We investigate balls-into-bins processes allocating m balls into n bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of d ≥ 2 randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when m ≈ n. In this paper we present the first tight analysis in the heavily loaded case, that is, when m ≫ n rather than m ≈ n. The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of m/n + O ( √ m ln n/n) with high probability. We show, however, that the multiple-choice processes are fundamentally different from the single-choice variant in that they have “short memory. ” The great consequence of this property is that the deviation of the multiple-choice processes from the optimal allocation (that is, the allocation in which each bin has either ⌊m/n ⌋ or ⌈m/n ⌉ balls) does not increase with the number of balls as in the case of the single-choice process. In particular, we investigate the allocation obtained by two different multiple-choice allocation schemes,

Abstract. We investigate randomized processes underlying load balancing based on the multiple-choice paradigm: m balls have to be placed in n bins, and each ball can be placed into one out of 2 randomly selected bins. The aim is to distribute the balls as evenly as possible among the bins. Previously, it was known that a simple process that places the balls one by one in the least loaded bin can achieve a maximum load of m/n + Θ(log log n) with high probability. Furthermore, it was known that it is possible to achieve (with high probability) a maximum load of at most ⌈m/n ⌉ +1using maximum flow computations. In this paper, we extend these results in several aspects. First of all, we show that if m ≥ cn log n for some sufficiently large c, thenaperfect distribution of balls among the bins can be achieved (i.e., the maximum load is ⌈m/n⌉) with high probability. The bound for m is essentially optimal, because it is known that if m ≤ c ′ n log n for some sufficiently small constant c ′ , the best possible maximum load that can be achieved is ⌈m/n ⌉ +1with high probability. Next, we analyze a simple, randomized load balancing process based on a local search paradigm. Our first result here is that this process always converges to a best possible load distribution. Then, we study the convergence speed of the process. We show that if m is sufficiently large compared to n,thenno matter with which ball distribution the system starts, if the imbalance is ∆, then the process needs only ∆·n O(1) steps to reach a perfect distribution, with high probability. We also prove a similar result for m ≈ n, and show that if m = O(n log n / log log n), then an optimal load distribution (which has the maximum load of ⌈m/n ⌉ +1) is reached by the random process after a polynomial number of steps, with high probability.

We compare the long-term, steady-state performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks, to the performance of a path-selection algorithm based on the “balanced-allocation ” principle [3, 17]; we refer to this new algorithm as the Balanced Dynamic Alternative Routing (BDAR) algorithm. While DAR checks alternative routes sequentially until available bandwidth is found, the BDAR algorithm compares and chooses the best among a small number of alternatives. We show that, at the expense of a minor increase in routing overhead, the BDAR algorithm gives a substantial improvement in network performance, in terms both of network congestion and of bandwidth requirement.

Multiple-choice allocation algorithms have been studied intensively over the last decade. These algorithms have several applications in the areas of load balancing, routing, resource allocation and hashing. The underlying idea is simple and can be explained best in the balls-and-bins model: Instead of assigning balls (jobs, requests, or keys) simply at random to bins (machines, servers, or positions in a hash table), choose first a small set of bins at random, inspect these bins, and place the ball into one of the bins containing the smallest number of balls among them. The simple idea of first selecting a small set of alternatives at random and then making the final choice after careful inspection of these alternatives leads to great improvements against algorithms that place their decisions simply at random. We illustrate the power of this principle in terms of simple balls-and-bins processes. In particular, we study recently presented algorithms that treat bins asymmetrically in order to obtain a better load balancing. We compare the behavior of these asymmetric schemes with symmetric schemes and prove that the asymmetric schemes achieve a better load balancing than their symmetric counterparts. 1

Let G(n, m) be an undirected random graph with n vertices and m multiedges that may include loops, where each edge is realized by choosing its two vertices independently and uniformly at random with replacement from the set of all n vertices. The random graph G(n, m) is said to be k-orientable, where k ≥ 2 is an integer, if there exists an orientation of the edges such that the maximum out-degree is at most k. Let ck = sup {c: G(n, cn) is k-orientable w.h.p.}. We prove that for k large enough, 1 − 2 k exp −k + 1 + e −k/4) < ck/k < 1 − exp

ABSTRACT. Wavelength division multiplexing (WDM) offers a solution to the problem of exploiting the large bandwidth on optical links; it is the current favorite multiplexing technology for optical communication networks. Due to the high cost of an optical amplifier, it is desirable to strategically place the amplifiers throughout the network in a way that guarantees that all the signals are adequately amplified while minimizing the total number amplifiers being used. Previous studies all consider a star-based network. This paper demonstrates an original approach for solving the problem in switch-based WDM optical network assuming the traffic matrix is always the permutation of the nodes. First we formulate the problem by choosing typical permutations which can maximize traffic load on individual links; then a GA (Genetic Algorithm) is used to search for feasible amplifier placements. Finally, by setting up all the lightpaths without violating the power constaints we confirm the feasibility of the solution. Keywords: WDM, Switch-based optical network,

) Petra Berenbrink Dept. of Mathematics & Computer Science Paderborn University D-33095 Paderborn, Germany pebe@uni-paderborn.de Artur Czumaj y Department of Computer and Information Science New Jersey Institute of Technology University Heights, Newark, NJ 07102-1982, USA czumaj@cis.njit.edu Tom Friedetzky Institut fur Informatik Technische Universitat Munchen D-80290 Munchen, Germany friedetz@informatik.tu-muenchen.de Nikita D. Vvedenskaya Institute of Information Transmission Problems Russian Academy of Science Moscow 101447, Russia ndv@iitp.ru Abstract In recent years, the task of allocating jobs to servers has been studied with the \balls and bins" abstraction. Results in this area exploit the large decrease in maximum load that can be achieved by allowing each job (ball) a little freedom in choosing its destination server (bin). In this paper we examine an innite and parallel allocation process (see [ABS98]) which is related to the \balls and bins" abs...

We compare the long-term, steady-state performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks, to the performance of a path-selection algorithm based on the "balanced-allocation" principle [Y. Azer, A. Z. Broder, A. R. Karlin, and E. Upfal, SIAM J Comput 29(1) (2000), 180--200; M. Mitzenmacher, Ph.D. Thesis, University of California, Berkeley, August 1996]; we refer to this new algorithm as the Balanced Dynamic Alternative Routing (BDAR) algorithm. While DAR checks alternative routes sequentially until available bandwidth is found, the BDAR algorithm compares and chooses the best among a small number of alternatives. We show that, at the expense of a minor increase in routing overhead, the BDAR algorithm gives a substantial improvement in network performance, in terms both of network congestion and of bandwidth requirement. 2005 Wiley Periodicals, Inc. Random Struct. Alg., 26, 446--467, 2005 1.

Abstract. We consider a random sequence of calls between nodes in a complete network with link capacities. Each call first tries the direct link. If that link is saturated, then the ‘first-fit dynamic alternative routing algorithm ’ sequentially selects up to d random two-link alternative routes, and assigns the call to the first route with spare capacity on each link, if there is such a route. The ‘balanced dynamic alternative routing algorithm ’ simultaneously selects d random two-link alternative routes; and the call is accepted on a route minimising the maximum of the loads on its two links, provided neither of these two links is saturated. We determine the capacities needed for these algorithms to route calls successfully, and find that the second ‘balanced ’ algorithm requires a much smaller capacity. Our results strengthen and extend the discrete-time results presented in [10]. 1.

We consider an online routing problem in continuous time, where calls have Poisson arrivals and exponential durations. The first-fit dynamic alternative routing algorithm sequentially selects up to d random two-link routes between the two endpoints of a call, via an intermediate node, and assigns the call to the first route with spare capacity on each link, if there is such a route. The balanced dynamic alternative routing algorithm simultaneously selects d random two-link routes; and the call is accepted on a route minimising the maximum of the loads on its two links, provided neither of these two links is saturated. We determine the capacities needed for these algorithms to route calls successfully, and find that the balanced algorithm requires a much smaller capacity. 1