Summary: Ancient and new algorithms for load balancing in the L p norm
Adi Avidor \Lambda Yossi Azar y Jir'i Sgall z
July 7, 1997
We consider the online load balancing problem where there are m identical machines
(servers) and a sequence of jobs. The jobs arrive one by one and should be assigned
to one of the machines in an online fashion. The goal is to minimize the sum over all
machines of the squares of the loads, instead of the traditional maximum load.
We show that for the sum of the squares the greedy algorithm performs within 4=3 of
the optimum, and no online algorithm achieves a better competitive ratio. Interestingly,
we show that the performance of greedy is not monotone in the number of machines.
More specifically, the competitive ratio is 4=3 for any number of machines divisible by
3 but strictly less than 4=3 in all the other cases (although it approaches 4=3 for large
number of machines). To prove that greedy is optimal, we show a the lower bound of 4=3
for any algorithm for 3 machines. Surprisingly, we provide a new online algorithm that
performs within 4=3 \Gamma ffi of the optimum, for some fixed ffi ? 0, for any sufficiently large
number of machines. This implies that the asymptotic competitive ratio of our new
algorithm is strictly better than the competitive ratio of any possible online algorithm.
Such phenomena is not known to occur for the classic maximum load problem.
Minimizing the sum of the squares is equivalent to minimizing the load vector with