The Complexity of Deciding Strictly Non-Blocking Concentration and Generalized-Concentration Properties

1997 ◽  
Vol 08 (03) ◽  
pp. 237-252 ◽  
Author(s):  
H. K. Dai

Concentrators and generalized-concentrators are interconnection networks that provide respectively pairwise vertex-disjoint directed paths and trees to satisfy interconnection requests. An interconnection network is non-blocking in the strict sense if every compatible interconnection request can be satisfied by a path regardless of any existing interconnections. We present polynomial time computational complexity results for deciding the strictly non-blocking concentration and generalized-concentration properties with small depth, by using b-matching techniques.

Author(s):  
Abderezak Touzene ◽  
Khaled Day

We obtain the conditional fault-diameter of the square torus interconnection network under the condition of forbidden faulty sets (i.e. assuming that each non-faulty processor has at least one non-faulty neighbor). We show that under this condition, the square torus, whose connectivity is 4, can tolerate up to 5 faulty nodes without becoming disconnected. The conditional node connectivity is, therefore, 6. We also show that the conditional fault-diameter of the square torus is equal to the fault-free diameter plus two. With this result the torus joins a group of interconnection networks (including the hypercube and the star-graph) whose conditional fault-diameter has been shown to be only two units over the fault-free diameter. Two fault-tolerant routing algorithms are discussed based on the proposed vertex disjoint paths construction.  


2013 ◽  
Vol 321-324 ◽  
pp. 2715-2720
Author(s):  
Xin Yu ◽  
Gao Cai Wang ◽  
Yan Yu

Crossed cube is a variation of hypercube, but some properties of the former are superior to those of the latter. However, it is difficult to extend the scale of crossed cube networks. As a kind of hierarchical ring interconnection networks, crossed cube-connected ring interconnection network CRN can effectively overcome the disadvantage. Hence, it is a good topology for interconnection networks. In this paper, we prove that there exist n internally vertex-disjoint paths between any two vertexes in CRN, and analyze the lengths of the paths.


2019 ◽  
Vol 30 (08) ◽  
pp. 1301-1315 ◽  
Author(s):  
Liqiong Xu ◽  
Shuming Zhou ◽  
Weihua Yang

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A graph [Formula: see text] is said to be maximally local-connected if each pair of vertices [Formula: see text] and [Formula: see text] are connected by [Formula: see text] vertex-disjoint paths. In this paper, we show that Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected and are also [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have a triangle, [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have no triangles. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected if their corresponding transposition generating graphs have no triangles.


2012 ◽  
Vol 22 (01) ◽  
pp. 1250003
Author(s):  
IAIN A. STEWART

We equate a routing algorithm in a (faulty) interconnection network whose underlying graph is a k-ary n-cube or a hypercube, that attempts to route a packet from a fixed source node to a fixed destination node, with the sub-digraph of (healthy) links potentially usable by this routing algorithm as it attempts to route the packet. This gives rise to a naturally defined problem, parameterized by this routing algorithm, relating to whether a packet can be routed from a given source node to a given destination node in one of our interconnection networks in which there are (possibly exponentially many) faulty links. We show that there exist such problems that are PSPACE-complete (all are solvable in PSPACE) but that there are (existing and popular) routing algorithms for which the computational complexity of the corresponding problem is significantly easier (yet still computationally intractable).


1986 ◽  
Vol 9 (3) ◽  
pp. 323-342
Author(s):  
Joseph Y.-T. Leung ◽  
Burkhard Monien

We consider the computational complexity of finding an optimal deadlock recovery. It is known that for an arbitrary number of resource types the problem is NP-hard even when the total cost of deadlocked jobs and the total number of resource units are “small” relative to the number of deadlocked jobs. It is also known that for one resource type the problem is NP-hard when the total cost of deadlocked jobs and the total number of resource units are “large” relative to the number of deadlocked jobs. In this paper we show that for one resource type the problem is solvable in polynomial time when the total cost of deadlocked jobs or the total number of resource units is “small” relative to the number of deadlocked jobs. For fixed m ⩾ 2 resource types, we show that the problem is solvable in polynomial time when the total number of resource units is “small” relative to the number of deadlocked jobs. On the other hand, when the total number of resource units is “large”, the problem becomes NP-hard even when the total cost of deadlocked jobs is “small” relative to the number of deadlocked jobs. The results in the paper, together with previous known ones, give a complete delineation of the complexity of this problem under various assumptions of the input parameters.


2021 ◽  
Author(s):  
Karthik K ◽  
Sudarson Jena ◽  
Venu Gopal T

Abstract A Multiprocessor is a system with at least two processing units sharing access to memory. The principle goal of utilizing a multiprocessor is to process the undertakings all the while and support the system’s performance. An Interconnection Network interfaces the various handling units and enormously impacts the exhibition of the whole framework. Interconnection Networks, also known as Multi-stage Interconnection Networks, are node-to-node links in which each node may be a single processor or a group of processors. These links transfer information from one processor to the next or from the processor to the memory, allowing the task to be isolated and measured equally. Hypercube systems are a kind of system geography used to interconnect various processors with memory modules and precisely course the information. Hypercube systems comprise of 2n nodes. Any Hypercube can be thought of as a graph with nodes and edges, where a node represents a processing unit and an edge represents a connection between the processors to transmit. Degree, Speed, Node coverage, Connectivity, Diameter, Reliability, Packet loss, Network cost, and so on are some of the different system scales that can be used to measure the performance of Interconnection Networks. A portion of the variations of Hypercube Interconnection Networks include Hypercube Network, Folded Hypercube Network, Multiple Reduced Hypercube Network, Multiply Twisted Cube, Recursive Circulant, Exchanged Crossed Cube Network, Half Hypercube Network, and so forth. This work assesses the performing capability of different variations of Hypercube Interconnection Networks. A group of properties is recognized and a weight metric is structured utilizing the distinguished properties to assess the performance exhibition. Utilizing this weight metric, the performance of considered variations of Hypercube Interconnection Networks is evaluated and summed up to recognize the effective variant. A compact survey of a portion of the variations of Hypercube systems, geographies, execution measurements, and assessment of the presentation are examined in this paper. Degree and Diameter are considered to ascertain the Network cost. On the off chance that Network Cost is considered as the measurement to assess the exhibition, Multiple Reduced Hypercube stands ideal with its lower cost. Notwithstanding it, on the off chance that we think about some other properties/ scales/metrics to assess the performance, any variant other than MRH may show considerably more ideal execution. The considered properties probably won't be ideally adequate to assess the effective performance of Hypercube variations in all respects. On the off chance that a sensibly decent number of properties are utilized to assess the presentation, a proficient variation of Hypercube Interconnection Network can be distinguished for a wide scope of uses. This is the inspiration to do this research work.


2002 ◽  
Vol 03 (01n02) ◽  
pp. 49-65 ◽  
Author(s):  
NADER F. MIR

A thorough routing analysis of a switching network called the spherical switching network for high-speed applications is presented in this paper. The spherical switching network has a cyclic, regular, and highly expandable structure with a simple self-routing scheme. The network is constructed with fixed-size switch elements regardless of the size of the network. Each switch element consists of a carefully-selected sized 9 input/output crossbar and a local controller. One of the nine pairs of links is external and carries the external traffic, and the other eight pairs are internal. The contention resolution in each switch element is based on deflection of losing packets and incremental priority of packets. The switch elements do not utilize any buffering within the network. The analysis shows that this network clearly outperforms typical interconnection networks currently being deployed in practical switches and routers such as Banyan network. In order to keep the number of deflections low, each incoming external link is connected to a buffer with flow control capabilities. Due to the special arrangement of interconnections in the network, a much larger number of shortest paths between each pair of source/destination exists. The related analysis for finding the number of hops and shortest paths appear in this paper.


2020 ◽  
Author(s):  
Vinícius De Souza Carvalho ◽  
Cândida Nunes Da Silva ◽  
Orlando Lee

 Given a digraph D, a coloring 𝒞 of D is a partition of V(D) into stable sets. The k-norm of 𝒞 is defined as ΣC∈𝒞 min{|C|, k}. A coloring of D with minimum k-norm has its k-norm noted by χk(D). A (path)-k-pack of a digraph D is a set of k vertex-disjoint (directed) paths of D. The weight of a k-pack is the number of vertices covered by the k-pack. We denote by λk(D) the weight of a maximum k-pack. Linial conjectured that χk(D) ≤ λk(D) for every digraph. Such conjecture remains open, but has been proved for some classes of digraphs. We prove the conjecture for path-spine digraphs, defined as follows. A digraph D is path-spine if there exists a partition {X, Y} of V(D) such that D[X] has a Hamilton path and every arc in D[Y] belongs to a single path Q. 


2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.


2013 ◽  
Vol 23 (06) ◽  
pp. 1521-1531 ◽  
Author(s):  
JONAH HOROWITZ

This paper examines the computational complexity of determining whether or not an algebra satisfies a certain Mal'Cev condition. First, we define a class of Mal'Cev conditions whose satisfaction can be determined in polynomial time (special cube term satisfying the DCP) when the algebra in question is idempotent and provide an algorithm through which this determination may be made. The aforementioned class notably includes near unanimity terms and edge terms of fixed arity. Second, we define a different class of Mal'Cev conditions whose satisfaction, in general, requires exponential time to determine (Mal'Cev conditions satisfiable by CPB0 operations).


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