Load Balancing in the Lp Norm

In the load balancing problem, there is a set of servers, and jobs arrive sequentially. Each job can be run on some subset of the servers, and must be assigned to one of them in an online fashion. Traditionally, the assignment of jobs to servers is measured by the L1 norm; in other words, an assignment of jobs to servers is quanti ed by the maximum load assigned to any server. In this measure the performance of the greedy load balancing algorithm may be a logarithmic factor higher than the o ine optimal [3]. In many applications, the L1 norm is not a suitable way to measure how well the jobs are balanced. If each job sees a delay that is proportional to the number of jobs on its server, then the average delay among all jobs is proportional to the sum of the squares of the numbers of jobs assigned to the servers. Minimizing the average delay is equivalent to minimizing the Euclidean (or L2) norm. For any xed p, 1 p <1, we show that the greedy algorithm performs within a constant factor of the o ine optimal with respect to the Lp norm. The constant grows linearly with p, which is best possible, but does not depend on the number of servers and jobs. Also a liated with MIT Lab. for Computer Science. Supported by ARPA/Army contract DABT63{93{C{0038, NSF grant CCR{9114440, and ARPA contract N00014{J{92{1799. ySupported in part by Allon Fellowship and by the Israel Science Foundation. Supported in part by ARO grant DAAH04{93{G{0076. xSupported in part by NSF grant CCR{9101385. {Supported in part by an IBM Fellowship, by NSF grant CCR{9007851, by ARO grant DAAH04{93{G{0076, and by AFOSR grant F49620{94{1{0217. This work was done while the author was visiting Duke University from Brown University. kSupported in part by NSF grant CCR{9007851 and by ARO grant DAAH04{93{G{0076.