2 Disjoint Paths Problem in Oriented Mesh Network (original) (raw)

Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph

Journal of the ACM, 1978

^SSTnXcr. Gwen a graph G = (V, E) and four verttces s~, tx, s~, and t2, the problem of finding two disjoint paths, P~ from s~ to tt and P2 from s2 to t2, is considered This problem may arise as a transportation network problem and m printed clrcmts routing The relations between several vemons of the problem are discussed Efficient algorithms are gwen for the following special cases-acyche directed graphs and 3-connected planar and chordal graphs.

Edge-disjoint paths in planar graphs

Journal of Combinatorial Theory, Series B, 1985

Given a planar graph G = (V, E), find k edge-disjoint paths in G connecting k pairs of terminals specified on the outer face of G. Generalizing earlier results of Okamura and Seymour (J. Combin. Theory Ser. B 31 (1981) 75-81) and of the author (Combinatorics 2, No. 4 (1982) 361-371) we solve this problem when each node of G not on the outer face has even degree. The solution involves a good characterization for the solvability and the proof gives rise to an algorithm of complexity 0(1 F'1310gl I'(). In particular, the integral multicommodity flow problem is proved to belong to the problem class P when the underlying graph is outerplanar.

On the disjoint paths problem

Operations Research Letters, 2007

Using flow and matching algorithms to solve the problem of finding disjoint paths through a given node, and with a technique of Chekuri and Khanna, we give an O( √ n) approximation for the edge-disjoint paths problem in undirected graphs, directed acyclic graphs and directed graphs with edge capacity at least 2.

Constrained Disjoint Paths in Geometric Network

International Journal of Computational Intelligence and Applications, 2009

We address the problem of constructing a pair of node disjoint paths connecting two given nodes in a geometric network. We propose a simple algorithm for constructing such a path that does not use any complicated data structure which can be implemented very easily. We also consider two variations of disjoint path pair problems which we call narrow corridor problem and width bounded corridor problem and present efficient algorithms for solving them

CONSTRAINED DISJOINT PATHS IN GEOMETRIC NETWORKS

Finding Arc and Vertex-Disjoint Paths in Networks

2009 Eighth IEEE International Conference on Dependable, Autonomic and Secure Computing, 2009

Multipath Routing plays an important role in communication networks. Multiple disjoint paths can increase the effective bandwidth between pairs of vertices, avoid congestion in a network and reduce the probability of dropped packets. In this paper, we build mathematical models for arc-disjoint paths problem and vertex-disjoint paths problem respectively, and prove that they are both Linear Programming Problem. Then we propose polynomial algorithms for finding the shortest pair of arc and vertex-disjoint paths, both with the time complexity of () Om . Furthermore, we extend these algorithms to find any k disjoint paths, whose sum-weight is minimized in time () O km .

A software package of algorithms and heuristics for disjoint paths in Planar Networks

Discrete Applied Mathematics, 1999

We present a package for algorithms on planar networks. This package comes with a graphical user interface, which may be used for demonstrating and animating algorithms. Our focus so far has been on disjoint path problems. However, the package is intended to serve as a general framework, wherein algorithms for various problems on planar networks may be integrated and visualized. For this aim, the structure of the package is designed so that integration of new algorithms and even new algorithmic problems amounts to applying a short "recipe." The package has been used to develop new variations of well-known disjoint path algorithms, which heuristically optimize additional NP-hard objectives such as the total length of all paths. We will prove that the problem of ÿnding edge-disjoint paths of minimum total length in a planar graph is NP-hard, even if all terminals lie on the outer face, the Eulerian condition is fulÿlled, and the maximum degree is four. Finally, as a demonstration how PlaNet can be used as a tool for developing new heuristics for NP-hard problems, we will report on results of experimental studies on e cient heuristics for this problem. ? @uni-konstanz.d, WWW http:== www.informatik.uni-konstanz.de/ {brandes, wagner,weihe} (U. Brandes, D. Wagner, K. Weihe), neyer@inf.ethz.ch, WWW http:==www.inf.ethz.ch= personal=neyer (G. Neyer), schlicke@ifor.math.ethz.ch, WWW, http:==www.ifor.math.ethz.ch=sta =schlicke (W. Schlickenrieder).

Vertex Disjoint Paths in Upward Planar Graphs

Lecture Notes in Computer Science, 2014

The k-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed k when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter k) on this restricted class. Only recently, [13]. achieved a major breakthrough and answered the question positively. Despite the importance of this result (and the brilliance of their proof), it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has at least double exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller class of planar digraphs, the class of upward planar digraphs, a well studied class of planar graphs which can be drawn in a plane such that all edges are drawn upwards. We show that on the class of upward planar digraphs the problem (i) remains NP-complete and (ii) the problem is fixed-parameter tractable. While membership in FPT follows immediately from [13]'s general result, our algorithm has only single exponential parameter dependency compared to the double exponential parameter dependence for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in [13].

An Algorithm for Finding Two Node-Disjoint Paths in Arbitrary Graphs

Journal of Computing and Information Technology

Given two distinct vertices (nodes) source s and target t of a graph G = (V, E), the two node-disjoint paths problem is to identify two node-disjoint paths between s ∈ V and t ∈ V. Two paths are node-disjoint if they have no common intermediate vertices. In this paper, we present an algorithm with O(m)-time complexity for finding two node-disjoint paths between s and t in arbitrary graphs where m is the number of edges. The proposed algorithm has a wide range of applications in ensuring reliability and security of sensor, mobile and fixed communication networks.

A linear-time algorithm for edge-disjoint paths in planar graphs

Combinatorica, 1995

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the "classical" case where an instance additionally fulfills the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires O (nb/3(loglogn)l/3) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in an O(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.

2 Disjoint Paths Problem in Oriented Mesh Network (original) (raw)

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