191 research outputs found
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
The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree
In this paper we study the Steiner tree problem over a dynamic set of
terminals. We consider the model where we are given an -vertex graph
with positive real edge weights, and our goal is to maintain a tree
which is a good approximation of the minimum Steiner tree spanning a terminal
set , which changes over time. The changes applied to the
terminal set are either terminal additions (incremental scenario), terminal
removals (decremental scenario), or both (fully dynamic scenario). Our task
here is twofold. We want to support updates in sublinear time, and keep
the approximation factor of the algorithm as small as possible. We show that we
can maintain a -approximate Steiner tree of a general graph in
time per terminal addition or removal. Here,
denotes the stretch of the metric induced by . For planar graphs we achieve
the same running time and the approximation ratio of .
Moreover, we show faster algorithms for incremental and decremental scenarios.
Finally, we show that if we allow higher approximation ratio, even more
efficient algorithms are possible. In particular we show a polylogarithmic time
-approximate algorithm for planar graphs.
One of the main building blocks of our algorithms are dynamic distance
oracles for vertex-labeled graphs, which are of independent interest. We also
improve and use the online algorithms for the Steiner tree problem.Comment: Full version of the paper accepted to STOC'1
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
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
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
Prioritized Metric Structures and Embedding
Metric data structures (distance oracles, distance labeling schemes, routing
schemes) and low-distortion embeddings provide a powerful algorithmic
methodology, which has been successfully applied for approximation algorithms
\cite{llr}, online algorithms \cite{BBMN11}, distributed algorithms
\cite{KKMPT12} and for computing sparsifiers \cite{ST04}. However, this
methodology appears to have a limitation: the worst-case performance inherently
depends on the cardinality of the metric, and one could not specify in advance
which vertices/points should enjoy a better service (i.e., stretch/distortion,
label size/dimension) than that given by the worst-case guarantee.
In this paper we alleviate this limitation by devising a suit of {\em
prioritized} metric data structures and embeddings. We show that given a
priority ranking of the graph vertices (respectively,
metric points) one can devise a metric data structure (respectively, embedding)
in which the stretch (resp., distortion) incurred by any pair containing a
vertex will depend on the rank of the vertex. We also show that other
important parameters, such as the label size and (in some sense) the dimension,
may depend only on . In some of our metric data structures (resp.,
embeddings) we achieve both prioritized stretch (resp., distortion) and label
size (resp., dimension) {\em simultaneously}. The worst-case performance of our
metric data structures and embeddings is typically asymptotically no worse than
of their non-prioritized counterparts.Comment: To appear at STOC 201
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