NP-complete problems form an extensive equivalence class of combinatorial problems for which no nonenumerative algorithms are known. Our first result shows that determining a shortest-length schedule in an m-machine flowshop is NP-complete for m ≥ 3. (For m = 2, there is an efficient algorithm for finding such schedules.) The second result shows that determining a minimum mean-flow-time schedule in an m-machine flowshop is NP-complete for every m ≥ 2. Finally we show that the shortest-length schedule problem for an m-machine jobshop is NP-complete for every m ≥ 2. Our results are strong in that they hold whether the problem size is measured by number of tasks, number of bits required to express the task lengths, or by the sum of the task lengths.
Sequencing to minimize mean finishing time (or mean time in system) is not only desirable to the user, but it also tends to minimize at each point in time the storage required to hold incomplete tasks. In this paper a deterministic model of independent tasks is introduced and new results are derived which extend and generalize the algorithms known for minimizing mean finishing time. In addition to presenting and analyzing new algorithms it is shown that the most general mean-finishing-time problem for independent tasks is polynomial complete, and hence unlikely to admit of a non-enumerative solution.
The problem of evaluating arithmetic expressions on a machine with N > 1 general purpose registers is considered. It is initially assumed that no algebraic laws apply to the operators and operands in the expression. An algorithm for evaluation of expressions under this assumption is proposed, and it is shown to take the shortest possible number of instructions. It is then assumed that certain operators are commutative or both commutative and associative. In this case a procedure is given for finding an expression equivalent to a given one and having the shortest possible evaluation sequence. It is then shown that the algorithms presented here also minimize the number of storage references in the evaluation. '
We examine from a formal point of view some problems which arise in code optimization and present some of the results which can come from such an approach.Specifically, a set of transformations which characterize optimization algorithms for straight line code is presented.Then we present an algorithm which produces machine code for evaluating arithmetic expressions on machines with N ~ I general purpose registers.We can prove that this algorithm produces optimal code when the cost criterion is the length of machine code generated.
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