Showing papers in "Networks in 2005"

Design of Survivable Networks: A survey

Abstract: For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the advances of optical technologies and the explosive growth of the Internet, telecommunication networks have seen an important evolution and therefore designing survivable networks has become a major objective for telecommunication operators. Over the past years, much research has been carried out to devise efficient methods for survivable network models, and particularly cutting plane based algorithms. In this paper, we attempt to survey some of these models and the optimization methods used for solving them. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 1–21 2005

A hybrid genetic algorithm for the weight setting problem in OSPF-IS-IS routing

Abstract: Intradomain traffic engineering aims to make more efficient use of network resources within an autonomous system. Interior Gateway Protocols such as OSPF (Open Shortest Path First) and IS-IS (Intermediate System-Intermediate System) are commonly used to select the paths along which traffic is routed within an autonomous system. These routing protocols direct traffic based on link weights assigned by the network operator. Each router in the autonomous system computes shortest paths and creates destination tables used to direct each packet to the next router on the path to its final destination. Given a set of traffic demands between origin-destination pairs, the OSPF weight setting problem consists of determining weights to be assigned to the links so as to optimize a cost function, typically associated with a network congestion measure. In this article, we propose a genetic algorithm with a local improvement procedure for the OSPF weight-setting problem. The local improvement procedure makes use of an efficient dynamic shortest path algorithm to recompute shortest paths after the modification of link weights. We test the algorithm on a set of real and synthetic test problems, and show that it produces near-optimal solutions. We compare the hybrid algorithm with other algorithms for this problem illustrating its efficiency and robustness. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 36–56 2005(This research was done while the first author was a visiting scholar at the Internet and Network Systems Research Center at AT&T Labs Research (AT&T Labs Research Technical Report TD-5NTN5G).)

An approximation algorithm for Stackelberg network pricing

Abstract: We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of ${{1 \over 2}\log_2 m_T+1}$, where mT denotes the number of toll arcs. Finally, we show that the approximation is tight with respect to a natural relaxation by constructing a family of instances for which the relaxation gap is reached. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 57–67 2005

Strong formulations for network design problems with connectivity requirements

Abstract: The network design problem with connectivity requirements lNDCr includes as special cases a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edge-connectivity NDC problem: unitary problems requiring connected network designs, and nonunitary problems permitting nonconnected networks as solutions. We l1r present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation; l2r project out two classes of valid inequalities—partition inequalities, and combinatorial design inequalities—that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem; and l3r show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the power of flow-based formulations for network design problems with connectivity requirements. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45l2r, 61–79 2005

A network flow algorithm to minimize beam-on time for unconstrained multileaf collimator problems in cancer radiation therapy

Abstract: In this article, we study the modulation of intensity matrices arising in cancer radiation therapy using multileaf collimators. This problem can be formulated by decomposing a given m × n integer matrix into a positive linear combination of l0, 1r matrices with the strict consecutive 1's property in rows. We consider a special case in which no technical constraints have to be taken into account. In this situation, the rows of the intensity matrix are independent of each other and the problem is equivalent to decomposing m intensity rows—independent of each other—into positive linear combinations of l0, 1r rows with the consecutive 1's property. We demonstrate that this problem can be transformed into a minimum cost flow problem in a directed network that has the following special structures: l1r the network is acyclic; l2r it is a complete graph lthat is, there is an arc li, jr whenever i < jr; l3r each arc cost is 1; and l4r each arc is uncapacitated lthat is, it has infinite capacityr. We show that using this special structure, the minimum cost flow problem can be solved in Olnr time. Because we need to solve m such problems, the total running time of our algorithm is Olnmr, which is an optimal algorithm to decompose a given m × n integer matrix into a positive linear combination of l0, 1r matrices. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45l1r, 36–41 2005

A branch-and-price algorithm for the capacitatedp-median problem

Abstract: The capacitated p-median problem is the variation of the well-known p-median problem in which a demand is associated to each user, a capacity is associated to each candidate median, and the total demand of the users associated to the same median must not exceed its capacity. We present a branch-and-price algorithm, that exploits column generation, heuristics and branchand-bound to compute optimal solutions. We compare our branch-and-price algorithm with other methods proposed so far, and we present computational results both on test instances taken from the literature and on random instances with different values of the ratio between the number of medians and the number of users. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(3), 125–142 2005

A branch-and-price algorithm for the capacitated p-median problem

Abstract: The capacitated p-median problem is the variation of the well-known p-median problem in which a demand is associated to each user, a capacity is associated to each candidate median, and the total demand of the users associated to the same median must not exceed its capacity. We present a branch-and-price algorithm, that exploits column generation, heuristics and branch-and-bound to compute optimal solutions. We compare our branch-and-price algorithm with other methods proposed so far, and we present computational results both on test instances taken from the literature and on random instances with different values of the ratio between the number of medians and the number of users. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45l3r, 125–142 2005

Near-shortest and K-shortest simple paths

Abstract: We present a new algorithm for enumerating all near-shortest simple (loopless) s-t paths in a graph G = (V, E) with nonnegative edge lengths. Letting n = |V| and m = |E|, the time per path enumerated is O(nS(n, m)) given a user-selected shortest-path subroutine with complexity O(S(n, m)). When coupled with binary search, this algorithm solves the corresponding K-shortest paths problem (KSPR) in O(KnS(n, m)(log n+ log cmax)) time, where cmax is the largest edge length. This time complexity is inferior to some other algorithms, but the space complexity is the best available at O(m). Both algorithms are easy to describe, to implement and to extend to more general classes of graphs. In computational tests on grid and road networks, our best polynomial-time algorithm for KSPR appears to be at least an order of magnitude faster than the best algorithm from the literature. However, we devise a simpler algorithm, with exponential worst-case complexity, that is several orders of magnitude faster yet on those test problems. A minor variant on this algorithm also solves “KSPU,” which is analogous to KSPR but with loops allowed. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 98–109 2005

A branch-and-price approach for the maximum weight independent set problem

Abstract: The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied problems in combinatorial optimization. This article presents a novel approach to solve MWISP exactly by decomposing the original graph into vertex-induced subgraphs. The approach solves MWISP for the original graph by solving MWISP on the subgraphs to generate columns for a branch-and-price framework. The authors investigate different implementation techniques that can be associated with the approach, and offer computational results to identify the strengths and weaknesses of each implementation technique. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(4), 198–209 2005

Fixed-parameter tractability and data reduction for multicut in trees

Abstract: We study an NP-complete (and MaxSNP-hard) communication problem on tree networks, the so-called MULTICUT IN TREES: given an undirected tree and some pairs of nodes of the tree, find out whether there is a set of at most k tree edges whose removal separates all given pairs of nodes. MULTICUT has been intensively studied for trees as well as for general graphs mainly from the viewpoint of polynomial time approximation algorithms. By way of contrast, we provide a simple fixed-parameter algorithm for MULTICUT IN TREES showing fixed-parameter tractability with respect to parameter k. Moreover, based on some polynomial time data reduction rules, which appear to be of particular interest from an applied point of view, we show a problem kernel for MULTICUT IN TREES by an intricate mathematical analysis. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(3), 124–135 2005

Adversarial models for priority-based networks

Abstract: In this article, we propose several variations of the adversarial queueing model and address stability issues of networks and protocols in those proposed models. The first such variation is the priority model, which is directed at static network topologies and takes into account the case in which packets can have different priorities. Those priorities are assigned by an adversary at injection time. A second variation, the variable priority model, is an extension of the priority model in which the adversary may dynamically change the priority of packets at each time step. Two more variations, namely the failure model and the reliable model, are proposed to cope with dynamic networks. In the failure and reliable models the adversary controls, under different constraints, the failures that the links of the topology might suffer. Concerning stability of networks in the proposed adversarial models, we show that the set of universally stable networks in the adversarial model remains the same in the priority, variable priority, failure, and reliable models. From the point of view of protocols lor queueing policiesr, we show that several protocols that are universally stable in the adversarial queueing model remain so in the priority, failure, and reliable models. However, we show that the longest-in-system lLISr protocol, which is universally stable in the adversarial queueing model, is not universally stable in any of the other models we propose. Moreover, we show that no queueing policy is universally stable in the variable priority model. Finally, we analyze the problem of deciding stability of a given network under a fixed protocol. We provide a characterization of the networks that are stable under first-in-first-out lFIFOr and LIS in the failure model land therefore in the reliable and priority modelsr. This characterization allows us to show that the stability problem under FIFO and LIS in the failure model can be solved in polynomial time. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45l1r, 23–35 2005Part of this work was presented in 28th International Symposium on Mathematical Foundations of Computer Science, Bratislava, Slovakia, 2003. LNCS 2747:142–151, Springer.

On restricted connectivities of permutation graphs

Abstract: A permutation graph lor generalized prismr Gπ of a graph G is obtained by taking two disjoint copies of G and adding an arbitrary matching between the two copies. Permutation graphs can be seen as suitable models for building larger interconnection networks from smaller ones without increasing significantly their maximum transmission delays, in such a way that these larger networks are highly fault-tolerant. For permutations graphs, in this article we provide conditions that guarantee optimal values for two parameters of connectivity, λ′ and κ′. For a connected graph G the restricted edge-connectivity λ′lGr is defined as the minimum cardinality of a restricted edge-cut; that is, the minimum cardinality of a set S of edges such that G - S is not connected and S does not contain the set of incident edges of any vertex of the graph. A graph G is said to be λ′-optimal if λ′lGr e ξlGr, where ξlGr is the minimum edge-degree in G defined as ξlGr e minldlur p dlvr - 2 : uv ∈ ElGrr, and dlur denotes the degree of vertex u. Among other things, we prove that permutation graphs satisfy: minlλ′lGr p δlGr, 2λ′lGr, ξlGπrr ≤ λ′lGπr ≤ ξlGπr if vVlGr v ≥ ξlGr p 2. Furthermore, minl 2λ′lGr, ξlGπrr ≤ λ′lGπr ≤ ξlGπr if G is triangle-free. We also study the vertex case considering the restricted connectivity κ′lGr and relating it to the superconnectivity κ1lGr; the latter is defined as the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has at least two vertices. For instance, we prove that 2κlGr ≤ κ1lGr ≤ κ′lGπr ≤ ξlGπr if G is triangle-free and the permutation graph has no cycles of length five. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45l3r, 113–118 2005

Mutually independent hamiltonian paths in star networks

Abstract: Two hamiltonian paths P1 = 〈u1, u2,…,un(G)〉 and P2 = 〈v1, v2,…,vn(G)〉 of G from u to v are independent if u = u1 = v1, v = vn(G) = un(G), and vi ≠ ui for every 1 < i < n(G). A set of hamiltonian paths, lP1, P2,…,Pkr, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two nodes from different partite sets. A bipartite graph is k-mutually independent hamiltonian laceable if there exists k-mutually independent hamiltonian paths between any two nodes from distinct partite sets. The mutually independent hamiltonian laceability of a bipartite graph G, IHPL(G), is the maximum integer k such that G is k-mutually independent hamiltonian laceable. Let Sn denote the n-dimensional star graph. We prove that IHPL(S2) = 1, IHPL(S3) = 0, and IHPL(Sn) = n- 2 if n ≥ 4. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 110–117 2005

Minimal feedback vertex sets in directed split-stars

Abstract: In a graph G = (V,E), a subset F ⊂ V(G) is a feedback vertex set of G if the subgraph induced by V(G)bF is acyclic. In this article, we propose an algorithm for finding minimal feedback vertex sets of directed split-stars. Indeed, our algorithm can derive an upper bound on the size of the minimum feedback vertex set in directed split-stars. Moreover, a simple distributed algorithm is presented for obtaining such sets. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 218–223 2005

Matchings in 3-vertex-critical graphs: The even case

Abstract: A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G)and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G - v) < γ(G), for every v vertex in G. Comparatively little is known to date about the structure of γ-vertex-critical graphs, even in the case when γ = 3. In the present article, we begin the study of matchings in 3-vertex-critical graphs. In particular, we show that any 3-vertex-critical graph on an even number of vertices, which has no induced subgraph isomorphic to the bipartite graph K1,5 much have a perfect matching, whereas 3-vertex-critical even graphs in general need not contain such a matching. We close with a conjecture. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 210–213 2005

Connectedness of digraphs and graphs under constraints on the conditional diameter

Abstract: Given a digraph G with minimum degree δ and an integer 0≤ ν ≤ δ, consider every pair of vertex subsets V1 and V2 such that both the minimum out-degree of the induced subdigraph GlV1r and the minimum in-degree of GlV2r are at least ν. The conditional diameter Dν of G is defined as the maximum of the distances dlV1, V2r between any two such vertex subsets. Clearly, D0 is the standard diameter and D0 ≥ D1 ≥ ··· ≥ Dδ holds. In this article, we guarantee appropriate lower bounds for the connectivities and superconnectivities of a digraph G when Dν ≤ hleπr, hleπr being a function of the parameter eπ—which is related to the shortest paths in G. As a corollary of these results, we give some constraints of the kind Dν ≤ hleπr, which assure that the digraph is maximally connected, maximally edge-connected, superconnected, or edge-superconnected, extending other previous results of the same kind. Similar statements can be obtained for a graph as a direct consequence of those for its associated symmetric digraph. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45l2r, 80–87 2005

Greedy approximation algorithms for directed multicuts

Abstract: The Directed Multicut (DM) problem is: given a simple directed graph G = (V, E) with positive capacities ue on the edges, and a set K ⊆ V × V of ordered pairs of nodes of G, find a minimum capacity K-multicut; C ⊆ E is a K-multicut if in G - C there is no (s, t)-path for any (s, t) s K. In the uncapacitated case (UDM) the goal is to find a minimum size K-multicut. The best approximation ratio known for DM is $O(\min\{\sqrt{n},opt\})$ by Gupta, where n = |V|, and opt is the optimal solution value. All known nontrivial approximation algorithms for the problem solve large linear programs. We give the first combinatorial approximation algorithms for the problem. Our main result is an O(n2s3sopt1s3)-approximation algorithm for UDM, which improves the $O(\min\{opt,\sqrt{n}\})$-approximation for opt = Ω(n1s2+e). Combined with the article of Gupta, we get that UDM can be approximated within better than $O(\sqrt n)$, unless $opt={\tilde \Theta}(\sqrt n)$. We also give a simple and fast O(n2s3)-approximation algorithm for DM. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 214–217 2005

Balanced paths in acyclic networks: Tractable cases and related approaches

Abstract: Given a weighted acyclic network G and two nodes s and t in G, we consider the problem of computing k balanced paths from s to t, that is, k paths such that the difference in cost between the longest and the shortest path is minimized. The problem has several variants. We show that, whereas the general problem is solvable in pseudopolynomial time, both the arc-disjoint and the node-disjoint variants li.e., the variants where the k paths are required to be arc-disjoint and node-disjoint, respectivelyr are strongly NP-Hard. We then address some significant special cases of such variants, and propose exact as well as approximate algorithms for their solution. The proposed approaches are also able to solve versions of the problem in which k origin-destination pairs are provided, and a set of k paths linking the origin-destination pairs has to be computed in such a way to minimize the difference in cost between the longest and the shortest path in the set. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45l2r, 104-111 2005

A simple protocol for deniable authentication based on ElGamal cryptography

Abstract: Deniable authentication is different from traditional authentication in that the receiver cannot prove the source of a given message to a third party. This article presents a new deniable authentication protocol, based on ElGamal cryptography. The protocol meets the two properties of deniable authentication and can withstand person-in-middle (PIM) attacks. Compared with previous schemes, it is simpler and more efficient. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 193–194 2005

Nonblocking conditions of multicast three-stage interconnection networks

Abstract: This article considers three-stage switching networks able to support multicast traffic, that is, connections in which one inlet is connected to more than one output at the same time. The nonblocking conditions for this network are studied under the assumption of absence of any optimized routing of the connections inside the structure (the so-called strict-sense nonblocking networks). The theoretical nonblocking condition for such network under point-to-point traffic is the well known Clos condition. We give here the necessary and sufficient conditions for such network to be strict-sense nonblocking under multicast traffic. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(4), 163–170 2005

Efficient neighborhood search for the Probabilistic Pickup and Delivery Travelling Salesman Problem

Abstract: In this article the probabilistic Pickup and Delivery Travelling Salesman Problem is studied and an efficient neighborhood search procedure is developed. This procedure requires O(n3) computations for the evaluation of the neighborhood of a given solution while the straightforward approach has an O(n5)complexity. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 195–198 2005

Minimizing the cost of placing and sizing wavelength division multiplexing and optical crossconnect equipment in a telecommunications network

Abstract: Cost reduction is a major concern when designing optical fiber networks. Multiwavelength optical devices are new technology for increasing the capacity of fiber networks while reducing costs, when compared to installing traditional (e.g., SONET) equipment and new fiber. In this article we discuss the development of a metaheuristic method that seeks to optimize the location of Wavelength Division Multiplexing (WDM) and Optical Crossconnect (OXC) equipment in fiber networks. The procedure combines ideas from the scatter search, tabu search, and multistart methodologies. Computational experiments with both real-world and artificial data show the effectiveness of the proposed procedure. The experiments include a comparison with a permutation-based approach and with lower bounds generated with CPLEX. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 199–209 2005

Graphs, branchwidth, and tangles! Oh my!

Abstract: Branch decomposition-based algorithms have been used in practical settings to solve some NP-hard problems like the travelling salesman problem lTSPr and general minor containment. The notions of branch decompositions and branchwidth were introduced by Robertson and Seymour to assist in proving the Graph Minors Theorem. Given a connected graph G and a branch decomposition of G of width k where k is at least 3, a practical branch decomposition-based algorithm to test whether a graph has branchwidth at most k - 1 is given. The algorithm either constructs a branch decomposition of G of width at most k - 1 or constructs a tangle basis of order k, which offers a lower bound on the branchwidth of G. The algorithm is utilized repeatedly in a practical setting to find an optimal branch decomposition of a connected graph, whose branchwidth is at least 2, given an input branch decomposition of the graph from a heuristic. This is the first algorithm for the optimal branch decomposition problem for general graphs that has been shown to be practical. Computational results are provided to illustrate the effectiveness of finding optimal branch decompositions. A tangle basis is related to a tangle, a notion also introduced by Robertson and Seymour; however, a tangle basis is more constructive in nature. Furthermore, it is shown that a tangle basis of order k is coextensive to a tangle of order k. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45l2r, 55–60 2005

Competitive routing in multicast communications

Abstract: We consider competitive routing in multicast networks from a noncooperative game theoretical perspective. There are N users sharing a network, and each has to send a quantity of packets to a different set of addressees (each address must receive the same packets). To do this the user has only to send one copy of a packet, the network making the duplications of the packets at appropriate nodes (depending on the chosen trees). The routing choice of a user is how to split its flow between different multicast trees. We present different criteria for optimization of this type of game. We treat two specific networks and establish the uniqueness of the Nash equilibrium in these networks, as well as the uniqueness of link utilization at Nash equilibria for specific cost functions in networks with general topology. We also present a result for convergence to equilibria from an initial nonequilibrium state. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 22–35 2005

Optimal broadcasting with universal lists based on competitive analysis

Abstract: In this article we study a variant of broadcasting: each node has a predetermined ordered list of neighbors regardless of the node, called the source, from which the originating message is transmitted to all nodes in a network. Each node transmits a received message to its neighbors in order of the list. We propose a new measure of the efficiency of a Broadcasting scheme, which is obtained from competitive analysis, and we design new broadcasting schemes for lines, complete k-ary trees, grids, complete graphs, and hypercubes. In particular, we provide optimal broadcasting schemes for lines and grids. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 224–231 2005A preliminary version of this article appeared in the Proceedings of the 12th ISAAC Conference, Lecture Notes in Computer Science, vol. 2223, 74-85, 2001.

Dynamic programming algorithms for the conditional covering problem on path and extended star graphs

Abstract: The Conditional Covering Problem (CCP) is a facility location problem on a graph, where the set of nodes represents demand points and potential facility locations. The key aspect of the CCP is that each facility covers all nodes within a given facility-specific coverage radius, except for the node at which it is located. The objective of this problem is to minimize the sum of the facility location costs required to cover all demand points. We first discuss the worst-case complexity of the CCP by examining literature related to the total domination problem, which is a special case of the CCP. Next, we examine the special case of path graphs and provide an O(n2) algorithm for its solution. Finally, we leverage information obtained from this procedure to derive an optimal algorithm for “extended star” graphs (multiple paths having one node in common), without increasing the worst-case complexity of the algorithm. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(4), 177–185 2005

Maximal vertex-connectivity of

Abstract: The class of star graphs is a popular topology for interconnection networks. However, it has certain deficiencies. A class of generalization of star graphs called (n, k)-star graphs was introduced by Chiang and Chen to address these issues. In this article we will consider the vertex-connectivity of the directed (n, k)-star graph, , given by Cheng and Lipman, 8, and show that it is maximally connected. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(3), 154–162 2005

The strongly connected reliability of complete digraphs

Abstract: Given a digraph D, consider the model where each vertex is always operational, but the edges are independently operational with probability p. The strongly connected reliability of D, scRel(D,p), is the probability that the spanning subgraph of D consisting of the operational edges is strongly connected. One can view strongly connected reliability as the probability that any vertex can send information to any other vertex, given that edges fail independently. There are very few classes for which there is an efficient algorithm for calculating the strongly connected reliability. This article presents the fist polynomial time algorithm for computing the strongly connected reliability of complete digraphs, that is, digraphs in which every vertex is joined to every other vertex by exactly one edge (one in each direction). © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(3), 165168 2005

Reliable broadcasting in double loop networks

Abstract: Broadcast is the fundamental collective communication routine in which the same message is delivered from a single source to all the nodes in a network. The most efficient way to implement broadcast is through the construction of a broadcast tree. We introduce a revised definition of optimality for the i-port model broadcast tree based on fault-tolerance, and we construct optimal broadcast trees for interconnection networks modeled by Double Loop-Graphs for 1 ≤ i ≤ 4. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 88–97 2005

On large (Δ, 6)-Graphs

Abstract: In this article, a new method is proposed for obtaining large-diameter 6 graphs by replacing some vertices of a Moore bipartite diameter 6 graph by complete Kh graphs. These complete graphs are joined to the remaining nonmodified graph and to each other by means of new edges. More precisely, these new edges are those of a bipartite diameter 3 graph, together with some appropriate extra edges. The degree and the diameter of the graph so constructed coincide with those of the original one. The construction put forward here gives rise to some of the largest known diameter 6 graphs. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 82–87 2005