Scalable Network Architectures Using the Optical Transpose Interconnection System (OTIS)

The Optical Transpose Interconnection System (OTIS) proposed by Marsden et al. (Opt. Lett.18, 13 (July 1993), 1083–1085) makes use of free-space optical interconnects to augment an electronic system by adding nonlocal interconnections. In this paper, we show how these connections can be used to implement a large-scale system with a given network topology using small copies of a similar topology. In particular, we show that, using OTIS, an N² node 4-D mesh can be constructed from N copies of the N-node 2-D mesh, an N² node hypercube can be constructed from N copies of the N-node hypercube, and an (N², α², c/2) expander can be constructed from N copies of an (N, α, c) expanders, all with small slowdown. Finally, we show how this expander construction can be used to build multibutterfly networks in a scalable fashion.     

The Hamiltonicity of swapped (OTIS) networks built of Hamiltonian component networks

A two-level swapped (also known as optical transpose interconnect system, or OTIS) network with n² nodes is built of n copies of an n-node basis network constituting its clusters. A simple rule for intercluster connectivity (node j in cluster i connected to node i in cluster j for all i≠j) leads to regularity, modularity, packageability, fault tolerance, and algorithmic efficiency of the resulting networks. We prove that a swapped network is Hamiltonian if its basis network is Hamiltonian. This general closure property for Hamiltonicity under swap or OTIS composition replaces a number of proofs in the literature for specific basis networks and obviates the need for proving Hamiltonicity for many other basis networks of potential practical interest.     

Fully symmetric swapped networks based on bipartite cluster connectivity

The class of swapped or OTIS (optical transpose interconnect system) networks, built of n copies of an n-node cluster by connecting node i in cluster j to node j in cluster i for i≠j, has been studied extensively. One problem with such networks is that node i of cluster i has no intercluster link. This slight asymmetry complicates a number of algorithms and hinders both theoretical investigations and practical pursuits, such as building parallel node-disjoint paths for fault tolerance. We introduce biswapped networks that are fully symmetric and have cluster connectivity very similar to swapped/OTIS networks. We derive basic topological parameters, present a simple distributed shortest-path routing algorithm, and point to a number of other interesting properties under investigation for biswapped networks.     

Optical transpose k-ary n-cube networks

This paper derives a number of results related to the topological properties of OTIS k-ary n-cube interconnection networks. The basic topological metrics of size, degree, shortest distance, and diameter are obtained. Then results related to embedding in OTIS k-ary n-cubes of OTIS k-ary (n−1)-cubes, cycles, meshes, cubes, and spanning trees are derived. The OTIS k-ary n-cube is shown to be Hamiltonian. Minimal one-to-one routing and optimal broadcasting algorithms are proposed. The OTIS k-ary n-cube is shown to be maximally fault-tolerant. These results are derived based on known properties of k-ary n-cube networks and general properties of OTIS networks.     

The panpositionable panconnectedness of augmented cubes

A graph G is panconnected if, for any two distinct vertices x and y of G, it contains an [x, y]-path of length l for each integer l satisfying d_G(x, y) ? l ? ∣V(G)∣ − 1, where d_G(x, y) denotes the distance between vertices x and y in G, and V(G) denotes the vertex set of G. For insight into the concept of panconnectedness, we propose a more refined property, namely panpositionable panconnectedness. Let x, y, and z be any three distinct vertices in a graph G. Then G is said to be panpositionably panconnected if for any d_G(x, z) ? l₁ ? ∣V(G)∣ − d_G(y, z) − 1, it contains a path P such that x is the beginning vertex of P, z is the (l₁ + 1)th vertex of P, and y is the (l₁ + l₂ + 1)th vertex of P for any integer l₂ satisfying d_G(y, z) ? l₂ ? ∣V(G)∣ − l₁ − 1. The augmented cube, proposed by Choudum and Sunitha [6] to be an enhancement of the n-cube Q_n, not only retains some attractive characteristics of Q_n but also possesses many distinguishing properties of which Q_n lacks. In this paper, we investigate the panpositionable panconnectedness with respect to the class of augmented cubes. As a consequence, many topological properties related to cycle and path embedding in augmented cubes, such as pancyclicity, panconnectedness, and panpositionable Hamiltonicity, can be drawn from our results.     

A kind of conditional fault tolerance of alternating group graphs

A vertex subset F is a k-restricted vertex-cut of a connected graph G if G−F is disconnected and every vertex in G−F has at least k good neighbors in G−F. The cardinality of the minimum k-restricted vertex-cut of G is the k-restricted connectivity of G, denoted by κk(G). This parameter measures a kind of conditional fault tolerance of networks. In this paper, we show that for the n-dimensional alternating group graph AGn, κ²(AG₄)=4 and κ²(AGn)=6n−18 for n?5.     

The OTIS hyper hexa-cell optoelectronic architecture

Optical transpose interconnection system (OTIS) is an optoelectronic architecture that promises to be a great choice for future-generation parallel systems. OTIS combines the advantages of electronic and optical links, where electronic links are used for short distances which require low material cost, and optical links are used for long distances which provide high speed network with low power consumption. Taking into account the advantageous characteristics of OTIS and based on the attractive properties of hyper hexa-cell (HHC) interconnection topology from low diameter and good minimum node degree, this paper introduces a new optoelectronic architecture referred to as OTIS hyper hexa-cell (OHHC). This paper also provides an evaluation and a comparison of the new topology with OTIS-mesh in terms of the following topological properties: size, diameter, maximum and minimum node degree, bisection width, total cost and optical cost. The results of this study proved the excellence of the proposed OHHC over OTIS-mesh in terms of diameter, minimum node degree, bisection width, and optical cost.     

Hamiltonian properties of twisted hypercube-like networks with more faulty elements

Twisted hypercube-like networks (THLNs) are a large class of network topologies, which subsume some well-known hypercube variants. This paper is concerned with the longest cycle in an n-dimensional (n-D) THLN with up to 2n−9 faulty elements. Let G be an n-D THLN, n≥7. Let F be a subset of V(G)?E(G), |F|≤2n−9. We prove that G−F contains a Hamiltonian cycle if δ(G−F)≥2, and G−F contains a near Hamiltonian cycle if δ(G−F)≤1. Our work extends some previously known results.     

Simple and Improved Parameterized Algorithms for Multiterminal Cuts

Given a graph G=(V,E) with n vertices and m edges, and a subset T of k vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k≥3 but well-known to be polynomial-time solvable for k=2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O(2 ^l kT(n,m)) time and Vertex Multiterminal Cut can be solved in O(k ^l T(n,m)) time, where T(n,m)=O(min?(n ^2/3,m ^1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k: Edge 3-Terminal Cut can be solved in O(1.415 ^l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059 ^l T(n,m)), O(2.772 ^l T(n,m)), O(3.349 ^l T(n,m)) and O(3.857 ^l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: $O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m))Given a graph G=(V,E) with n vertices and m edges, and a subset T of k vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k≥3 but well-known to be polynomial-time solvable for k=2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O(2 ^l kT(n,m)) time and Vertex Multiterminal Cut can be solved in O(k ^l T(n,m)) time, where T(n,m)=O(min (n ^2/3,m ^1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k: Edge 3-Terminal Cut can be solved in O(1.415 ^l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059 ^l T(n,m)), O(2.772 ^l T(n,m)), O(3.349 ^l T(n,m)) and O(3.857 ^l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: O((min(?{2k},l)+1)^2k2^lT(n,m))O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m)) -time algorithm for Edge Multicut and O((2k) ^k+l/2 T(n,m))-time algorithm for Vertex Multicut.     

Path bipancyclicity of hypercubes

A bipartite graph is vertex-bipancyclic (respectively, edge-bipancyclic) if every vertex (respectively, edge) lies in a cycle of every even length from 4 to |V(G)| inclusive. It is easy to see that every connected edge-bipancyclic graph is vertex-bipancyclic. An n-dimensional hypercube, or n-cube denoted by Qn, is well known as bipartite and one of the most efficient networks for parallel computation. In this paper, we study a stronger bipancyclicity of hypercubes. We prove that every n-dimensional hypercube is (2n−4)-path-bipancyclic for n?3. That is, for any path P of length k with 1?k?2n−4 and any integer l with max{2,k}?l?²n−1, an even cycle C of length 2l can be found in Qn such that the path P is included in C for n?3.     

Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs

A bipartite graph G is bipancyclic if G has a cycle of length l for every even 4?l?|V(G)|. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length l of G. In this paper, we show that the bubble-sort graph Bn is bipancyclic for n?4 and also show that it is edge-bipancyclic for n?5. Assume that F is a subset of E(Bn). We prove that Bn−F is bipancyclic, when n?4 and |F|?n−3. Since Bn is a (n−1)-regular graph, this result is optimal in the worst case.     

Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithms

Motivated by the research in reconfigurable memory array structures, this paper studies the complexity and algorithms for the constrained minimum vertex cover problem on bipartite graphs (min-cvcb) defined as follows: given a bipartite graph G=(V,E) with vertex bipartition V=U∪L and two integers k_u and k_l, decide whether there is a minimum vertex cover in G with at most k_u vertices in U and at most k_l vertices in L. It is proved in this paper that the min-cvcb problem is NP-complete. This answers a question posed by Hasan and Liu. A parameterized algorithm is developed for the problem, in which classical results in matching theory and recently developed techniques in parameterized computation theory are nicely combined and extended. The algorithm runs in time O(1.26^k_u+k_l+(k_u+k_l)|G|) and significantly improves previous algorithms for the problem.     

Partitioning a Weighted Tree into Subtrees with?Weights in a Given Range

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are given integers such that 0≤l≤u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such a partition is called an (l,u)-partition. We deal with three problems to find an (l,u)-partition of a given graph: the minimum partition problem is to find an (l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l,u)-partition with a given number p of components. All these problems are NP-hard even for series-parallel graphs, but are solvable in linear time for paths. In this paper, we present the first polynomial-time algorithm to solve the three problems for arbitrary trees.     

Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

Assume that n is a positive integer with n?2. It is proved that between any two different vertices x and y of Qn there exists a path Pl(x,y) of length l for any l with h(x,y)?l?n²−1 and 2|(l−h(x,y)). We expect such path Pl(x,y) can be further extended by including the vertices not in Pl(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that for any two vertices x and z from different partite set of n-dimensional hypercube Qn, for any vertex y∈V(Qn)−{x,z}, and for any integer l with h(x,y)?l?n²−1−h(y,z) and 2|(l−h(x,y)), there exists a hamiltonian path R(x,y,z;l) from x to z such that dR(x,y,z;l)(x,y)=l. Moreover, for any two distinct vertices x and y of Qn and for any integer l with h(x,y)?l?²n−1 and 2|(l−h(x,y)), there exists a hamiltonian cycle S(x,y;l) such that dS(x,y;l)(x,y)=l.     

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u,v∈V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC(u,v)=max{dC(x,y)|x,y∈V(C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to |V(G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices u,v∈V(G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG(u,v),3}?l?|V(G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249-1251] proved that the n-dimensional Augmented cube, AQn, is pancyclic in the sense that a cycle of length l exists, 3?l?|V(AQn)|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n?2.     

Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V(G)| inclusive. It has been shown that Q_n is bipancyclic if and only if n?2. In this paper, we improve this result by showing that every edge of Q_n−E′ lies on a cycle of every even length from 4 to |V(G)| inclusive where E′ is a subset of E(Q_n) with |E′|?n−2. The result is proved to be optimal. To get this result, we also prove that there exists a path of length l joining any two different vertices x and y of Q_n when h(x,y)?l?|V(G)|−1 and l−h(x,y) is even where h(x,y) is the Hamming distance between x and y.     

Super restricted edge connected Cartesian product graphs

An edge-cut F of a connected graph G is called a restricted edge-cut if G−F contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity λ^′(G) of G. A graph G is said to be λ^′-optimal if λ^′(G)=ξ(G), where ξ(G) is the minimum edge-degree of G. A graph is said to be super-λ^′ if every minimum restricted edge-cut isolates an edge. This article gives a sufficient condition for Cartesian product graphs to be super-λ^′. Using this result, certain classes of networks which are recursively defined by the Cartesian product can be simply shown to be super-λ^′.     

Restricted arc-connectivity of digraphs

Since interconnection networks are often modeled by graphs or digraphs, the edge-connectivity of a graph or arc-connectivity of a digraph are important measurements for fault tolerance of networks.The restricted edge-connectivity λ^′(G) of a graph G is the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G−S. A connected graph G is called λ^′-connected, if λ^′(G) exists.In 1988, Esfahanian and Hakimi [A.H. Esfahanian, S.L. Hakimi, On computing a conditional edge-connectivity of a graph, Inform. Process. Lett. 27 (1988), 195-199] have shown that each connected graph G of order n?4, except a star, is λ^′-connected and satisfies λ^′(G)?ξ(G), where ξ(G) is the minimum edge-degree of G.If D is a strongly connected digraph, then we call in this paper an arc set S a restricted arc-cut of D if D−S has a non-trivial strong component D₁ such that D−V(D₁) contains an arc. The restricted arc-connectivity λ^′(D) is the minimum cardinality over all restricted arc-cuts S.We observe that the recognition problem, whether λ^′(D) exists for a strongly connected digraph D is solvable in polynomial time. Furthermore, we present some analogous results to the above mentioned theorem of Esfahanian and Hakimi for digraphs, and we show that this theorem follows easily from one of our results.     

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R₁×R₂×?×Rk, where each Ri is a closed interval on the real line of the form [ai,ai+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph.It is known that for a graph G, . Recently it has been shown that for a graph G, cub(G)?4(Δ+1)lnn, where n and Δ are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G=(A∪B,E) with |A|=n₁, |B|=n₂, n₁?n₂, and Δ^′=min{ΔA,ΔB}, where ΔA=maxa∈Ad(a) and ΔB=maxb∈Bd(b), d(a) and d(b) being the degree of a and b in G, respectively, cub(G)?2(Δ^′+2)⌈lnn₂⌉. We also give an efficient randomized algorithm to construct the cube representation of G in 3(Δ^′+2)⌈lnn₂⌉ dimensions. The reader may note that in general Δ^′ can be much smaller than Δ.