{jcrichigno, bbaran}@cnc.una.py http://www.una.py. Abstract. Multicast routing ..... A. Tanenbaum, Computer Networks, Prentice Hall, 2003. 2. V. Kompella, J.
A Multicast Routing Algorithm Using Multiobjective Optimization Jorge Crichigno, Benjamín Barán P. O. Box 1439 - Universidad Nacional de Asunción Asunción –Paraguay. Tel/Fax: (+595-21) 585619 {jcrichigno, bbaran}@cnc.una.py http://www.una.py
Abstract. Multicast routing problem in computer networks, with more than one objective to consider, like cost and delay, is usually treated as a mono-objective Optimization Problem, where the cost of the tree is minimized subject to a priori restrictions on the delays from the source to each destination. This paper presents a new multicast algorithm based on the Strength Pareto Evolutionary Algorithm (SPEA), which simultaneously optimizes the cost of the tree, the maximum end-to-end delay and the average delay from the source node to each destination node. Simulation results show that the proposed algorithm is able to find Pareto optimal solutions. In addition, they show that for the problem of minimum cost with constrained end-to-end delay, the proposed algorithm provides better solutions than other well-known alternatives as Shortest Path and KPP algorithms.
1
Introduction
Multicast consists of concurrent data transmission from a source to a subset of all possible destinations in a computer network [1]. In recent years, multicast routing algorithms have become more important due the increased use of new point to multipoint applications, such as radio and TV transmission, on-demand video, teleconferences and e-learning. Such applications have an important quality-ofservices (QoS) parameter, which is the end-to-end delay along the individual paths from the source to each destination. Another important consideration in multicast routing is the cost tree. It is given by the sum of the costs of its links. Most algorithms dealing with cost of a tree and delay from source to each destination, address multicast routing as a mono-objective optimization problem, minimizing the cost subjected to an end-to-end delay restriction. In [2], Kompella et al. present an algorithm (KPP) based on dynamic programming that minimizes the cost of the tree with a bounded end-to-end delay. For the same problem, Ravikumar et al. [3] present a method based on a simple genetic algorithm. This work was improved in turn by Zhengying et al. [4] and Araujo et al. [5]. The main disadvantage with this approach is the necessity of an a priory upper bound for the delay that may discard solutions of very low cost with a delay only slightly larger than a predefined upper bound. In contrast to the mono-objective algorithms, a MultiObjective Evolutionary Algorithm
(MOEA) simultaneously optimizes several objective functions; therefore, they can consider end-to-end delay as a new objective function. Multiobjective Evolutionary Algorithms provide a way to solve a multiobjective problem (MOP), finding a whole set of Pareto solutions in only one run [6]. This paper presents a new approach to solve the multicast routing problem based on a MOEA called the Strength Pareto Evolutionary Algorithm (SPEA) [6]. The remainder of this paper is organized as follow. A general definition of a multiobjective optimization problem is presented in Section 2. The problem formulation and the objective functions are given in Section 3. The proposed algorithm is explained in Section 4. Experimental results are shown in Section 5. Finally, the conclusions are presented in Section 6.
2
Multiobjective Optimization Problem
A general Multiobjective Optimization Problem (MOP) includes a set of n decision variables, k objective functions, and m restrictions. Objective functions and restrictions are functions of decision variables. This can be expressed as: Optimize y = f(x) = (f1(x), f2(x), ..., fk(x)) Subject to e(x) = (e1(x), e2(x), ... ,em(x)) ≥ 0 where x = (x1, x2, ..., xn) ∈ X is the decision vector, and y = (y1, y2, ... , yk) ∈ Y is the objective vector. X denotes the decision space while the objective space is denoted by Y. The set of restrictions e(x) ≥ 0 determines the set of feasible solutions Xf and its corresponding set of objective vectors Yf. The problem consists of finding x that optimizes f(x). In general, there is no unique “best” solution but a set of solutions. Thus, a new concept of optimality is established for MOPs. Given u, v ∈ X, f(u)=f(v) iff ∀i ∈ {1,2,...,k}: fi(u)=fi(v); f(u) ≤ f(v) iff ∀i ∈ {1,2,...,k}: fi(u) ≤ fi(v); f(u)
