New algorithms and lower bounds for the parallel evaluation of certain rational expressions

This paper presents new algorithms for the parallel evaluation of certain polynomial expression. In particular, for the parallel evaluation of x<supscrpt>n</supscrpt>,we introduce an algorithm which takes two steps of <underline>parallel division</underline> and [log<subscrpt>2</subscrpt>n] steps of parallel addition, while the usual algorithm takes [log<subscrpt>2</subscrpt>n] steps of parallel multiplication. Hence our algorithm is faster than the usual algorithms when multiplication takes more time than addition. Similar algorithm for the evaluation of other polynomial expressions are also introduced. Lower bounds on the time needed for the parallel evaluation of rational expressions are given. All the algorithms presented in the paper are shown to be asymptotically optimal. Moreover, we prove that by using parallelism the evaluation of any first order rational recurrence, e.g., [equation], and any <underline>non-linear</underline> polynomial recurrence can be sped up at most by a constant factor, no matter how many processors are used.