# 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.

## Keywords

Parallel Algorithm Active Object Color Class Virtual Processor Active Request## Preview

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## References

- [1]M. Ajtai and M. Ben-Or, A Theorem on Probabilistic Constant Depth Computations,
*in*Proc. 16th Annual ACM Symposium on Theory of Computing (1984), pp. 471–474.Google Scholar - [2]B. Awerbuch and Y. Shiloach, New Connectivity and MSF Algorithms for Ultracomputer and PRAM, in Proc. International Conference on Parallel Processing, 1983, pp. 175–179.Google Scholar
- [3]H. Bast, personal communication.Google Scholar
- [4]H. Bast, M. Dietzfelbinger, and T. Hagerup, A Parallel Real-Time Dictionary, in preparation.Google Scholar
- [5]H. Bast and T. Hagerup, Fast and Reliable Parallel Hashing, manuscript. A preliminary version appears in Proc. 3rd Annual ACM Symposium on Parallel Algorithms and Architectures (1991), pp. 50–61.Google Scholar
- [6]O. Berkman and U. Vishkin, Recursive Tree Parallel Data-Structure,
*in*Proc. 30th Annual Symposium on Foundations of Computer Science (1989), pp. 196–202.Google Scholar - [7]M. Dietzfelbinger and F. Meyer auf der Heide, A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, in Proc. 17th International Colloquium on Automata, Languages and Programming (1990), Springer Lecture Notes in Computer Science, Vol. 443, pp. 6–19.Google Scholar
- [8]M. L. Fredman, J. Komlós, and E. Szemerédi, Storing a Sparse Table with
*O*(1) Worst Case Access Time,*J. ACM***31**(1984), pp. 538–544.Google Scholar - [9]J. Gil, Y. Matias, and U. Vishkin, Towards a Theory of Nearly Constant Time Parallel Algorithms, in Proc. 32nd Annual Symposium on Foundations of Computer Science (1991), pp. 698–710.Google Scholar
- [10]M. T. Goodrich, Using Approximation Algorithms to Design Parallel Algorithms that May Ignore Processor Allocation,
*in*Proc. 32nd Annual Symposium on Foundations of Computer Science (1991), pp. 711–722.Google Scholar - [11]M. T. Goodrich, Fast Parallel Approximation Algorithms for Problems with Near-Logarithmic Lower Bounds, Tech. Rep. no. JHU-91/13 (1991), Johns Hopkins University, Baltimore, MD.Google Scholar
- [12]V. Grolmusz and P. Ragde, Incomparability in Parallel Computation,
*in*Proc. 28th Annual Symposium on Foundations of Computer Science (1987), pp. 89–98.Google Scholar - [13]T. Hagerup, Constant-Time Parallel Integer Sorting,
*in*Proc. 23rd Annual ACM Symposium on Theory of Computing (1991), pp. 299–306.Google Scholar - [14]T. Hagerup, Fast Parallel Space Allocation, Estimation and Integer Sorting, Tech. Rep. no. MPI-I-91-106 (1991), Max-Planck-Institut für Informatik, Saarbrücken.Google Scholar
- [15]T. Hagerup, On Parallel Counting, presented at the Dagstuhl Workshop on Randomized Algorithms, Schloß Dagstuhl, June, 1991.Google Scholar
- [16]T. Hagerup, Fast Parallel Generation of Random Permutations,
*in*Proc. 18th International Colloquium on Automata, Languages and Programming (1991), Springer Lecture Notes in Computer Science, Vol. 510, pp. 405–416.Google Scholar - [17]T. Hagerup, Fast and Optimal Simulations between CRCW PRAMs, these proceedings.Google Scholar
- [18]T. Hagerup and C. Rüb, A Guided Tour of Chernoff Bounds,
*Inform. Proc. Lett.***33**(1990), pp. 305–308.Google Scholar - [19]Y. Matias and U. Vishkin, Converting High Probability into Nearly-Constant Time — with Applications to Parallel Hashing,
*in*Proc. 23rd Annual ACM Symposium on Theory of Computing (1991), pp. 307–316.Google Scholar - [20]C. McDiarmid, On the Method of Bounded Differences,
*in*Surveys in Combinatorics, 1989, ed. J. Siemons, London Math. Soc. Lecture Note Series 141, Cambridge University Press, pp. 148–188.Google Scholar - [21]I. Parberry, Parallel Complexity Theory, Pitman, London, 1987.Google Scholar
- [22]P. Ragde, The Parallel Simplicity of Compaction and Chaining,
*in*Proc. 17th International Colloquium on Automata, Languages and Programming (1990), Springer Lecture Notes in Computer Science, Vol. 443, pp. 744–751.Google Scholar - [23]R. Raman, The Power of Collision: Randomized Parallel Algorithms for Chaining and Integer Sorting,
*in*Proc. 10th Conference on Foundations of Software Technology and Theoretical Computer Science (1990), Springer Lecture Notes in Computer Science, Vol. 472, pp. 161–175.Google Scholar - [24]L. Stockmeyer, The Complexity of Approximate Counting,
*in*Proc. 15th Annual ACM Symposium on Theory of Computing (1983), pp. 118–126.Google Scholar