5,538 research outputs found
Exact Distance Oracles for Planar Graphs with Failing Vertices
We consider exact distance oracles for directed weighted planar graphs in the
presence of failing vertices. Given a source vertex , a target vertex
and a set of failed vertices, such an oracle returns the length of a
shortest -to- path that avoids all vertices in . We propose oracles
that can handle any number of failures. More specifically, for a directed
weighted planar graph with vertices, any constant , and for any , we propose an oracle of size
that answers queries in
time. In particular, we show an
-size, -query-time
oracle for any constant . This matches, up to polylogarithmic factors, the
fastest failure-free distance oracles with nearly linear space. For single
vertex failures (), our -size,
-query-time oracle improves over the previously best
known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for , . For multiple failures, no planarity exploiting
results were previously known
Exact Distance Oracles for Planar Graphs
We present new and improved data structures that answer exact node-to-node
distance queries in planar graphs. Such data structures are also known as
distance oracles. For any directed planar graph on n nodes with non-negative
lengths we obtain the following:
* Given a desired space allocation , we show how to
construct in time a data structure of size that answers
distance queries in time per query.
As a consequence, we obtain an improvement over the fastest algorithm for
k-many distances in planar graphs whenever .
* We provide a linear-space exact distance oracle for planar graphs with
query time for any constant eps>0. This is the first such data
structure with provable sublinear query time.
* For edge lengths at least one, we provide an exact distance oracle of space
such that for any pair of nodes at distance D the query time is
. Comparable query performance had been observed
experimentally but has never been explained theoretically.
Our data structures are based on the following new tool: given a
non-self-crossing cycle C with nodes, we can preprocess G in
time to produce a data structure of size that can
answer the following queries in time: for a query node u, output
the distance from u to all the nodes of C. This data structure builds on and
extends a related data structure of Klein (SODA'05), which reports distances to
the boundary of a face, rather than a cycle.
The best distance oracles for planar graphs until the current work are due to
Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For
and space , we essentially improve the query
time from to .Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on
Discrete Algorithms, SODA 201
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
- …