On Algorithms for the Minimum Link Disjoint Paths Problem

Consider the problem of routing from a single source node to multiple target nodes with the additional condition that these disjoint paths be the shortest. This problem is harder than the standard one-to-many routing in that such paths do not always exist. Various sufficient and necessary conditions have been found to determine when such paths exist for some interconnection networks. And when these conditions do hold, the problem of finding such paths can be reduced to the problem of finding a disjoint ordering of sets. We study the problem of finding a disjoint ordering of sets $X_1$, $X_2$, $X_s$ where $X_i \subseteq \{1, 2, \cdots, n\}$ and $s \leq n$. We present an $O(n^3)$ algorithm for doing so, under certain conditions, thus improving the previously known $O(n^4)$ algorithm, and consequently, improving the corresponding one-to-many routing algorithms for finding disjoint and shortest paths.

The metacube interconnection network introduced a few years ago has some very interesting properties: it has a short diameter similar to the hypercube, and its degree is much lower than that of a hypercube of the same size. In this paper, we describe an efficient algorithm for finding disjoint paths between one source node and at most m+k target nodes in a metacube MC(k, m) excluding MC(*, 1), MC(2, 2), MC(3, 2) and MC(3, 3). We show that we can find m+k disjoint paths between the source node and the m+k targets of length at most metacube diameter plus (k + 4) with time complexity of order of metacube degree times its diameter.

Disjoint paths and connected subgraphs for H-free graphs

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.

Many fundamental problems in the area of distributed systems such as security, reliable routing, network survivability and broadening available bandwidth can be addressed through the use of disjoint paths between communication endpoints. Therefore, distributed solutions to the disjoint paths problem are of crucial importance to these fundamental problems. Disjoint paths naturally exist in traditional networks; however, these networks provide no guarantees regarding the presence and the number of available disjoint paths. On the other hand, network topologies structured based on Caley graphs such as hypercube, star networks and their variations possess many desirable properties related to disjoint paths. Therefore, it is anticipated that backbones of future networks will be structured following some Caley graph topologies. In this paper, we present a simple novel stabilizing and inherently stabilizing algorithm to route messages over all node-disjoint paths of optimal length from one non-faulty process to another in at most n+1 rounds in an n-dimensional hypercube network in the presence of crash failures. The proposed algorithm can tolerate up to 2n-⌈log (n+1)⌉ process/link failures in the network and a maximum of ⌊(L-2)/3⌋ process/link failures on each disjoint path, where L is the number of processes on a disjoint path and the distance between any two failed processes/links is at least three. The proposed algorithm tolerates a large number of process and link failures, while delivering all n messages over optimal-length disjoint paths in the presence of maximum permissible process/link failures on each path. This is achieved by a simple uniform distributed algorithm using only local knowledge of failure locations.

In this paper, we investigate efficient approaches and algorithms for protecting multicast sessions against any single link failure while establishing multicast sessions in WDM mesh networks. Since a single failure may affect whole nodes in a multicast group and causes severe service disruption and a lot of traffic loss, protecting critical multicast sessions against link failure such as fiber cut becomes important in WDM optical networks. One of the most efficient algorithms is optimal path pair-shared disjoint paths (OPP-SDP). In this algorithm every source-destination (SD) pair has the optimal path pair (working and protection path) between the source and destination node. Since degree of sharing among the paths is essential to reduce the total cost and blocking probability, we propose the longest path first-shared disjoint paths (LPF-SDP) algorithm which decides the priority of selection among SD pairs in a resource-saving manner. Our LPF-SDP is shown to outperform over OPP-SDP in terms of degree of sharing and blocking probability.

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

A new polynomial time algorithm is presented for finding two vertex-disjoint paths between two specified pairs of vertices on an undirected graph. An application of the two disjoint path algorithm to the automatic wire routing is also discussed.

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.

Automation production systems and computer integrated manufacturing: Mikell P. Groover

Given a set of sources and a set of sinks in the two dimensional grid of size n, the disjoint paths (DP) problem is to connect every source to a distinct sink by a set of edge-disjoint paths. Let v be the total number of sources and sinks. In [3], Chan and Chin showed that without loss of generality we can assume v ≤ n ≤ 4v2. They also showed how to compress the grid optimally to a dynamic network (structure of the network may change depending on the paths found currently) of size O(√nv), and solve the problem in O(√nv3/2) time using augmenting path method in maximum flow. In this paper, we improve the time complexity of solving the DP problem to O(n3/4v3/4). The factor of improvement is as large as √v when n is Θ(v), while it is at least 4√v for n is Θ(v2).

We consider routing in survivable networks that provide protection against node or link failures. In these networks resilience against failures is provided by routing connections on pairs of disjoint paths called primary and backup paths. The primary path of a connection carries its traffic under normal circumstances and in the eventuality of a network failure effecting the primary path the connection traffic (all or some portion of it) is rerouted over its backup path. In an online setting as connection requests arrive a pair of disjoint primary and backup paths of least total cost (under some link cost metric) are selected to route the connections. In many situations the cost metric used for the primary path differs from the cost metric used for the backup path. Also in many realistic settings these two cost metrics are related to each other. In this paper we study the problem of finding a pair of edge or node disjoint paths of least total cost where the cost of the primary path is the total cost of its links while the cost for the backup path is α times the sum of the cost of its links, for some given α < 1. We show that the problem is hard to approximate to within a factor $$\Omega((\frac{1}{\alpha})^{1-\epsilon})$$ for any positive $$\epsilon \le 1$$ . In addition we show that the problem is complete for a set of hard to approximate problems. On the positive side we show that a simple algorithm achieves an approximation ratio of $$O(\frac{1}{\alpha})$$ for the problem.

Given k + 1 pairs of vertices (s 1,s 2),(u 1,v 1),...,(u k ,v k ) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (u l ,v l ) with 1 ≤ l≤ k, a tuple (s 1,t 1,s 2,t 2) with {t 1,t 2} = {u l ,v l } in constant time such that there are two disjoint paths p 1, from s 1 to t 1, and p 2, from s 2 to t 2, if such a tuple exists. Disjoint can mean vertex- as well as edge-disjoint. As an application we show that the presented data structure can be used to improve the previous best known running time O(mn) for the so called 2-disjoint paths problem on directed acyclic graphs to O(m(log2 + m/n n) + nlog3 n). In this problem, given a tuple (s 1,s 2,t 1,t 2) of four vertices, we want to construct two disjoint paths p 1, from s 1 to t 1, and p 2, from s 2 to t 2, if such paths exist.

The problem of finding the maximum number of vertex-disjoint uni-color paths in an edge-colored graph (MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (distance from disjoint paths and size of vertex cover), and that is not FPT-approximable. Moreover, we introduce a new variant of the problem, called MaxCDDP, whose goal is to find the maximum number of vertex-disjoint and color-disjoint uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDP is already hard on graphs at distance two from disjoint paths.

Given a source node and a node set, the node-to-set disjoint paths problem is to find disjoint paths from the source node to all nodes from the node set. In this paper, we propose an improved algorithm for the node-to-set disjoint paths problem on bi-rotator graphs. Our result improves the complexity of previous result from O(n5) to O(n3logn), where n is the dimension of the input graph.