Constant Depth Prefix Circuits With Unbounded Fan-In

Abstract

It is well known that among the three classes of the PRAM models, namely, CRCW, CREW, and EREW, the CRCW models are the weakest, in the sense that, they permit concurrent read/write by processors. Accordingly, algorithms on the CRCW model mainly concentrate on the core computations without much ado about data access. Consequently, this model, at least in principle, allows for the design of the fastest algorithm for a problem. It is intriguing to ask how fast prefixes can be computed on the CRCW models. Since CRCW models are equivalent to the unbounded fan-in circuits (refer to Chapter 2), the task of developing the fastest algorithms for the prefix problems is pursued in the context of the unbounded fan-in circuits. Recall from Chapter 2, that while the standard measures, such as, size and depth are still used to quantify the goodness of unbounded fanin circuits, the size of the circuit is measured by the total number of edges incident on all of its operation nodes, instead of by the number of operations nodes. It turns out that the size and depth of unbounded fanin circuits for computing prefixes, depends critically on the structure of the underlying semigroup from which the input elements are drawn. The principal result of this concluding Chapter may be stated as follows: There exists unbounded fan-in parallel prefix circuits of constant depth and polynomial size if, and only if, the underlying semigroup is group free. The proof of this result involves a very clever synthesis of a number of ideas drawn from different directions — structure of group free semigroups, their relations to a special class of regular sets, called non-counting regular sets, the relation of this latter class of regular sets to yet another class of regular sets defined by star-free regular expressions, and the design of a special class of finite state deterministic automata called RS machines that accept star-free regular expressions. In this context, it is convenient to define the notion of small circuits as the class of circuits with constant depth and polynomial size.