A parallel task is one that can be executed by multiple processors. All the processors allotted to a task are required to execute that task in unison and simultaneously. A non-malleable parallel task is one that requires a specific number of processors for a specific units of time, while a malleable task is one that can be executed on any number of processors, with its execution time being a function of the number of processors allotted to it. A malleable task is said to have a linear speedup if the task execution time describes a linear speedup up to some specified maximum degree of parallelism, and is constant thereafter.
This thesis deals three different problems, and presents approximation algorithms for solving each of the problems. We first consider the problem of minimizing makespan on hypercube systems. For non-malleable tasks, we propose an algorithm with an approximation factor of 1.875. This is the first algorithm achieving an approximation factor less than 2 by a constant for this problem. For non-malleable tasks with linear speedups, we propose another algorithm with an approximation factor of $\frac{5}{3}$ We also consider on-line scheduling problems, and present some partial results.
Second, we consider the parallel tasks which are associated with individual deadlines on a PRAM. The objective is to maximize the sum of the works of the tasks that are completed before their deadlines in the schedule. For non-malleable tasks, we present an algorithm with an approximation factor of 5+ε epsilon for any constant epsilon ε > 0, and, for malleable tasks with linear speedups, present another algorithm with an approximation factor of 4.5.
Third, we introduce a new kind of parallel tasks of which the execution times depend on not only the number of processors but also the processors themselves on which the tasks are to be executed. This problem captures such aspects of the real systems as the non-homogeneity of the processors or the data locality between the processors. We present approximation algorithms for solving this problem on the several parallel systems including PRAMs, 1-dimensional meshes, and 2-dimensional meshes.
