5,632 research outputs found
Rerouting shortest paths in planar graphs
A rerouting sequence is a sequence of shortest st-paths such that consecutive
paths differ in one vertex. We study the the Shortest Path Rerouting Problem,
which asks, given two shortest st-paths P and Q in a graph G, whether a
rerouting sequence exists from P to Q. This problem is PSPACE-hard in general,
but we show that it can be solved in polynomial time if G is planar. To this
end, we introduce a dynamic programming method for reconfiguration problems.Comment: submitte
Distance Preserving Graph Simplification
Large graphs are difficult to represent, visualize, and understand. In this
paper, we introduce "gate graph" - a new approach to perform graph
simplification. A gate graph provides a simplified topological view of the
original graph. Specifically, we construct a gate graph from a large graph so
that for any "non-local" vertex pair (distance higher than some threshold) in
the original graph, their shortest-path distance can be recovered by
consecutive "local" walks through the gate vertices in the gate graph. We
perform a theoretical investigation on the gate-vertex set discovery problem.
We characterize its computational complexity and reveal the upper bound of
minimum gate-vertex set using VC-dimension theory. We propose an efficient
mining algorithm to discover a gate-vertex set with guaranteed logarithmic
bound. We further present a fast technique for pruning redundant edges in a
gate graph. The detailed experimental results using both real and synthetic
graphs demonstrate the effectiveness and efficiency of our approach.Comment: A short version of this paper will be published for ICDM'11, December
201
Dual Failure Resilient BFS Structure
We study {\em breadth-first search (BFS)} spanning trees, and address the
problem of designing a sparse {\em fault-tolerant} BFS structure, or {\em
FT-BFS } for short, resilient to the failure of up to two edges in the given
undirected unweighted graph , i.e., a sparse subgraph of such that
subsequent to the failure of up to two edges, the surviving part of
still contains a BFS spanning tree for (the surviving part of) . FT-BFS
structures, as well as the related notion of replacement paths, have been
studied so far for the restricted case of a single failure. It has been noted
widely that when concerning shortest-paths in a variety of contexts, there is a
sharp qualitative difference between a single failure and two or more failures.
Our main results are as follows. We present an algorithm that for every
-vertex unweighted undirected graph and source node constructs a
(two edge failure) FT-BFS structure rooted at with edges. To
provide a useful theory of shortest paths avoiding 2 edges failures, we take a
principled approach to classifying the arrangement these paths. We believe that
the structural analysis provided in this paper may decrease the barrier for
understanding the general case of faults and pave the way to the
future design of -fault resilient structures for . We also provide
a matching lower bound, which in fact holds for the general case of
and multiple sources . It shows that for every , and
integer , there exist -vertex graphs with a source set
of cardinality for which any FT-BFS structure rooted
at each , resilient to up to -edge faults has
edges
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