Maximum Number Of Paths Research Articles

One-to-one node disjoint paths on divide-and-swap cubes

In massive parallel and distributed systems, high fault-tolerance and reliability are extremely important for interconnection networks. Node-disjoint paths play a key role in improving the performance, reliability and scalability of networks.The hypercube, as well as many hypercube-like variants, are among the most popular and attractive network architectures. The divide-and-swap cube DS C n is a newly proposed hypercube variant which has many desirable properties. In this work, we show that there exist d + 1 node-disjoint paths between any two distinct nodes in divide-and-swap cubes. Since the connectivity of the network is d + 1, d + 1 is the maximum number of paths that can be constructed between any two distinct nodes.

Hop-Constrained Oblivious Routing

We prove the existence of an oblivious routing scheme that is -competitive in terms of , thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize , defined as follows: The dilation is the maximum path hop length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have within a factor of the best possible value. More precisely, for any integer hop constraint , this oblivious routing scheme selects paths of length at most and is -competitive in terms of congestion in comparison to the best possible congestion achievable via paths of length at most hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002; Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008], which are -competitive in terms of congestion but are not competitive in terms of dilation.

Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

Some properties of generalized convexity for sets and functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based on both interpolation conditions and shape knowledge. It is proved that the approximant objects preserve both the high-order convexity and some extremum properties of the exact reliability polynomials. It leads to pointing out the area of the network where the maximum number of paths is achieved. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.

Flow characteristics analysis for multi-path hydrogen supply within proton exchange membrane fuel cell stack

The flow rate distribution of hydrogen between single cells has a significant impact on the performance and durability of the entire proton exchange membrane fuel cell stack, especially for open-cathode stacks. However, few studies consider the effects of fuel distribution on output performance, pressure drop distribution, and temperature distribution simultaneously at the stack level. To investigate the effect of hydrogen distribution on the stack output characteristics, several different multi-path hydrogen supply modes are proposed and explored in this study through the combination of flow network models and experiments. Considering hydrogen reaction consumption in the calculation models. Parameters such as stack voltage, flow uniformity index, dimensionless pressure drop, and temperature uniformity index under different modes are quantitatively compared. The results show that Triple-path hydrogen supply mode has the highest stack voltage of 5.278 V at rated condition, 4% improvement over the lowest Z-shape mode. While Quad-path hydrogen supply mode has the best flow rate distribution and single-cell voltage uniformity, with a maximum voltage difference of only 0.073 V between single cells in the stack. Multi-path hydrogen supply modes provide more uniform pressure drop distribution over the conventional U and Z-shape (with a single path), but the relationship between output performance and the number of hydrogen supply paths is not simply linearly correlated. Furthermore, the whole stack temperature decreases with the increase of hydrogen supply path, but the temperature uniformity is not optimal with the maximum number of paths.

Grid spanners with low forwarding index for energy efficient networks

A routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect toR (or simply the forwarding index with respect to R) π(G,R) of G is the maximum number of paths in R passing through any edge of G. The forwarding indexπ(G) of G is the minimum π(G,R) over all routings R’s of G. This parameter has been studied for different graph classes (Xu and Xu, 2012; Bouabdallah and Sotteau, 1993; Fernandez de la Vega and Gordone, 1992; de la Vega and Manoussakis, 1992). Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.

Grid spanners with low forwarding index for energy efficient networks

A routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to R) π(G,R) of G is the maximum number of paths in R passing through any edge of G. The forwarding indexπ(G) of G is the minimum π(G,R) over all routings R's of G. This parameter has been studied for different graph classes [Xu, J.-M. and M. Xu, The forwarding indices of graphs – a survey, CoRR abs/1204.2604 (2012)], [Bouabdallah, A. and D. Sotteau, On the edge forwarding index problem for small graphs, Networks23 (1993), pp. 249–255], [Fernandez de la Vega, W. and L. M. Gordones, The forwarding indices of random graphs, Random Structures and Algorithms 3 (1992), pp. 107–116], [de la Vega, W. F. and Y. Manoussakis, The forwarding index of communication networks with given connectivity, Discrete Appl. Math. 37–38 (1992), pp. 147–155]. Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.

Survivable Multipath Routing of Anycast and Unicast Traffic in Elastic Optical Networks

In this paper, we focus on the survivability of elastic optical networks (EONs) that jointly support two types of traffic demands: unicast and anycast. To provide network survivability, we apply multipath routing; i.e., we allow the splitting of a demand into a number of routing paths if the paths' combination guarantees the realization of a specific demand volume in the case of a single link failure. We formulate the corresponding optimization problem as an integer linear program (ILP) and propose a survivable multipath allocation (SMA) algorithm to solve the problem in a reasonable amount of time. Next, we perform numerical experiments to compare the efficiency (ability to provide a good-quality solution in a reasonable amount of time) of the ILP model and SMA as well as to evaluate the impact of survivable multipath routing on the objective defined as a maximum spectrum usage in EONs. Our results show that the SMA method finds good-quality solutions in a reasonable amount of time and that survivable multipath routing in EONs requires additional spectrum resources, up to 45%. However, the amount of additional resources depends on the required protection level, amount of anycast traffic, the maximum number of paths used for demand realization, and the considered network topology.

Approximating minimum cost source location problems with local vertex-connectivity demands

In the source location problem, the goal is to compute a minimum cost source set in a graph so that the connectivity between the source set and each vertex is at least the demand of the vertex. In this paper, we consider the problem for undirected graphs, and the connectivity between a source set S and a vertex v is defined as the maximum number of paths between v and S no two of which have common vertex except v. We give an O(d^@?logd^@?)-approximation algorithm for the problem with maximum demand d^@?. We also define a variant of the source location problem and give an approximation algorithm for it.

The forwarding indices of graphs - a survey

A routing R of a connected graph G of order n is a collection of n(n 1) simple paths connecting every ordered pair of vertices of G. The vertex-forwarding index (G;R) of G with respect to a routing R is defined as the maximum number of paths in R passing through any vertex of G. The vertex-forwarding index (G) of G is defined as the minimum (G;R) over all routings R of G. Similarly, the edge-forwarding index (G;R) of G with respect to a routing R is the maximum number of paths in R passing through any edge of G. The edge-forwarding index (G) of G is the minimum (G;R) over all routings R of G. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention for over twenty years. This paper surveys some known results on these forwarding indices, further research problems and several conjectures, also states some diculty and relations to other topics in graph theory.

A polynomial-time approximation scheme for embedding hypergraph in a cycle

We consider the problem of embedding hyperedges of a hypergraph as paths in a cycle such that the maximum congestion, namely the maximum number of paths that use any single edge in a cycle, is minimized. The minimum congestion hypergraph embedding in a cycle problem is known to be NP-hard and its graph version, the minimum congestion graph embedding in a cycle , is solvable in polynomial-time. Furthermore, for the graph problem, a polynomial-time approximation scheme for the weighted version is known. For the hypergraph model, several approximation algorithms with a ratio of two have been previously published. A recent paper reduced the approximation ratio to 1.5. We present a polynomial-time approximation scheme in this article, settling the debate regarding whether the problem is polynomial-time approximable.

Efficient algorithms for wavelength assignment on trees of rings

A fundamental problem in communication networks is wavelength assignment (WA): given a set of routing paths on a network, assign a wavelength to each path such that the paths with the same wavelength are edge-disjoint, using the minimum number of wavelengths. The WA problem is NP-hard for a tree of rings network which is well used in practice. In this paper, we give an efficient algorithm which solves the WA problem on a tree of rings with an arbitrary (node) degree using at most 3 L wavelengths and achieves an approximation ratio of 2.75 asymptotically, where L is the maximum number of paths on any link in the network. The 3 L upper bound is tight since there are instances of the WA problem that require 3 L wavelengths even on a tree of rings with degree four. We also give a 3 L and 2-approximation (resp. 2.5-approximation) algorithm for the WA problem on a tree of rings with degree at most six (resp. eight). Previous results include: 4 L (resp. 3 L ) wavelengths for trees of rings with arbitrary degrees (resp. degree at most eight), and 2-approximation (resp. 2.5-approximation) algorithm for trees of rings with degree four (resp. six).

Forwarding index of cube-connected cycles

For a given connected graph G of order v , a routing R in G is a set of v ( v − 1 ) elementary paths specified for every ordered pair of vertices in G . The vertex (resp. edge) forwarding index of G is the maximum number of paths in R passing through any vertex (resp. edge) in G . Shahrokhi and Székely [F. Shahrokhi, L.A. Székely, Constructing integral flows in symmetric networks with application to edge forwarding index problem, Discrete Applied Mathematics 108 (2001) 175–191] obtained an asymptotic formula for the edge forwarding index of n -dimensional cube-connected cycle C C C n as 5 4 n 2 2 n ( 1 − o ( 1 ) ) . This paper determines the vertex forwarding index of C C C n as 7 4 n 2 2 n ( 1 − o ( 1 ) ) asymptotically.

The forwarding indices of augmented cubes

For a given connected graph G of order n, a routing R in G is a set of n ( n − 1 ) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of G is the maximum number of paths in R passing through any vertex (resp. edge) in G. Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2002) 71–84] proposed a variant of the hypercube Q n , called the augmented cube AQ n and presented a minimal routing algorithm. This paper determines the vertex and the edge forwarding indices of AQ n as 2 n 9 + ( − 1 ) n + 1 9 + n 2 n 3 − 2 n + 1 and 2 n − 1 , respectively, which shows that the above algorithm is optimal in view of maximizing the network capacity.

Efficient algorithms for minimum congestion hypergraph embedding in a cycle

The minimum congestion hypergraph embedding in a cycle (MCHEC) problem is to embed the hyperedges of a hypergraph as paths in a cycle with the same node set such that the maximum congestion (the maximum number of paths that use any single edge in the cycle) is minimized. The MCHEC problem has many applications, including optimizing communication congestions in computer networks and parallel computing. The problem is NP-hard. In this paper, we give a 1.8-approximation algorithm for the MCHEC problem. This improves the previous 2-approximation results. Our algorithm has the optimal time complexity O(mn) for a hypergraph with m hyperedges and n nodes. We also propose an algorithm which finds an embedding with the optimal congestion L* for the MCHEC problem in O(n(nL*)/sup L*/) time. This improves the previous O((mn)/sup L*+1/) time algorithm.

Randomized path coloring on binary trees

Motivated by the problem of WDM routing in all-optical networks, we study the following NP-hard problem. We are given a directed binary tree T and a set R of directed paths on T. We wish to assign colors to paths of R, in such way that no two paths that share a directed arc of T are assigned the same color, and the total number of colors used is minimized. Our results are expressed in terms of the depth of the tree and of the maximum load l of R, i.e. the maximum number of paths that go through a directed arc of T. So far, only deterministic greedy algorithms have been presented for the problem. The best known algorithm colors any set R of maximum load l using at most 5l/3 colors. Alternatively, we say that this algorithm has performance ratio 5 3 . It is also known that no deterministic greedy algorithm can achieve a performance ratio better than 5 3 . In this paper we define the class of greedy algorithms that use randomization. We study their limitations and prove that, with high probability, randomized greedy algorithms cannot achieve a performance ratio better than 3 2 when applied for binary trees of depth Ω(l) , and 1.293− o(1) when applied for binary trees of constant depth. Exploiting inherent properties of randomized greedy algorithms, we obtain the first randomized algorithm for the problem that uses at most 7l/5+ o(l) colors for coloring any set of paths of maximum load l on binary trees of depth O(l 1/3−ε) , with high probability. We also present an existential upper bound of 7l/5+ o(l) that holds on any binary tree. For the analysis of our bounds we develop tail inequalities for random variables following hypergeometrical probability distributions that might be of their own interest.

Static and dynamic low-congested interval routing schemes

Interval routing schemes (IRS) have been extensively investigated in the past years with special emphasis on shortest paths. Besides their theoretical interest, IRS have practical applications, as they have been implemented with wormhole routing in the last generation of INMOS transputer router chips. In this paper we consider IRS that are optimal with respect to the congestion of the induced path system. In fact, wormhole routing is strongly influenced by the maximum number of paths that share a physical link and from low to moderate congestion it outperforms the packet switching technique. We provide a general framework able to deal with the various congestion issues in IRS. In fact, we will distinguish between static cases, in which the source–destination configurations are fixed, and dynamic cases, where they vary over time. All these situations can be handled in a unified setting, thanks to the notion of competitiveness introduced in this paper. We first give some general results not related to specific traffic demands. Then, in the one-to-all communication pattern, we show that constructing competitive IRS for a given network is an intractable problem, both for the static and the dynamic case, that is when the root vertex is fixed and when it can change along the time, respectively. Finally, both for one-to-all and all-to-all communication patterns, we provide nicely competitive k-IRS for relevant topologies. Networks considered are chains, trees, rings, chordal rings and multi-dimensional grids and tori. We consider both the directed congestion case, in which there are pairwise opposite unidirectional links connecting two neighbor processors, and the undirected congestion case, in which two neighbors are connected by a single bi-directional link.

Analysis of Routing Issues in the Design of Packet Networks

An analysis of several routing issues that have an impact on the topological design of nonhierarchical packet networks is presented. Routing constraints that arise due to the limits on maximum number of hops and maximum number of paths and due to the virtual circuit mode of operation have an impact on critical network design parameters such as trunk costs, topological connectivity, and average packet delay. The analysis is conducted through a design algorithm that configures cost-effective networks subject to an end-to-end mean delay constraint and specified routing constraints. The analysis indicates that significant savings in trunk costs can be realized when the hop and path constraints are relaxed from one to three. Beyond these limits, however, the reduction in network cost is marginal. Furthermore, it is more cost effective to allow flow bifurcation over multiple paths than to allow longer paths. Virtual circuit routing constraints have a significant impact on network costs under highly unbalanced loads; however, their impact is minimal under more realistk traffic patterns. The analysis also shows that network designs based solely on a network average delay constraint may lead to high end-to-end delays for some node pairs. The algorithm described here is capable of controlling the end-to-end delays for all node pairs at little or no additional costs.

Digraphs with maximum number of paths and cycles

AbstractWe construct a digraph with the maximum number of simple paths between two specified vertices, for a digraph with a given number of edges. The following cases are considered: digraphs with parallel edges, acyclic simple digraphs and general simple digraphs. The corresponding extremal digraphs are the tri‐chains, the (deficient) Fibonacci digraphs and (if our conjecture is true) the 3‐diamond strings, respectively. The similarity of these three families of digraphs is discussed. The related problem of digraphs with the maximum number of simple cycles for a given number of edges is considered too.

A floating via router based on a saturated zone strategy

This letter describes a new routing algorithm which is oriented towards floating via technology in a two-sided environment. The algorithm builds up "saturated zones" of interconnections and then combines these saturated zones by four tactics which attempt to complete a maximum number of paths selected from a restricted set of path shapes. A router based on this algorithm has been implemented, and has averaged 99.4 percent completion on 24 Illiac IV boards, and attained 100 percent completion on a large printed wiring board. The router averages about one via per wire, uses modest storage, and is generally economical.