20,877 research outputs found
Finding Simple Shortest Paths and Cycles
The problem of finding multiple simple shortest paths in a weighted directed
graph has many applications, and is considerably more difficult than
the corresponding problem when cycles are allowed in the paths. Even for a
single source-sink pair, it is known that two simple shortest paths cannot be
found in time polynomially smaller than (where ) unless the
All-Pairs Shortest Paths problem can be solved in a similar time bound. The
latter is a well-known open problem in algorithm design. We consider the
all-pairs version of the problem, and we give a new algorithm to find
simple shortest paths for all pairs of vertices. For , our algorithm runs
in time (where ), which is almost the same bound as
for the single pair case, and for we improve earlier bounds. Our approach
is based on forming suitable path extensions to find simple shortest paths;
this method is different from the `detour finding' technique used in most of
the prior work on simple shortest paths, replacement paths, and distance
sensitivity oracles.
Enumerating simple cycles is a well-studied classical problem. We present new
algorithms for generating simple cycles and simple paths in in
non-decreasing order of their weights; the algorithm for generating simple
paths is much faster, and uses another variant of path extensions. We also give
hardness results for sparse graphs, relative to the complexity of computing a
minimum weight cycle in a graph, for several variants of problems related to
finding simple paths and cycles.Comment: The current version includes new results for undirected graphs. In
Section 4, the notion of an (m,n) reduction is generalized to an f(m,n)
reductio
A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph
We present an algorithm for the k shortest simple path problem on weighted
directed graphs (kSSP) that is based on Eppstein's algorithm for a similar
problem in which paths are allowed to contain cycles. In contrast to most other
algorithms for kSSP, ours is not based on Yen's algorithm and does not solve
replacement path problems. Its worst-case running time is on par with
state-of-the-art algorithms for kSSP. Using our algorithm, one may find O(m)
simple paths with a single shortest path tree computation and O(n + m)
additional time per path in well-behaved cases, where n is the number of nodes
and m is the number of edges. Our computational results show that on random
graphs and large road networks, these well-behaved cases are quite common and
our algorithm is faster than existing algorithms by an order of magnitude.
Further, the running time is far better predictable due to very small
dispersion
Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model
Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of
minimum weight in a graph . This is a fundamental graph problem with
classical sequential algorithms that run in and
time where and . In recent years this problem
has received significant attention in the context of hardness through fine
grained sequential complexity as well as in design of faster sequential
approximation algorithms.
For computing minimum weight cycle in the distributed CONGEST model,
near-linear in lower and upper bounds on round complexity are known for
directed graphs (weighted and unweighted), and for undirected weighted graphs;
these lower bounds also apply to any -approximation algorithm.
This paper focuses on round complexity bounds for approximating MWC in the
CONGEST model: For coarse approximations we show that for any constant , computing an -approximation of MWC requires rounds on weighted undirected graphs and on directed graphs, even
if unweighted. We complement these lower bounds with sublinear
-round algorithms for approximating MWC close to a factor
of 2 in these classes of graphs.
A key ingredient of our approximation algorithms is an efficient algorithm
for computing -approximate shortest paths from sources in
directed and weighted graphs, which may be of independent interest for other
CONGEST problems. We present an algorithm that runs in rounds if and rounds if , and this round
complexity smoothly interpolates between the best known upper bounds for
approximate (or exact) SSSP when and APSP when
Efficiently listing bounded length st-paths
The problem of listing the shortest simple (loopless) -paths in a
graph has been studied since the early 1960s. For a non-negatively weighted
graph with vertices and edges, the most efficient solution is an
algorithm for directed graphs by Yen and Lawler
[Management Science, 1971 and 1972], and an algorithm for
the undirected version by Katoh et al. [Networks, 1982], both using
space. In this work, we consider a different parameterization for this problem:
instead of bounding the number of -paths output, we bound their length. For
the bounded length parameterization, we propose new non-trivial algorithms
matching the time complexity of the classic algorithms but using only
space. Moreover, we provide a unified framework such that the solutions to both
parameterizations -- the classic -shortest and the new length-bounded paths
-- can be seen as two different traversals of a same tree, a Dijkstra-like and
a DFS-like traversal, respectively.Comment: 12 pages, accepted to IWOCA 201
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
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