Overlay multicast constructs a multicast delivery tree among end hosts. Unlike traditional IP multicast, the nonleaf nodes in the tree are normal end hosts, which are potentially more susceptible to failures than routers and may leave the multicast group voluntarily. In these cases, all downstream nodes will be affected. Thus an important problem in overlay multicast is how to recover from node departures in order to minimize the disruption of service to those affected nodes. In this paper, we propose a proactive approach to restore overlay multicast trees. Rather than letting downstream nodes try to find a new parent after a node departure, each non-leaf node precalculates a parent-to-be for each of its children. When this nonleaf node is gone, all its children can find their respective new parents immediately. The salient feature of the approach is that each non-leaf node can compute a rescue plan for its children independently, and in most cases, rescue plans from multiple non-leaf nodes can work together for their children when they fail or leave at the same time. We develop a protocol for nodes to communicate with new parents so that the delivery tree can be quickly restored. Extensive simulations demonstrate that our proactive approach can recover from node departures 5 times faster than reactive methods in some cases, and 2 times faster on average.

The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degree-constrained minimum spanning tree problem. For a weighted, undirected graph G(V, E), this problem seeks to identify the shortest spanning tree whose node degrees do not exceed an upper bound d 2. Within the EA, a candidate spanning tree is simply represented by its set of edges. Special initialization, crossover, and mutation operators are used to generate new, always feasible candidate solutions. In contrast to previous spanning tree representations, the proposed approach provides substantially higher locality and is nevertheless computationally efficient; an offspring is always created in O(|V time. In addition, it is shown how problemdependent heuristics can be effectively incorporated into the initialization, crossover, and mutation operators without increasing the time-complexity. Empirical results are presented for hard problem instances with up to 500 vertices. Usually, the new approach identifies solutions superior to those of several other optimization methods within few seconds. The basic ideas of this EA are also applicable to other network optimization tasks.

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using adoptions to meet the degree constraints is considered. A novel network-flow based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previously obtained. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds for any algorithm at all, then it also holds for our algorithm. The performance guarantee is the following. If the degree constraint d(v) for each v is at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 \Gamma min n d(v)\Gamma2 deg T (v)\Gamma2 : deg T (v) ? 2 o ; where deg T (v) is the initial degree of v. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. Choosing T to be a minimum spanning tree yields approximation algorithms for the general problem on geometric graphs with distances induced by various Lp norms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.

Designing a storage area network (SAN) fabric requires devising a set of hubs, switches and links to connect hosts to their storage devices. The network must be capable of simultaneously meeting specified data flow requirements between multiple host-device pairs, and it must do so cost-effectively, since large-scale SAN fabrics can cost millions of dollars. Given that the number of data flows can easily number in the hundreds, simple overprovisioned manual designs are often not attractive: they can cost significantly more than they need to, may not meet the performance needs, may expend valuable resources in the wrong places, and are subject to the usual sources of human error. Producing SAN fabric designs automatically can address these difficulties, but it is a non-trivial problem: it extends the NP-hard minimum-cost fixed-charge multicommodity network flow problem to include degree constraints, node capacities, node costs, unsplittable flows, and other requirements. Nonetheless, we present here two efficient algorithms for automatic SAN design. We show that these produce cost-effective SAN designs in very reasonable running times, and explore how the two algorithms behave over a range of design problems.

is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degree-constrained minimum spanning tree problem. For a connected, weighted graph, this problem seeks to identify the shortest spanning tree whose degree does not exceed an upper bound k 2. In the coding, chromosomes are strings of numerical weights associated with the target graph's vertices. The weights temporarily bias the graph's edge costs, and an extension of Prim's algorithm, applied to the biased costs, identifies the feasible spanning tree a chromosome represents. This decoding algorithm enforces the degree constraint, so that all chromosomes represent valid solutions and there is no need to discard, repair, or penalize invalid chromosomes. On a set of hard graphs whose unconstrained minimum spanning trees are of high degree, a genetic algorithm that uses this coding identifies degree-constrained minimum spanning trees that are on average shorter than those found by several competing algorithms.

The fundamental design choices in an evolutionary algorithm are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in evolutionary algorithms for network design problems directly as sets of their edges, and we describe initialization, recombination, and mutation operators for this representation. The operators offer

Abstract: Minimum Spanning Tree (MST) problem is of high importance in network optimization. The multi-criteria MST (mc-MST) is a more realistic representation of the practical problem in the real-world, but it is difficult for the traditional network optimiza- tion technique to deal with. In this paper, a genetic algorithm (GA) approach is developed to deal with this problem. Without neglecting its network topology, the proposed method adopts the Prfifer number as the tree encoding and applies the Multiple Criteria Decision Making (MCDM) technique and nondominated sorting technique to make the GA search give out all Pareto optimal solutions either focused on the region near the ideal point or distributed all along the Pareto frontier. Compared with the enumeration method of Pareto optimal solution, the numerical analysis shows the efficiency and effectiveness of the GA approach on the mc-MST problem.

. Given an undirected graph with weights associated with its edges, the degreeconstrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph, subject to constraints on node degrees. We propose a variable neighborhood search heuristic for the degree-constrained minimum spanning tree problem, based on a dynamic neighborhood model and using a variable neighborhood descent iterative improvement algorithm for local search. Computational experiments illustrating the effectiveness of the approach on benchmark problems are reported. Key words. Combinatorial optimization, degree-constrained minimum spanning tree, local search, metaheuristics, variable neighborhoods 1. Introduction. Let G = (V; E) be a connected undirected graph, where V is the set of nodes and E denotes the set of edges. Given a non-negative weight function w : E ! IR + associated with its edges and a non-negative integer valued degree function b : V ! IN associated with its nodes, th...

Finding the degree-constrained minimum spanning tree (d-MST) of a graph is a well studied NP-hard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the d-MST, either using heuristic or evolutionary approaches. The basis of this encoding is a spanning-tree construction algorithm which we call the Randomised Primal Method (RPM), based on the well-known Prim's algorithm [6], and an extension [4] which we call `d-Prim's'. We describe a novel encoding for spanning trees, which involves using the RPM to interpret lists of potential edges to include in the growing tree. We also describe a random graph generator which produces particularly challenging d-MST problems. On these and other problems, we find that an evolutionary algorithm (EA) using the RPM encoding outperforms the previous best published technique from the operations research literature, and also outperforms simulated...

Finding the degree-constrained minimum spanning tree (d-MST) of a graph is a wellstudied NP-hard problem of importance in communications network design and other network-related problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a novel tree construction algorithm called the Randomised Primal Method (RPM) which builds degree-constrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multi-start hillclimbing, and a genetic algorithm. While other researchers have mainly concentrated on finding spanning trees in Euclidean graphs, we consider the more general case of random graph problems. We describe two random graph generators which produce particularly challenging d-MST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multi-start hillclimbing. Our experimental ...