A Faster Algorithm for Finding Disjoint Ordering of Sets

Nowadays, the development of robots has created problems whose solution should be found. One of these problems is the search for disjoint paths on polygons with a minimum number of links. The article describes cases when there are several sources and targets at the polygon. The special cases of 2×2 and 3×3 links are considered. The intersection solution in the case of 2×2 is achieved by minimizing the sum of the Euclidean link lengths. If the paths with minimal Euclidean distances intersect, then there is no solution, otherwise solution exists. This method can be extended to the 2×N case. The 3×3 case is solved using an hourglass model. First and third links are taut-string links. The middle link is selected within the side walls of the hourglass. For the case of M×N links, a heuristic algorithm for finding disjoint paths is given. The article also describes an algorithm for constructing forbidden regions that are used in each case.

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.

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.

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.

The multiple pairs shortest path problem (MPSP) arises in many applications where the shortest paths and distances between only some specific pairs of origin-destination (OD) nodes in a network are desired. The traditional repeated single-source shortest path (SSSP) and all pairs shortest paths (APSP) algorithms often do unnecessary computation to solve the MPSP problem. We propose a new shortest path algorithm to save computational work when solving the MPSP problem. Our method is especially suitable for applications with fixed network topology but changeable arc lengths and desired OD pairs. Preliminary computational experiments demonstrate our algorithm’s superiority on airline network problems over other APSP and SSSP algorithms.

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.

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.

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.

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

We propose a simple and effective heuristic that allows fast generation of a large set of shortest path alternatives in weighted directed graphs. The heuristic is based on existing deviation path algorithms for exact k shortest paths. It precalculates a backward shortest path tree and thus avoids doing many shortest path computations, but as a result it does not necessarily find the exact set of k shortest paths. Computational results on real-world road networks are reported. Our tests show that the quality of the paths produced by the heuristic is most satisfactory: typically, the kth path found by the heuristic is less than 1% longer than the exact kth shortest path, for values of k up to 10,000. Moreover, the heuristic runs very fast. We also show how the heuristic can be enhanced to an exact k shortest paths algorithm, which performs well in comparison with the existing exact k shortest path algorithms.

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.

Finding shortest distances and paths have always been a crucial area of study. In this article, we have proposed a linear time pre-processing method for optimization of shortest path and distance algorithms, for an undirected weighted graph with non-negative edges. In this approach, certain parts of a graph are grouped together as subgraphs termed as optimized deterministic routing areas (ODRAs). Each ODRA has a unique representative element called optimized proxy. The input graph is reduced by replacing ODRAs with optimized proxies and the new reduced graph is used for the shortest path and distance queries. This is a linear time pre-processing stage which helps in significant decrease in computation time for finding shortest distances and paths in graphs. The lower bound and upper bound of time complexity of the proposed algorithm is O(n/2) and O(n) respectively.

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).