76,517 research outputs found
Approximating the Diameter of Planar Graphs in Near Linear Time
We present a -approximation algorithm running in
time for finding the diameter of an undirected
planar graph with non-negative edge lengths
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
Consider a directed or an undirected graph with integral edge weights from
the set [-W, W], that does not contain negative weight cycles. In this paper,
we introduce a general framework for solving problems on such graphs using
matrix multiplication. The framework is based on the usage of Baur-Strassen's
theorem and of Strojohann's determinant algorithm. It allows us to give new and
simple solutions to the following problems:
* Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time
algorithm for finding shortest cycles in undirected and directed graphs. For
directed graphs (and undirected graphs with non-negative weights) this matches
the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the
other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously
known for undirected graphs with negative weights. Furthermore our algorithm
for a given directed or undirected graph detects whether it contains a negative
weight cycle within the same running time.
* Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega})
time algorithm for computing a diameter and radius of an undirected or directed
graphs. To the best of our knowledge no algorithm with this running time was
known for undirected graphs with negative weights.
* Finding Minimum Weight Perfect Matchings -- We present an
\tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect
matchings in undirected graphs. This resolves an open problem posted by
Sankowski in 2006, who presented such an algorithm but only in the case of
bipartite graphs.
In order to solve minimum weight perfect matching problem we develop a novel
combinatorial interpretation of the dual solution which sheds new light on this
problem. Such a combinatorial interpretation was not know previously, and is of
independent interest.Comment: To appear in FOCS 201
Complexity of diameter on AT-free graphs is linear
We develop a linear time algorithm for finding the diameter of an asteroidal
triple-free (AT-free) graph. Furthermore, we update the definition of polar
pairs and develop new properties of polar pairs for (weak) dominating pair
graphs. We prove that the problem of computing a simplicial vertex in a general
graph can be accomplished in O(n^2) based on an existing reduction to the
problem of finding diameter in an AT-free graph. We improve the best-known
run-time complexities of several graph theoretical problems
Algorithms for Finding Diameter Cycles of Biconnected Graphs
In this paper, we first coin a new graph theoretic problem called the diameter cycle problem with numerous applications. A longest cycle in a graph G = (V, E) is referred to as a diameter cycle of G iff the distance in G of every vertex on the cycle to the rest of the on-cycle vertices is maximal. We then present two algorithms for finding a diameter cycle of a biconnected graph. The first algorithm is an abstract intuitive algorithm that utilizes a brute-force mechanism for expanding an initial cycle by repeatedly replacing paths on the cycle with longer paths. The second algorithm is a concrete algorithm that uses fundamental cycles in the expansion process and has the time and space complexity of O(n^6) and O(n^2), respectively. To the best of our knowledge, this problem was neither defined nor addressed in the literature. The diameter cycle problem distinguishes itself from other cycle finding problems by identifying cycles that are maximally long while maximizing the distances between vertices in the cycle. Existing cycle finding algorithms such as fundamental and longest cycle algorithms do not discover cycles where the distances between vertices are maximized while also maximizing the length of the cycle
A Near-Optimal Algorithm for Finding an Optimal Shortcut of a Tree
We consider the problem of finding a shortcut connecting two vertices of a graph that minimizes the diameter of the resulting graph. We present an O(n^2 log^3 n)-time algorithm using linear space for the case that the input graph is a tree consisting of n vertices. Additionally, we present an O(n^2 log^3 n)-time algorithm using linear space for a continuous version of this problem
Fast Computation of Small Cuts via Cycle Space Sampling
We describe a new sampling-based method to determine cuts in an undirected
graph. For a graph (V, E), its cycle space is the family of all subsets of E
that have even degree at each vertex. We prove that with high probability,
sampling the cycle space identifies the cuts of a graph. This leads to simple
new linear-time sequential algorithms for finding all cut edges and cut pairs
(a set of 2 edges that form a cut) of a graph.
In the model of distributed computing in a graph G=(V, E) with O(log V)-bit
messages, our approach yields faster algorithms for several problems. The
diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain
simple O(Diam)-time distributed algorithms to find all cut edges,
2-edge-connected components, and cut pairs, matching or improving upon previous
time bounds. Under natural conditions these new algorithms are universally
optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain
a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this
is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A
simple extension of our work yields the first distributed algorithm with
sub-linear time for 3-edge-connected components. The basic distributed
algorithms are Monte Carlo, but they can be made Las Vegas without increasing
the asymptotic complexity.
In the model of parallel computing on the EREW PRAM our approach yields a
simple algorithm with optimal time complexity O(log V) for finding cut pairs
and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200
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