Pair Of Disjoint Paths Research Articles

Enhanced FRER Mechanism in Time-Sensitive Networking for Reliable Edge Computing.

Time-Sensitive Networking (TSN) and edge computing are promising networking technologies for the future of the Industrial Internet. TSN provides a reliable and deterministic low-latency communication service for edge computing. The Frame Replication and Elimination for Reliability (FRER) mechanism is important for improving the network reliability of TSN. It achieves high reliability by transmitting identical frames in parallel on two disjoint paths, while eliminating duplicated frames at the destination node. However, there are two problems with the FRER mechanism. One problem is that it does not consider the path reliability, and the other one is that it is difficult to find two completely disjoint path pairs in some cases. To solve the above problems, this paper proposes a method to find edge-disjoint path pairs considering path reliability for FRER in TSN. The method includes two parts: one is building a reliability model for paths, and the other one is computing a working path and a redundant path with the Edge-Disjoint Path Pairs Selection (EDPPS) algorithm. Theoretical and simulation results show that the proposed method effectively improves path reliability while reducing the delay jitter of frames. Compared with the traditional FRER mechanism, the proposed method reduces delay jitter by 15.6% when the network load is 0.9.

Minimum Cost Survivable Routing Algorithms for Generalized Diversity Coding

Generalized diversity coding is a promising proactive recovery scheme against single edge failures for unicast connections in transport networks. At the source node, the user data is split into two parts, and their bitwise XOR is computed as a third redundancy sub-flow. In order to guarantee instantaneous failure recovery without costly node upgrades, the network must ensure that any two of the three sub-flows reach the destination node in case of a single edge failure only by allowing flow duplication or merging identical flows, and avoiding any coding operation in the core network. In this paper, we investigate the corresponding routing problem to calculate capacity-efficient routes for these sub-flows. We propose a polynomial-time algorithm for topologies without capacity constraints on the links and without capability limitations of the nodes. We show that with node limitations the presented algorithm (as well as a minimum cost disjoint path-pair) provides a 4/3-approximation for the routing problem. Furthermore, we formulate an integer linear program to provide a minimum cost solution with arbitrary constraints in general graphs and we propose a polynomial-time algorithm in directed acyclic graphs. Our simulation results suggest that with upgrading only a small set of core network nodes with flow duplication and merging capabilities most of the benefits of generalized diversity coding can be achieved.

An efficient Manhattan‐distance‐constrained disjoint paths algorithm for incomplete mesh network

SummaryFinding link/node‐disjoint paths between a pair of nodes is capable of providing Quality of Service and reliable routing which is very critical for mesh‐connected Network‐on‐Chip. State‐of‐art works usually aim at random topologies and multiple constraints. Therefore, it is difficult to optimize their time complexity. In this paper, MDPPIM (Manhattan‐distance‐constrained Disjoint Path Pair Problem in Incomplete Mesh) problem is presented based on Network‐on‐Chip application scenarios. Then, PCDP (Path‐Counting Disjoint Path) algorithm is proposed to solve the MDPPIM problem with low time complexity. Compared with previous disjoint path algorithms, PCDP algorithm does not have to use Dijkstra's algorithm to find a shortest path. It is optimized according to the feature of incomplete mesh such as the regularity of mesh and Manhattan‐distance constraint. Therefore, it is with low time complexity. Numerical results demonstrate the proposed PCDP algorithm's effectiveness and low time complexity.

Minimum-Weight Link-Disjoint Node-“Somewhat Disjoint” Paths

Network survivability has been recognized as an issue of major importance in terms of security, stability and prosperity. A crucial research problem in this context is the identification of suitable pairs of disjoint paths. Here, “disjointness” can be considered in terms of either nodes or links. Accordingly, several studies have focused on finding pairs of either link or node disjoint paths with a minimum sum of link weights. In this paper, we investigate the gap between the optimal node-disjoint and link-disjoint solutions. Specifically, we formalize several optimization problems that aim at finding minimum-weight link-disjoint paths while restricting the number of its common nodes. We establish that some of these variants are computationally intractable, while for other variants we establish polynomial-time algorithmic solutions. Finally, through extensive simulations, we show that, by allowing link-disjoint paths share a few common nodes, a major improvement is obtained in terms of the quality (i.e., total weight) of the solution.

Efficient heuristics for determining node‐disjoint path pairs visiting specified nodes

A new recursive heuristic is proposed to calculate a shortest simple path, from a source node to a destination node, that visits a specified set of nodes in a network. To provide survivability to failures along the path, the proposed heuristic is modified to ensure that the calculated path can be protected by a node‐disjoint backup path. Additionally, the case when both paths in the disjoint path pair are required to visit specific sets of nodes is studied and effective heuristics are proposed. An evaluation of the solutions of the heuristics is conducted by comparing with results from an integer linear programming (ILP) formulation for each of the considered problems, and also with previous heuristics. The ILP solver may require a significant amount of time to obtain a solution, especially in large networks, which justifies the need for effective, computationally efficient heuristics for solving these problems. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(4), 292–307 2017

A network-wide exact optimization approach for multiobjective routing with path protection in multiservice multiprotocol label switching networks

ABSTRACTA multiobjective routing model for multiprotocol label switching networks with multiple service types and path protection is presented in this article. The routing problem is formulated as a biobjective integer program, where the considered objectives are formulated according to a network-wide optimization approach, i.e. the objective functions of the route optimization problem depend explicitly on all traffic flows in the network. A disjoint path pair is considered for each traffic trunk, which guarantees protection to the associated connection. A link-path formulation is proposed for the problem, in which a set of possible pairs of paths is devised in advance for each traffic trunk. An exact method (based on the classical constraint method for solving multiobjective problems) is developed for solving the formulated problem. An extensive experimental study, with results on network performance measures in various randomly generated networks, is also presented and discussed.

Diversity coding-based survivable routing with QoS and differential delay bounds

Survivable routing with instantaneous recovery gained much attention in the last decade, as in optical backbone networks even the shortest disruption of a connection may cause tremendous loss of data. Recently, strict delay requirements emerge with the growing volume of multimedia and video streaming applications, which have to be ensured both before and after a failure. Diversity coding provides a nice trade-off between the simplicity of dedicated protection and bandwidth-efficiency of network coding to ensure instantaneous recovery for the connections. Hence, in this paper we thoroughly investigate the optimal structure of diversity coding-based survivable routing, which has a well-defined acyclic structure of subsequent paths and disjoint path-pairs between the communication end-points. We define the delay of these directed acyclic graphs, and investigate the effect of Quality-of-Service and differential delay bounds on the solution cost. Complexity analysis and integer linear programs are provided to solve these delay aware survivable routing problems. We discuss their approximability and provide some heuristic algorithms, too. Thorough experiments are conducted to demonstrate the benefits of diversity coding on randomly generated and real-world optical topologies.

Algorithms for determining a node-disjoint path pair visiting specified nodes

The problem of calculating the shortest path that visits a given set of nodes is at least as difficult as the traveling salesman problem, and it has not received much attention. Nevertheless an efficient integer linear programming (ILP) formulation has been recently proposed for this problem. That ILP formulation is firstly adapted to include the constraint that the obtained path can be protected by a node-disjoint path, and secondly to obtain a pair of node disjoint paths, of minimal total additive cost, each having to visit a given set of specified nodes. Computational experiments show that these approaches, namely in large networks, may fail to obtain solutions in a reasonable amount of time. Therefore heuristics are proposed for solving those problems, that may arise due to network management constraints. Extensive computational results show that they are capable of finding a solution in most cases, and that the calculated solutions present an acceptable relative error regarding the cost of the obtained path or pair of paths. Further the CPU time required by the heuristics is significantly smaller than the required by the used ILP solver.

Combinatorial optimization in networks with Shared Risk Link Groups

The notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In this paper, we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles.

Maximally node and SRLG-disjoint path pair of min-sum cost in GMPLS networks: a lexicographic approach

Path protection is a fast and capacity-efficient approach for increasing the availability of end-to-end connections. However, sometimes it is not possible to obtain a fully disjoint path pair. In this case, it may be admissible to consider a path pair which is as disjoint as possible, and thus provide the best (in a certain sense) level of the single-fault protection that can be ensured using this type of approach. A shared risk link group (SRLG) is a group of links which have a common risk of failure. Two new heuristics for solving the min-sum maximally node and SRLG-disjoint path pair are presented. The relative performance of the new heuristics and also of two other previously proposed heuristics is evaluated using four different networks. Results, regarding accuracy and execution time of the studied heuristics, show that one of the new proposed algorithms can be a good compromise for use in the Generalized Multi-protocol Label Switching control plane.

Survivable Path Sets: A New Approach to Survivability in Multilayer Networks

We consider the problem of survivability in multilayer networks. In single-layer networks, a pair of disjoint paths can be used to provide protection for a source–destination pair. However, this approach cannot be directly applied to layered networks where risk-disjoint paths may not always exist. In this paper, we take a new approach, which is based on finding a set of paths that may not be disjoint but together will survive any single risk. We start with two-layered communication networks, where the risks are fiber failures. We prove that in general, finding the minimum survivable path set (MSPS) is NP-hard, whereas if we restrict the length of paths the problem can be solved in polynomial time. We formulate the problem as an integer linear program (ILP), and use this formulation to develop heuristics and approximation algorithms. Moreover, we study the minimum cost survivable path set problem, where the cost is the number of fibers used, and thus, nonadditive. Finally, we generalize the survivability problem to the networks with more than two layers. By applying our algorithms for survivable path set, we assess the survivability of communication networks that operate relying on power from a power grid.

Survivable Inter-Domain Routing Based on Topology Aggregation With Intra-Domain Disjointness Information in Multi-Domain Optical Networks

In a network in which multiple domains are defined due to geographical and/or administrative reasons, only a limited amount of domain information is exchanged by domain service providers. Topology aggregation is a method used to facilitate this limited information exchange. The amount of information provided for each domain may vary based on the technical and management decisions taken by the service provider. For instance, some domains may choose to provide only a single shortest path between two border nodes, while another may be able to provide a pair of disjoint paths with minimum total cost. In such cases, end-to-end protected path routing needs to facilitate and use different amounts of domain information provided by domain service providers in order to find the best solution. In this work, we propose several approaches that help find a pair of disjoint end-to-end paths that may traverse multiple domains from source to destination and result in minimum total cost. These approaches include methods for inter-domain information exchange that carry costs of disjoint paths within a domain. The performance of minimizing the total cost of a pair of end-to-end paths is investigated. Finally, the blocking probabilities of these various approaches due to the existence of trap topologies in the network are also discussed.

Survivable Inter-Domain Routing Based on Topology Aggregation With Intra-Domain Disjointness Information in Multi-Domain Optical Networks

In a network in which multiple domains are defined due to geographical and/or administrative reasons, only a limited amount of domain information is exchanged by domain service providers. Topology aggregation is a method used to facilitate this limited information exchange. The amount of information provided for each domain may vary based on the technical and management decisions taken by the service provider. For instance, some domains may choose to provide only a single shortest path between two border nodes, while another may be able to provide a pair of disjoint paths with minimum total cost. In such cases, end-to-end protected path routing needs to facilitate and use different amounts of domain information provided by domain service providers in order to find the best solution. In this work, we propose several approaches that help find a pair of disjoint end-to-end paths that may traverse multiple domains from source to destination and result in minimum total cost. These approaches include methods for inter-domain information exchange that carry costs of disjoint paths within a domain. The performance of minimizing the total cost of a pair of end-to-end paths is investigated. Finally, the blocking probabilities of these various approaches due to the existence of trap topologies in the network are also discussed.

On Finding Min-Min Disjoint Paths

The Min-Min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-hard and admits no K-approximation for any K>1 in the general case (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006). In this paper, we first show that Bhatia et al.’s NP-hardness proof (Bhatia et al. in J. Comb. Optim. 12:83–96, 2006), a claim of correction to Xu et al.’s proof (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006), for the edge-disjoint Min-Min problem in the general undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in the proof of Bhatia et al. (J. Comb. Optim. 12:83–96, 2006). We then gave a correct proof of NP-hardness of this problem in undirected graphs. Finally we give a polynomial-time algorithm for the vertex disjoint Min-Min problem in planar graphs by showing that the vertex disjoint Min-Min problem is polynomially solvable in st-planar graph G=(V,E) whose corresponding auxiliary graph G(V,E∪{e(st)}) can be embedded into a plane, and a planar graph can be decomposed into several st-planar graphs whose Min-Min paths collectively contain a Min-Min disjoint-path pair between s and t in the original graph G. To the best of our knowledge, these are the first polynomial algorithms for the Min-Min problems in planar graphs.

On the complexity of the edge-disjoint min–min problem in planar digraphs

The min–min problem of finding a disjoint path pair with the length of the shorter path minimized is known to be NP-complete (Xu et al., 2006) [1]. In this paper, we prove that in planar digraphs the edge-disjoint min–min problem remains NP-complete and admits no K-approximation for any K>1 unless P=NP. As a by-product, we show that this problem remains NP-complete even when all edge costs are equal (i.e., stronglyNP-complete). To our knowledge, this is the first NP-completeness proof for the edge-disjoint min–min problem in planar digraphs.

An Algorithm for Enumerating SRLG Diverse Path Pairs

Telecommunication networks are intrinsically multi-layered, a single failure at a lower level usually corresponds to a multi-failure scenario at an upper layer. In this context, the concept of shared risk link group (SRLG) allows an upper layer to select, for a given active path (AP), a backup path (BP), which avoids every SRLG that may involve the selected AP, in the event of a failure. That is a SRLG diversepath set maybe defined as a set of paths, between an origin and a destination, such that no pair of paths can be simultaneously affected by any given failure (or risk) in a single failure scenario. Firstly we present the formulation of the SRLG di- verse path pair calculation problem in a directed network. An algorithm for enumerating SRLG diverse paths, by non decreasing cost of their total (additive) cost will be presented, which is based on an algorithm proposed for generating minimal cost node disjoint path pairs. The SRLG diverse path pairs may be node or arc disjoint, with or without length constraints. Computational results will be presented to show the efficiency of the proposed algorithm for obtaining node or arc disjoint SRLG diverse path pairs in undirected networks.

An effective algorithm for obtaining the whole set of minimal cost pairs of disjoint paths with dual arc costs

In telecommunication networks design the problem of obtaining optimal (arc or node) disjoint paths, for increasing network reliability, is extremely important. The problem of calculating k c disjoint paths from s to t (two distinct nodes), in a network with k c different (arbitrary) costs on every arc such that the total cost of the paths is minimised, is NP-complete even for k c =2. When k c =2 these networks are usually designated as dual arc cost networks. In this paper we propose an exact algorithm for finding the whole set of arc-disjoint path pairs, with minimal cost in a network with dual arc costs. The correctness of the algorithm is based on a condition which guarantees that the optimal path pair cost has been obtained and on a proposition which guarantees that at the end of the algorithm all the optimal pairs have been obtained. The optimality condition is based on the calculation of upper and lower bounds on the optimal cost. Extensive experimentation is presented to show the effectiveness of the algorithm.

CONSTRAINED DISJOINT PATHS IN GEOMETRIC NETWORKS

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 easily implemented. 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.

An effective algorithm for obtaining the minimal cost pair of disjoint paths with dual arc costs

Routing optimisation in some types of networks requires the calculation of the minimal cost pair of disjoint paths such that the cost functions associated with the arcs in the two paths are different. An exact algorithm for solving this NP-complete problem is proposed, based on a condition which guarantees that the optimal path pair cost has been obtained. This optimality condition is based on the calculation of upper and lower bounds on the optimal cost. A formal proof of the correctness of the algorithm is described. Extensive experimentation is presented to show the effectiveness of the algorithm, including a comparison with an integer linear programming formulation.

Efficient calculation of the most reliable pair of link disjoint paths in telecommunication networks

In transmission networks an important routing problem is to find a pair of link disjoint paths which optimises some performance measure. In this paper the problem of obtaining the most reliable pair of link disjoint paths, assuming the reliability of the links are known, is considered. This is a non-linear optimisation problem. It is further introduced the constraint that the length of the paths should not exceed a certain number of links, which makes the efficient resolution of the problem more complex. In this paper two variants of a novel exact algorithm for finding the most reliable pair of link disjoint paths with lengths constraints, are presented. Also a computational study on the performance of the variants, is described.