Maximum Number Of Disjoint Paths Research Articles

A formal approach to Menger's theorem

Menger's graph theorem equates the minimum size of a separating set for non-adjacent vertices a and b with the maximum number of disjoint paths between a and b. By capturing separating sets as models of an entailment relation, we take a formal approach to Menger's result. Upon showing that inconsistency is characterised by the existence of suficiently many disjoint paths, we recover Menger's theorem by way of completeness.

Research of the Secure Routing Method of Confidential Messages on Disjoint Paths

The article proposes a secure routing method of confidential messages in a telecommunication network by disjoint paths and its research results. The presented method of disjoint paths secure routing of confidential messages is based on the sequential solution of two optimization problems: calculating disjoint routes and secure balancing of confidential message fragments on a set of precalculated paths. An optimization model was chosen to determine the set of routes, namely, to calculate the maximum number of disjoint paths, including communication links with minimum compromise probability. Additionally, a model of fragmented transmission using a precalculated set of disjoint paths was selected according to the SPREAD mechanism and Shamir’s scheme to ensure the minimum possible probability of message compromise. The study of the proposed secure routing method on the selected network configuration for three different variants of the link compromise probabilities has been conducted. The study results confirmed the effectiveness of the proposed secure routing method and the efficiency of the underlying optimization models to determine both the set of disjoint paths and the order of secure balancing of confidential message fragments. Prospects for further research in the field of secure routing in telecommunication networks are related to considering the link compromise probabilities and other essential indicators of network security.

Model of secure routing based on determining the maximum number of disjoint paths to minimize the probability of compromise the confidential messages

Model of secure routing based on determining the maximum number of disjoint paths to minimize the probability of compromise the confidential messages

On the Kőnig‐Egerváry theorem for ‐paths

AbstractThe famous Kőnig‐Egerváry theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order equals the minimum order of a set of vertices intersecting all paths of order . For , we give complete structural descriptions of the graphs in . Furthermore, for odd , we give a complete structural description of the graphs in that contain no cycle of order less than . For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order and minimum sets of vertices intersecting all paths of order .

Organizing Multipath Routing in Cloud Computing Environments

One of the objectives of organizing cloud systems is to ensure effective access to remote resources by optimizing traffic engineering (TE) procedures. This paper considers the traffic engineering problem in a cloud environment by using a multipath routing technique. The multipath routing algorithm is used to identify the maximum number of disjoint paths in the graph which overcomes the problem in the junction area estimation process. So, the algorithm forms a plurality of non-overlapping and partially intersecting paths between any two nodes is proposed. Finally, the conditions for the formation of multipath virtual channels to ensure minimum build-time posts for the parallel transmission of its parts are also discussed.

OFDP: a distributed algorithm for finding disjoint paths with minimum total length in wireless sensor networks

This paper investigates the MINimum-length-$$k$$k-Disjoint-Paths (MIN-$$k$$k-DP) problem: in a sensor network, given two nodes $$s$$s and $$t$$t, a positive integer $$k$$k, finding $$k$$k (node) disjoint paths connecting $$s$$s and $$t$$t with minimum total length. An efficient distributed algorithm named Optimally-Finding-Disjoint-Paths (OFDP) is proposed for this problem. OFDP guarantees correctness and optimality, i.e., (1) it will find $$k$$k disjoint paths if there exist $$k$$k disjoint paths in the network or the maximum number of disjoint paths otherwise; (2) the disjoint paths it outputs do have minimum total length. To the best of our knowledge, OFDP is the first distributed algorithm that can solve the MIN-$$k$$k-DP problem with correctness and optimality guarantee. Compared with the existing centralized algorithms which also guarantee correctness and optimality, OFDP is shown to be much more efficient by simulation results.

DFDP: A Distributed Algorithm for Finding Disjoint Paths in Wireless Sensor Networks with Correctness Guarantee

In wireless sensor networks, routing messages through multiple (node) disjoint paths between two sensor nodes is a promising way to increase robustness, throughput, and load balance. This paper proposes an efficient distributed algorithm named distributedly finding disjoint paths (DFDP) to find k disjoint paths connecting two given nodes s and t. A set of paths connecting s and t are disjoint if any two of them do not have any common nodes except s and t. Unlike the existing distributed algorithms, DFDP guarantees correctness; that is, it will output k disjoint paths if there exist k disjoint paths in the network or the maximum number of disjoint paths otherwise. Compared with the centralized algorithms which also guarantee correctness, DFDP is shown to have much better efficiency and load balance by theory analysis and simulation results.

Efficient distributed algorithm for correctly finding disjoint paths in wireless sensor networks

Routing messages through multiple (node) disjoint paths connecting two sensor nodes is a promising way to increase robustness, throughput and load balance of sensor networks. This paper proposes a distributed algorithm to find k disjoint paths connecting two given nodes s and t. A set of paths connecting s and t are disjoint if any two of them do not have any common nodes besides s and t. Unlike the existing distributed algorithms, the proposed algorithm can guarantee answer correctness, i.e. it will output k disjoint paths if there exist k disjoint paths in the network or the maximum number of disjoint paths otherwise. Compared to the centralised algorithms which can also guarantee answer correctness, this algorithm is shown to have much lower communication cost, which is confirmed by experimental results. To the best of our knowledge, this algorithm is the first distributed algorithm which can guarantee answer correctness.

A Linear Time Algorithm for the Arc Disjoint Menger Problem in Planar Directed Graphs

Given a graph G=(V,E) and two vertices s,t ∈ V , s\neq t , the Menger problem is to find a maximum number of disjoint paths connecting s and t . Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or arc disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the arc disjoint version is Weihe's general maximum flow algorithm for planar networks [W1], which has running time \bf O (|V| log |V|) . Here we present a linear time, i.e., asymptotically optimal, algorithm for the arc disjoint version in planar directed graphs.

Efficient Algorithms for Finding the Maximum Number of Disjoint Paths in Grids

In a rectangular grid, given two sets of nodes, S (sources) and T (sinks), of size N2 each, the disjoint paths (DP) problem is to connect as many nodes in S to the nodes in T using a set of “disjoint” paths. (Both edge-disjoint and vertex-disjoint cases are considered in this paper.) Note that in this DP problem, a node in S can be connected to any node in T. Although in general the sizes of S and T do not have to be the same, algorithms presented in this paper can also find the maximum number of disjoint paths pairing nodes in S and T. We use the network flow approach to solve this DP problem. By exploiting all the properties of the network, such as planarity and regularity of a grid, integral flow, and unit capacity source/sink/flow, we can optimally compress the size of the working grid (to be defined) from O(N2) to O(N1.5) and solve the problem in O(N2.5) time for both the edge-disjoint and vertex-disjoint cases, an improvement over the straightforward approach which takes O(N3) time.

Maximum number of disjoint paths connecting specified terminals in a graph

Suppose G is a finite graph and let T ⊆ V ( G ) be a subset of its vertices throughout referred to as terminals . The problem MAXIMUM DISJOINT T -PATHS (MDT-P) is to find a family of maximum pairwise disjoint paths connecting pairs of terminals. MDT-P is a nontrivial extension of the maximum matching problem; the latter being equivalent to the special case T = V . We present a dual problem of minimum odd T-disconnector , an appropriate minimax theorem and an O( n 4 ) algorithm constructing optimal solutions for both primal and dual problems.

A generalization of Menger's theorem for certain unicyclic graphs

LetG be a unicyclic graph such that at most two points on its cycle have degrees ?3. Then the minimum number of points separating any independent setS of points inG is the maximum number of disjoint paths between the points of S.

A graph-theoretic characterization for the rank of the transfer matrix of a structured system

In this paper structured systems are considered and the generic rank of the transfer matrix of such systems is introduced. It is shown that this rank equals the maximum number of vertex disjoint paths from the input vertices to the output vertices in the graph that can be associated to the structured system. This maximum number of disjoint paths can be calculated using techniques from combinatorics. As an application a structural version of the well-known almost disturbance decoupling problem is proposed.

Disjoint path properties of the data manipulator network family☆

A critical component of any parallel processing system is the interconnection network that provides communications among the system's processors and memories. The data manipulator (gamma) network family is an important class of multistage interconnection networks that is being studied by various researchers. One interesting property of the data manipulator network family is the existence of multiple paths through the network for most source/ destination pairs. The condition that must be present to have disjoint paths through the network for a given source/ destination pair is shown, where disjoint paths are multiple paths with no links in common. It is derived that the maximum number of disjoint paths for any source/destination pair is two and a method for finding the routing tags that specify these paths is given. For source/destination pairs that have disjoint paths, a single fault cannot prevent communication between that source/ destination pair. The effect of a fault in a given stage of the network on the number of source/destination pairs that can be connected is also discussed. All results are proven mathematically.

Disjoint path properties of the data manipulator network family

A critical component of any parallel processing system is the interconnection network that provides communications among the system's processors and memories. The data manipulator (gamma) network family is an important class of multistage interconnection networks that is being studied by various researchers. One interesting property of the data manipulator network family is the existence of multiple paths through the network for most source/ destination pairs. The condition that must be present to have disjoint paths through the network for a given source/ destination pair is shown, where disjoint paths are multiple paths with no links in common. It is derived that the maximum number of disjoint paths for any source/destination pair is two and a method for finding the routing tags that specify these paths is given. For source/destination pairs that have disjoint paths, a single fault cannot prevent communication between that source/ destination pair. The effect of a fault in a given stage of the network on the number of source/destination pairs that can be connected is also discussed. All results are proven mathematically.

The complexity of finding maximum disjoint paths with length constraints

AbstractThe following problem is considered: Given an integer K, a graph G with two distinct vertices s and t, find the maximum number of disjoint paths of length K from s to t. The problem has several variants: the paths may be vertex‐disjoint or edge‐disjoint, the lengths of the paths may be equal to K or bounded by K, the graph may be undirected or directed. It is shown that except for small values of K all the problems are NP‐complete. Assuming P ≠ NP, for each problem, the largest value of K for which the problem is not NP‐complete is found. Whenever a polynomial algorithm exists, an efficient algorithm is described.

Graphs as models of communication network vulnerability: Connectivity and persistence

AbstractIt is well known that the maximum connectivity k of a graph G with p points and q lines is given by [2 q/p]. This is restated in two useful alternative forms which minimize q given p and k, and which maximize p in terms of q and k. We define the persistence of a graph as the smallest number of points whose removal increases the diameter. It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them. A similar result is obtained for line‐persistence and it is shown that these invariants are independent of each other.