The log-star revolution

Abstract

The last approximately one year has witnessed a dramatic change in the way theoreticians think about computing on the randomized concurrent-read concurrent-write parallel random access machine (CRCW PRAM). Today we have superfast algorithms that were inconceivable a few years ago. Many of these having running times of the form O((log*n)c), for some small constant c ε N, the name “log-star revolution” seems appropriate. This paper tries to put some of the most important results obtained next to each other and to explain their significance. In order to keep the exposition properly focussed, we restrict our attention to problems of a very fundamental nature that, in an ideal environment, would be handled by the operating system of a parallel machine rather than by each applications programmer: Processor allocation, memory allocation and the implementation of a particular conflict resolution rule for concurrent writing. The main contention of the paper is that the theoretical groundwork for providing such an ideal environment has been laid. Our goal is to provide the reader with an appreciation of the log-star revolution and to enable him or her to carry on the torch of revolution by exploring the largely unchartered new territories. The emphasis is on ideas, not on rigor. Warning: This paper deals exclusively with the randomized CRCW PRAM, a species of parallel machines that may never have any close relation to the realities of parallel computing. If you, dear reader, are very practically inclined, perhaps you should better stop reading here in order to not feel cheated later. The author confesses to a mathematical fascination with the subject rather than to a firm belief in any future practical impact (the possibility, definitely, is there). Having thus apologized once for playing a rather esoteric game, the author cordially invites the reader to accept the rules and to enjoy the game.