6,512 research outputs found
Approximating Approximate Distance Oracles
Given a finite metric space (V,d), an approximate distance oracle is a data structure which, when queried on two points u,v in V, returns an approximation to the the actual distance between u and v which is within some bounded stretch factor of the true distance. There has been significant work on the tradeoff between the important parameters of approximate distance oracles (and in particular between the size, stretch, and query time), but in this paper we take a different point of view, that of per-instance optimization. If we are given an particular input metric space and stretch bound, can we find the smallest possible approximate distance oracle for that particular input? Since this question is not even well-defined, we restrict our attention to well-known classes of approximate distance oracles, and study whether we can optimize over those classes.
In particular, we give an O(log n)-approximation to the problem of finding the smallest stretch 3 Thorup-Zwick distance oracle, as well as the problem of finding the smallest Pv{a}trac{s}cu-Roditty distance oracle. We also prove a matching Omega(log n) lower bound for both problems, and an Omega(n^{frac{1}{k}-frac{1}{2^{k-1}}}) integrality gap for the more general stretch (2k-1) Thorup-Zwick distance oracle. We also consider the problem of approximating the best TZ or PR approximate distance oracle with outliers, and show that more advanced techniques (SDP relaxations in particular) allow us to optimize even in the presence of outliers
Approximate Distance Oracles for Planar Graphs with Improved Query Time-Space Tradeoff
We consider approximate distance oracles for edge-weighted n-vertex
undirected planar graphs. Given fixed epsilon > 0, we present a
(1+epsilon)-approximate distance oracle with O(n(loglog n)^2) space and
O((loglog n)^3) query time. This improves the previous best product of query
time and space of the oracles of Thorup (FOCS 2001, J. ACM 2004) and Klein
(SODA 2002) from O(n log n) to O(n(loglog n)^5).Comment: 20 pages, 9 figures of which 2 illustrate pseudo-code. This is the
SODA 2016 version but with the definition of C_i in Phase I fixed and the
analysis slightly modified accordingly. The main change is in the subsection
bounding query time and stretch for Phase
Fast and Compact Exact Distance Oracle for Planar Graphs
For a given a graph, a distance oracle is a data structure that answers
distance queries between pairs of vertices. We introduce an -space
distance oracle which answers exact distance queries in time for
-vertex planar edge-weighted digraphs. All previous distance oracles for
planar graphs with truly subquadratic space i.e., space
for some constant ) either required query time polynomial in
or could only answer approximate distance queries.
Furthermore, we show how to trade-off time and space: for any , we show how to obtain an -space distance oracle that answers
queries in time . This is a polynomial
improvement over the previous planar distance oracles with query
time
Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs
Let be a graph where each vertex is associated with a label. A
Vertex-Labeled Approximate Distance Oracle is a data structure that, given a
vertex and a label , returns a -approximation of
the distance from to the closest vertex with label in . Such
an oracle is dynamic if it also supports label changes. In this paper we
present three different dynamic approximate vertex-labeled distance oracles for
planar graphs, all with polylogarithmic query and update times, and nearly
linear space requirements
Approximate Distance Sensitivity Oracles in Subquadratic Space
An -edge fault-tolerant distance sensitive oracle (-DSO) with stretch
is a data structure that preprocesses a given undirected,
unweighted graph with vertices and edges, and a positive integer
. When queried with a pair of vertices and a set of at most
edges, it returns a -approximation of the --distance in . We
study -DSOs that take subquadratic space. Thorup and Zwick [JACM 2015]
showed that this is only possible for . We present, for any
constant and , and any , an -DSO with stretch that takes
space and has an query time.
We also give an improved construction for graphs with diameter at most . For
any constant , we devise an -DSO with stretch that takes
space and has query time,
with a preprocessing time of . Chechik, Cohen, Fiat,
and Kaplan [SODA 2017] presented an -DSO with stretch and
preprocessing time , albeit with a super-quadratic
space requirement. We show how to reduce their preprocessing time to
.Comment: accepted at STOC 202
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