Affiliation:
1. Max-Planck Institut fuer Informatik, Saarbruecken, Germany
2. Indian Institute of Technology Delhi, New Delhi, India
Abstract
Let
G
= (
V
,
E
) be an undirected graph on
n
vertices, and let δ(
u
,
v
) denote the distance in
G
between two vertices
u
and
v
. Thorup and Zwick showed that for any positive integer
t
, the graph
G
can be preprocessed to build a data structure that can efficiently report
t
-approximate distance between any pair of vertices. That is, for any
u
,
v
∈
V
, the distance reported is at least δ(
u
,
v
) and at most
t
δ(
u
,
v
). The remarkable feature of this data structure is that, for
t
≥3, it occupies subquadratic space, that is, it does not store all-pairs distances explicitly, and still it can answer any
t
-approximate distance query in constant time. They named the data structure “approximate distance oracle” because of this feature. Furthermore, the trade-off between the stretch
t
and the size of the data structure is essentially optimal.In this article, we show that we can actually construct approximate distance oracles in expected
O
(
n
2
) time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first expected linear-time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph. A (2, 1) spanner of an undirected unweighted graph
G
= (
V
,
E
) is a subgraph (
V
, Ê), Ê ⊆
E
, such that for any two vertices
u
and
v
in the graph, their distance in the subgraph is at most 2δ(
u
,
v
) + 1.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
58 articles.
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