Affiliation:
1. Ben-Gurion University of the Negev, Beer-Sheva, Israel
2. University of Michigan, Ann Arbor, MI, USA
Abstract
Thorup and Zwick [2001a] proposed a landmark distance oracle with the following properties. Given an
n
-vertex undirected graph
G
= (
V
,
E
) and a parameter
k
= 1, 2, …, their oracle has size
O
(
kn
1 + 1/
k
), and upon a query (
u
,
v
) it constructs a path Π between
u
and
v
of length δ(
u
,
v
) such that
d
G
(
u
,
v
) ⩽ δ(
u
,
v
) ⩽ (2
k
− 1)
d
G
(
u
,
v
). The query time of the oracle from Thorup and Zwick [2001a] is
O
(
k
) (in addition to the length of the returned path), and it was subsequently improved to
O
(1) [Wulff-Nilsen 2012; Chechik 2014]. A major drawback of the oracle of Thorup and Zwick [2001a] is that its space is Ω(
n
· log
n
). Mendel and Naor [2006] devised an oracle with space
O
(
n
1 + 1/
k
) and stretch
O
(
k
), but their oracle can only report distance estimates and not actual paths. In this article, we devise a path-reporting distance oracle with size
O
(
n
1 + 1/
k
), stretch
O
(
k
), and query time
O
(
n
ϵ
), for an arbitrarily small constant ϵ > 0. In particular, for
k
= log
n
, our oracle provides logarithmic stretch using linear size. Another variant of our oracle has size
O
(
n
loglog
n
), polylogarithmic stretch, and query time
O
(loglog
n
).
For unweighted graphs, we devise a distance oracle with multiplicative stretch
O
(1), additive stretch
O
(β(
k
)), for a function β(·), space
O
(
n
1 + 1/
k
), and query time
O
(
n
ϵ
), for an arbitrarily small constant ϵ > 0. The tradeoff between multiplicative stretch and size in these oracles is far below Erdős’s girth conjecture threshold (which is stretch 2
k
− 1 and size
O
(
n
1 + 1/
k
)). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch β(
k
) and size
O
(
n
1 + 1/
k
). A similar type of tradeoff was exhibited by a construction of (1 + ϵ, β)-spanners due to Elkin and Peleg [2001]. However, so far (1 + ϵ, β)-spanners had no counterpart in the distance oracles’ world.
An important novel tool that we develop on the way to these results is a distance-preserving path-reporting oracle. We believe that this oracle is of independent interest.
Funder
Danish National Research Foundation
NSF
Center for Massive Algorithms
Binational Science Foundation
Lynn and William Frankel Center for Computer Science
Israeli Academy of Science
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference40 articles.
1. Ittai Abraham and Cyril Gavoille. 2011. On approximate distance labels and routing schemes with affine stretch. In DISC. 404--415. Ittai Abraham and Cyril Gavoille. 2011. On approximate distance labels and routing schemes with affine stretch. In DISC. 404--415.
2. On space-stretch trade-offs
3. Ittai Abraham and Ofer Neiman. 2012. Using petal-decompositions to build a low stretch spanning tree. In STOC. 395--406. 10.1145/2213977.2214015 Ittai Abraham and Ofer Neiman. 2012. Using petal-decompositions to build a low stretch spanning tree. In STOC. 395--406. 10.1145/2213977.2214015
4. R. Agarwal. 2014a. Personal communication. (2014). R. Agarwal. 2014a. Personal communication. (2014).
Cited by
9 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献
1. Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n);2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS);2023-11-06
2. New Algorithms for All Pairs Approximate Shortest Paths;Proceedings of the 55th Annual ACM Symposium on Theory of Computing;2023-06-02
3. Approximate distance oracles with improved stretch for sparse graphs;Theoretical Computer Science;2023-01
4. Having Hope in Hops: New Spanners, Preservers and Lower Bounds for Hopsets;2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS);2022-10
5. Computing Constrained Shortest-Paths at Scale;Operations Research;2022-01