Algorithm For Disjoint Paths Research Articles

Shortest node-to-node disjoint paths algorithm for symmetric networks

Shortest node-to-node disjoint paths algorithm for symmetric networks

BCCC Disjoint Path Construction Algorithm and Fault-Tolerant Routing Algorithm under Restricted Connectivity

Connectivity in large-scale data center networks is a critical indicator to evaluate network state. A feasible and performance-guaranteed algorithm enables us to find disjoint paths between network vertices to ensure effective data transfer and to maintain the normal operation of network in case of faulty nodes. As an important data center network, BCube Connected Crossbars (BCCC) has many excellent properties that have been widely studied. In this paper, we first propose a vertex disjoint path algorithm with the time complexity of O(nk) in BCCC, where n denotes a switch connected to n servers and k denotes dimension. Then, we prove that the restricted connectivity of BCCC(n,k). Finally, we present an O(knκ1(G)) fault-free algorithm in BCCC, where κ1(G) is the restricted connectivity. This algorithm can obtain a fault-free path between any two distinct fault-free vertices under the condition that each vertex has at least one fault-free neighbor in the BCCC and a set of faulty vertices F with |F|<κ1(G).

Disjoint paths and connected subgraphs for H-free graphs

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct vertex pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for H-free graphs. If k is fixed, we obtain the k-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every k≥1. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of k-Disjoint Connected Subgraphs for H-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for H-free graphs as for Disjoint Paths. Moreover, we give exact algorithms for Disjoint Paths and Disjoint Connected Subgraphs on graphs with n vertices and m edges that have running times of O(2nn2k) and O(3nkm), respectively.

Reliable Communication in Transmission Grids based on Nondisjoint Path Aggregation Using Software-Defined Networking

In electrical transmission grids, the redundant communication paths nondisjointly overlapping at links can be established between certain substations and the control center to guarantee reliable packet delivery under link failures. However, the generation of nondisjoint paths with multiplicative and concave constraints is with the NP-complete complexity and the failovers can lead to out-of-order packets. This paper presents an OpenFlow-based nondisjoint path aggregation mechanism to heuristically compute the constrained nondisjoint paths in a centralized fashion and reorganize the out-of-order packets at edge switches. The solution is evaluated through simulations of the IEEE 30-bus network scenario under 2% link failure rate and the result confirms its effectiveness: the packet delivery success rate is significantly improved in comparison with the current nondisjoint and disjoint path algorithms. TCP throughput is improved by 121.62% with the packet reordering. Also, the memory usage for the packet reordering buffer is reduced by 76.563% compared with the distributed cognitive packet network.

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.

New algorithms for maximum disjoint paths based on tree-likeness

We study the classical {mathsf {NP}}-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is {2^{varOmega (sqrt{log n})}}, assuming {mathsf {NP}not subseteq mathsf {DTIME}(n^{mathcal {O}(log n)})}. This constitutes a significant gap to the best known approximation upper bound of {mathcal {O}(sqrt{n})} due to Chekuri et al. (Theory Comput 2:137–146, 2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica 7(4):365–374, 1987) introduce the technique of randomized rounding for LPs; their technique gives an {mathcal {O}(1)}-approximation when edges (or nodes) may be used by {mathcal {O}left( log n/log log nright) } paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set numberr of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:For MaxEDP, we give an {mathcal {O}(sqrt{r} log ({k}r))}-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio {mathcal {O}(sqrt{n})} due to Chekuri et al., as {rle n}.Further, we show how to route {varOmega ({text {OPT}}^{*})} pairs with congestion bounded by {mathcal {O}(log (kr)/log log (kr))}, strengthening the bound obtained by the classic approach of Raghavan and Thompson.For MaxNDP, we give an algorithm that gives the optimal answer in time {(k+r)^{mathcal {O}(r)}cdot n}. This is a substantial improvement on the run time of {2^kr^{mathcal {O}(r)}cdot n}, which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is {mathsf {NP}}-hard even for {r=1}, and MaxNDP is {mathsf {W}[1]}-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless {mathsf {FPT}= mathsf {W}[1]} and that our approximability results are relevant even for very small constant values of r.

Energy aware routing with link disjoint backup paths

Energy aware routing has attracted a wide spread attention in the last decade. However, relatively concentrated traffic after applying energy aware strategies makes the network be vulnerable to link failure and sudden traffic bursts. It is a challenge to take enhancing network reliability into account while keeping low energy consumption. Routing each source-to-terminal request via two disjoint paths can be a trade-off. In this paper, we propose a problem called Energy Aware Routing with Link Disjoint Backup Paths (EAR-LDBP) to minimize router power consumption when backup sharing is not allowed during off-peak hours. Each request is routed via an active path. A link disjoint backup path is computed for each request and can be used to route traffic in case of single link failure on the active path. A 0–1 integer linear programming model that can guarantee the network resilience to single link failure is formulated to minimize the power of used links and nodes while putting idle ones to sleep under constraints of link utilization and a link disjoint backup path for each request. We analyze the complexity of EAR-LDBP and show its Np-hardness. We prove that there is no polynomial-time constant-factor approximation algorithm even for two requests. Four different versions of EAR-LDBP (link/node-disjoint, link cost only/link and node cost) are shown to be equivalent and polynomial-time reducible to each other. Then two algorithms are proposed to solve this problem: 2 Stage Greedy Algorithm (2SG) and Greedy Two Link Disjoint Paths Algorithm (G2DP) with an iterative deletion strategy to improve the solution. For comparison purpose, we also use the state of the art reliable energy aware routing algorithm 2DP-NF in unsplittable way. Simulation results on synthetic networks and real network of Internet2 show the effectiveness of our algorithms. Particularly, our algorithms can save more energy (up to 39%) while using less computation time as compared to 2DP-NF (less than 1% at best).

Many-to-many disjoint paths in hypercubes with faulty vertices

This paper considers the problem of many-to-many disjoint paths in the hypercube Qn with f faulty vertices and obtains the following result. For any integer k with 1≤k≤n−1 and any two sets S and T of k fault-free vertices in different partite sets of Qn(n≥2), if f≤2n−2k−2 and each fault-free vertex has at least two fault-free neighbors, then there exist k fully disjoint fault-free paths linking S and T which contain at least 2n−2f vertices. A linear algorithm for finding such disjoint paths is also given. This result improves some known results in a sense.

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.

A Novel Shared Protection Scheme Based on Aggregate Wavelength in High Speed Networks

We propose novel analytical model of dynamic link cost evaluation in IP over WDM networks. We suggest disjoint path algorithm for the primary and backup path based on wavelength aggregate information, to provide shared backup. We show the optimality of pair selected because of joint optimization of the pair paths. The shareable capacity factor is introduced to establish the effect of load balancing on resources. We compared our simulation results with that of separate protection at connection and showed improvement on resource utilization performance of the network. We also study the blocking probability of proposed scheme.

A QoS Routing Protocol for Mobile Ad Hoc Networks Based on Multipath

With the development of Ad Hoc networks, the demand of QoS becomes more and more than ever before, it is more important to provide QoS guarantee for multimedia application. Most of present QoS routing algorithm based on single path, which do not take full advantage of resource of networks. This paper proposes a QoS-based multipath routing protocol for Ad Hoc network. Routing mechanism is described in detail and node disjoint path algorithm is a key. New routing protocol takes bandwidth and delay constraint into account, it can find several paths to provide QoS guarantee. Also, it can simultaneously use the node disjoint paths to transmit application data flow. The results of experiment show new protocol can significantly increase the packet delivery ratio and decrease the average delay, the performance is better than other protocols. This paper provides a good idea to study further QoS routing protocol.

DISJOINT-PATHS AND FAULT-TOLERANT ROUTING ON RECURSIVE DUAL-NET

The recursive dual-net is a newly proposed interconnection network for massive parallel computers. The recursive dual-net is based on recursive dual-construction of a symmetric base network. A k-level dual-construction for k > 0 creates a network containing (2n0)2k/2 nodes with node-degree d0 + k, where n0 and d0 are the number of nodes and the node-degree of the base network, respectively. The recursive dual-net is node and edge symmetric and can contain huge number of nodes with small node-degree and short diameter. Disjoint-paths routing and fault-tolerant routing are fundamental and critical issues for the performance of an interconnection network. In this paper, we propose efficient algorithms for disjoint-paths and fault-tolerant routings on the recursive dual-net.

이븐 연결망 Ed의 에지 중복 없는 스패닝 트리를 구성하는 알고리즘

이븐 연결망은 고장허용 다중컴퓨터에 대한 하나의 모형으로 제안된 연결망으로, 간단한 라우팅 알고리즘, 최대고장허용도, 노드 중복 없는 경로와 같은 여러 가지 유용한 성질과 알고리즘들이 분석되었다. 기존에 발표된 라우팅 알고리즘과 노드 중복 없는 경로를 구성하는 알고리즘은 최적임이 증명되었다. 하지만 아직까지 이븐 연결망에서 에지 중복 없는 스패닝 트리를 구성하는 기법은 소개되지 않았다. 에지 중복 없는 스패닝 트리는 상호연결망의 고장허용도의 성능 향상과 효율적인 방송 기법을 분석하기 위해서 사용되는 매우 유용한 기법이다. 기존에 발표된 라우팅 알고리즘 또는 노드 중복 없는 경로를 구성하는 알고리즘은 라우팅 또는 노드 중복 없는 경로를 위한 알고리즘으로 에지 중복 없는 스패닝 트리를 구성하기 위해 적용될 수 없는 알고리즘이다. 본 논문에서는 이븐 연결망 <TEX>$E_d$</TEX>에서 에지 중복 없는 스패닝 트리를 구성하는 알고리즘을 제안한다. Even networks were introduced as a class of fault-tolerant multiprocessor networks and analyzed so many useful properties and algorithms such as simple routing algorithms, maximal fault tolerance, node disjoint path. Introduced routing algorithms and node disjoint path algorithms are proven to be optimal. However, it has not been introduced to constructing scheme for edge-disjoint spanning trees in even networks. The design of edge-disjoint spanning trees is a useful scheme to analyze for measuring the efficiency of fault tolerant of interconnection network and effective broadcasting. Introduced routing algorithm or node disjoint path algorithm are for the purpose of routing or node disjoint path hence they are not applicable to constitute edge disjoint spanning tree. In this paper, we show a construction algorithm of edge-disjoint spanning trees in even network <TEX>$E_d$</TEX>.

An evolutionary programming algorithm for finding constrained optimal disjoint paths for multihop communication networks

We present an evolutionary programming algorithm (EP) for finding hop count and bandwidth constrained cost optimal disjoint paths in multihop communication networks. In general, multi-constrained path selection is an NP complete problem. Our proposed algorithm can be used for real-time online network applications, which can initiate and process a number of calls simultaneously on disjoint paths, without overloading the network. This algorithm also requires a small memory space and low execution time. The proposed algorithm, which maintains a balance between finding an optimal shortest path and CPU mean execution time, also generates parallel suboptimal paths during the process of generating the best path. One of these suboptimal paths can be used as a backup path if it is link disjoint with all the primary paths (best paths) of the concurrent requests. Thus, the proposed algorithm provides a limited degree of reliability for routing of packets as well.

ρp-Cycle based tree protection of optical multicast traffic for combined link and node failure recovery in WDM mesh networks

The extension of the p-cycle concept to tree protection enables p-cycles to protect the multicast traffic for combined link and node failure recovery. In this paper, we present an efficiency-score based heuristic algorithm of p-cycle based tree protection (ESHT). Simulation results show that the capacity efficiency of ESHT is close to that of optimal path pair (OPP) algorithm, while the blocking performance of ESHT is in between OPP-shared disjoint paths (OPP-SDP) algorithm and OPP shared disjoint segments (OPP-SDS) algorithm. However, p-cycle based protection approaches offer much faster restoration speed, because p-cycles are pre-cross-connected.

A localized algorithm for finding disjoint paths in wireless sensor networks

This letter presents a localized algorithm that finds multiple node-disjoint paths in wireless sensor networks. The algorithm needs only local topology knowledge and provides automatic path restoration. We describe the algorithm, give the proof of correctness, and evaluate its performance using simulation. We conclude that the proposed algorithm is more suitable for wireless sensor networks than the existing distributed algorithms

An algorithm for disjoint paths in bubble‐sort graphs

AbstractAn n‐dimensional bubble‐sort graph is regular and symmetric. It has n! nodes and (n−1)n!/2 edges while its connectivity and diameter are n−1 and n(n−1)/2, respectively. Bubble‐sort graphs are attracting attention because of their simple, symmetric, and recursive structure. In this paper, for an n‐bubble‐sort graph, we give an O(n5)‐time algorithm that solves the node‐to‐set disjoint paths problem: Given a source node s and a set D = d1, d2, …, dk (s ∉ D) of k destination nodes in a k‐connected graph G, find k paths from s to di(1≤i≤k) that are node‐disjoint except for s. Once these k paths are obtained, they achieve fault tolerance; that is, at least one path can survive with k−1 faulty components. We also show that the total length of n−1 paths given by the algorithm is O(n3). Computer experiment results show that the average time complexity of the algorithm and the average total length of the paths given by the algorithm are O(n4.7) and O(n3.0), respectively. © 2006 Wiley Periodicals, Inc. Syst Comp Jpn, 37(12): 27–32, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20518

Finding Minimum Energy Disjoint Paths in Wireless Ad-Hoc Networks

We develop algorithms for finding minimum energy disjoint paths in an all-wireless network, for both the node and link-disjoint cases. Our major results include a novel polynomial time algorithm that optimally solves the minimum energy 2 link-disjoint paths problem, as well as a polynomial time algorithm for the minimum energy k node-disjoint paths problem. In addition, we present efficient heuristic algorithms for both problems. Our results show that link-disjoint paths consume substantially less energy than node-disjoint paths. We also found that the incremental energy of additional link-disjoint paths is decreasing. This finding is some what surprising due to the fact that in general networks additional paths are typically longer than the shortest path. However, in a wireless network, additional paths can be obtained at lower energy due to the broadcast nature of the wireless medium. Finally, we discuss issues regarding distributed implementation and present distributed versions of the optimal centralized algorithms presented in the paper.

Restoration mechanisms for handling channel and link failures in optical WDM networks: tunable laser-based switch architectures and performance analysis

In this paper, we study restoration mechanisms to handle channel and link failures in an optical wavelength division multiplexed (WDM) wavelength-routed wide-area backbone network based on a mesh topology. The solution uses a small number of tunable lasers to provide restoration capability. We consider two types of failures: link failures and individual channel ( or wavelength) failures that occur when one or more transceivers fail at a node that is the source of lightpath(s) or due to a failure in an intermediate node's optical switch fabric. We use the restoration mechanism that attempts to find alternate paths and resources after failure occurs. In our proposed mechanism, restoration is first attempted using the tunable lasers to transmit on the failed wavelengths. If all the failed lightpaths cannot be restored using the tunable lasers, unused wavelengths on the same link are used, if optical wavelength conversion is available. For the remaining lightpaths requiring restoration, two different link-level restoration mechanisms called redirection algorithm (RDA) and disjoint path algorithm (DPA) are used. Results based on discrete-event simulations to understand the performance of the proposed mechanisms, in terms of restoration efficiency and restoration times, are presented. The results show that for networks of varying size and node degree with 32 wavelengths on each link, using as few as eight tunable lasers per link provides good restoration efficiency under moderate traffic load. The performance of the proposed algorithms is compared to an earlier restoration mechanism based on broadcast, and it is seen that the proposed mechanism performs better, by offering both lower restoration times and higher restoration efficiency even with a small number of lasers. The impact of the number of tunable lasers on the performance is studied for failures occurring simultaneously on two links. It is seen that for a small number of such channel failures, as few as four tunable lasers per link are sufficient to recover from failures on a single link and on two links.

Provisioning of survivable multicast sessions against single link failures in optical WDM mesh networks

In this paper, we investigate approaches and algorithms for establishing a multicast session in a mesh network while protecting the session against any single link failure, e.g., a fiber cut in an optical network. First, we study these approaches and algorithms to protect a single multicast tree in a mesh network and then extend it to dynamically provision survivable multicast connections (where connections come and go) in an optical wavelength-division multiplexing (WDM) network. We propose two new and efficient approaches for protecting a multicast session: 1) segment protection in which we protect each segment in the primary tree separately (rather than the entire tree) and allow these backup segments to share edges with the other existing primary and backup segments and 2) the path-pair protection in which we find a path-pair (disjoint primary and backup paths) to each destination and allow a new path pair to share edges with already-found path pairs. Unlike previous schemes, such as finding link-disjoint trees and arc-disjoint trees, our new schemes 1) guarantee a solution where previous schemes fail and 2) find an efficient solution requiring less network resources. We study these approaches and algorithms systematically, starting with the existing approaches such as fully link-disjoint and arc-disjoint trees and then presenting our new and efficient proposed approaches, such as segment-disjoint and path-disjoint schemes for protecting multicast connections. Our most efficient algorithm, based on the path-pair protection scheme, called optimal path-pair-based shared disjoint paths (OPP-SDP) algorithm, finds a solution if such a solution exists and outperforms all the other schemes in terms of network cost. We also show that OPP-SDP performs close to the optimal solution obtained by solving a mathematical formulation of the problem expressed as an integer linear program. Building upon the study on protecting a single tree, we perform simulations, employing the above protection schemes, to study dynamic provisioning of survivable multicast sessions (where sessions come and go) in a WDM mesh network. Our simulations show that the most efficient scheme, OPP-SDP, has minimum blocking probability.