585,006 research outputs found
Structural factoring approach for analyzing stochastic networks
The problem of finding the distribution of the shortest path length through a stochastic network is investigated. A general algorithm for determining the exact distribution of the shortest path length is developed based on the concept of conditional factoring, in which a directed, stochastic network is decomposed into an equivalent set of smaller, generally less complex subnetworks. Several network constructs are identified and exploited to reduce significantly the computational effort required to solve a network problem relative to complete enumeration. This algorithm can be applied to two important classes of stochastic path problems: determining the critical path distribution for acyclic networks and the exact two-terminal reliability for probabilistic networks. Computational experience with the algorithm was encouraging and allowed the exact solution of networks that have been previously analyzed only by approximation techniques
Motion Planning for Unlabeled Discs with Optimality Guarantees
We study the problem of path planning for unlabeled (indistinguishable)
unit-disc robots in a planar environment cluttered with polygonal obstacles. We
introduce an algorithm which minimizes the total path length, i.e., the sum of
lengths of the individual paths. Our algorithm is guaranteed to find a solution
if one exists, or report that none exists otherwise. It runs in time
, where is the number of robots and is the total
complexity of the workspace. Moreover, the total length of the returned
solution is at most , where OPT is the optimal solution cost. To
the best of our knowledge this is the first algorithm for the problem that has
such guarantees. The algorithm has been implemented in an exact manner and we
present experimental results that attest to its efficiency
Finding k-Dissimilar Paths with Minimum Collective Length
Shortest path computation is a fundamental problem in road networks. However,
in many real-world scenarios, determining solely the shortest path is not
enough. In this paper, we study the problem of finding k-Dissimilar Paths with
Minimum Collective Length (kDPwML), which aims at computing a set of paths from
a source s to a target t such that all paths are pairwise dissimilar by at
least \theta and the sum of the path lengths is minimal. We introduce an exact
algorithm for the kDPwML problem, which iterates over all possible s-t paths
while employing two pruning techniques to reduce the prohibitively expensive
computational cost. To achieve scalability, we also define the much smaller set
of the simple single-via paths, and we adapt two algorithms for kDPwML queries
to iterate over this set. Our experimental analysis on real road networks shows
that iterating over all paths is impractical, while iterating over the set of
simple single-via paths can lead to scalable solutions with only a small
trade-off in the quality of the results.Comment: Extended version of the SIGSPATIAL'18 paper under the same titl
Finding detours is fixed-parameter tractable
We consider the following natural "above guarantee" parameterization of the
classical Longest Path problem: For given vertices s and t of a graph G, and an
integer k, the problem Longest Detour asks for an (s,t)-path in G that is at
least k longer than a shortest (s,t)-path. Using insights into structural graph
theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on
undirected graphs and actually even admits a single-exponential algorithm, that
is, one of running time exp(O(k)) poly(n). This matches (up to the base of the
exponential) the best algorithms for finding a path of length at least k.
Furthermore, we study the related problem Exact Detour that asks whether a
graph G contains an (s,t)-path that is exactly k longer than a shortest
(s,t)-path. For this problem, we obtain a randomized algorithm with running
time about 2.746^k, and a deterministic algorithm with running time about
6.745^k, showing that this problem is FPT as well. Our algorithms for Exact
Detour apply to both undirected and directed graphs.Comment: Extended abstract appears at ICALP 201
The edge-disjoint path problem on random graphs by message-passing
We present a message-passing algorithm to solve the edge disjoint path
problem (EDP) on graphs incorporating under a unique framework both traffic
optimization and path length minimization. The min-sum equations for this
problem present an exponential computational cost in the number of paths. To
overcome this obstacle we propose an efficient implementation by mapping the
equations onto a weighted combinatorial matching problem over an auxiliary
graph. We perform extensive numerical simulations on random graphs of various
types to test the performance both in terms of path length minimization and
maximization of the number of accommodated paths. In addition, we test the
performance on benchmark instances on various graphs by comparison with
state-of-the-art algorithms and results found in the literature. Our
message-passing algorithm always outperforms the others in terms of the number
of accommodated paths when considering non trivial instances (otherwise it
gives the same trivial results). Remarkably, the largest improvement in
performance with respect to the other methods employed is found in the case of
benchmarks with meshes, where the validity hypothesis behind message-passing is
expected to worsen. In these cases, even though the exact message-passing
equations do not converge, by introducing a reinforcement parameter to force
convergence towards a sub optimal solution, we were able to always outperform
the other algorithms with a peak of 27% performance improvement in terms of
accommodated paths. On random graphs, we numerically observe two separated
regimes: one in which all paths can be accommodated and one in which this is
not possible. We also investigate the behaviour of both the number of paths to
be accommodated and their minimum total length.Comment: 14 pages, 8 figure
The permutation-path coloring problem on trees
AbstractIn this paper we first show that the permutation-path coloring problem is NP-hard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and for circular permutations, which are permutations that contain exactly one cycle, on trees with maximum degree greater than or equal to 4. We calculate a lower bound on the average complexity of the permutation-path coloring problem on arbitrary networks. Then we give combinatorial and asymptotic results for the permutation-path coloring problem on linear networks in order to show that the average number of colors needed to color any permutation on a linear network on n vertices is n/4+o(n). We extend these results and obtain an upper bound on the average complexity of the permutation-path coloring problem on arbitrary trees, obtaining exact results in the case of generalized star trees. Finally we explain how to extend these results for the involutions-path coloring problem on arbitrary trees
Shortest Path Distance in Manhattan Poisson Line Cox Process
While the Euclidean distance characteristics of the Poisson line Cox process
(PLCP) have been investigated in the literature, the analytical
characterization of the path distances is still an open problem. In this paper,
we solve this problem for the stationary Manhattan Poisson line Cox process
(MPLCP), which is a variant of the PLCP. Specifically, we derive the exact
cumulative distribution function (CDF) for the length of the shortest path to
the nearest point of the MPLCP in the sense of path distance measured from two
reference points: (i) the typical intersection of the Manhattan Poisson line
process (MPLP), and (ii) the typical point of the MPLCP. We also discuss the
application of these results in infrastructure planning, wireless
communication, and transportation networks
- …