Recall that in the basic Load Balancing Problem from Section 11.1, were interested in placing jobs on machines so as to minimize the makespan the maximum load on any one machine. In a number of applications, it is natural to consider cases in which you have access to machines with different amounts of processing power, so that a given job may complete more quickly on one of your machines than on another. The question then becomes: How should you allocate jobs to machines in these more heterogeneous systems?

Heres a basic model that exposes these issues. Suppose you have

a system that consists of m slow machines and k fast machines. The

fast machines can perform twice as much work per unit time as the

slow machines. Now youre given a set of n jobs; job i takes time ti to

process on a slow machine and time 1t to process on a fast machine. 2i

You want to assign each job to a machine; as before, the goal is to minimize the makespanthat is the maximum, over all machines, of the total processing time of jobs assigned to that machine.

Give a polynomial-time algorithm that produces an assignment of jobs to machines with a makespan that is at most three times the opti- mum.