient query high-girth regular expanders fact show incompressibility bound set disjointness infinite query time space complexity
