988 research outputs found
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Space and Time Efficient Parallel Graph Decomposition, Clustering, and Diameter Approximation
We develop a novel parallel decomposition strategy for unweighted, undirected
graphs, based on growing disjoint connected clusters from batches of centers
progressively selected from yet uncovered nodes. With respect to similar
previous decompositions, our strategy exercises a tighter control on both the
number of clusters and their maximum radius.
We present two important applications of our parallel graph decomposition:
(1) -center clustering approximation; and (2) diameter approximation. In
both cases, we obtain algorithms which feature a polylogarithmic approximation
factor and are amenable to a distributed implementation that is geared for
massive (long-diameter) graphs. The total space needed for the computation is
linear in the problem size, and the parallel depth is substantially sublinear
in the diameter for graphs with low doubling dimension. To the best of our
knowledge, ours are the first parallel approximations for these problems which
achieve sub-diameter parallel time, for a relevant class of graphs, using only
linear space. Besides the theoretical guarantees, our algorithms allow for a
very simple implementation on clustered architectures: we report on extensive
experiments which demonstrate their effectiveness and efficiency on large
graphs as compared to alternative known approaches.Comment: 14 page
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
Treewidth distance on phylogenetic trees
In this article we study the treewidth of the display graph, an auxiliary graph structure obtained from the fusion of phylogenetic (i.e., evolutionary) trees at their leaves. Earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar. Here we further expand upon this relationship. We analyse a number of reduction rules, commonly used in the phylogenetics literature to obtain fixed parameter tractable algorithms. In some cases (the subtree reduction) the reduction rules behave similarly with respect to treewidth, while others (the cluster reduction) behave very differently, and the behaviour of the chain reduction is particularly intriguing because of its link with graph separators and forbidden minors. We also show that the gap between treewidth and Tree Bisection and Reconnect (TBR) distance can be infinitely large, and that unlike, for example, planar graphs the treewidth of the display graph can be as much as linear in its number of vertices. A number of other auxiliary results are given. We conclude with a discussion and list a number of open problems
Shorter Labeling Schemes for Planar Graphs
An \emph{adjacency labeling scheme} for a given class of graphs is an algorithm that for every graph from the class, assigns bit strings (labels) to vertices of so that for any two vertices , whether and are adjacent can be determined by a fixed procedure that examines only their labels. It is known that planar graphs with vertices admit a labeling scheme with labels of bit length . In this work we improve this bound by designing a labeling scheme with labels of bit length . In graph-theoretical terms, this implies an explicit construction of a graph on vertices that contains all planar graphs on vertices as induced subgraphs, improving the previous best upper bound of . Our scheme generalizes to graphs of bounded Euler genus with the same label length up to a second-order term. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time
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