The Parallel Complexity of Element Distinctness is $\Omega ( \sqrt{\log n} )

• Published in: SIAM journal on discrete mathematics Vol. 1; no. 3; pp. 399 - 410
• Main Authors: Ragde, Prabhakar, Steiger, William, Szemerédi, Endre, Wigderson, Avi
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.08.1988
• Subjects: Algorithms
• DOI: 10.1137/0401040
• ISSN: 0895-4801

We consider the problem of element distinctness. Here $n$ synchronized processors, each given an integer input, must decide whether these integers are pairwise distinct, while communicating via an infinitely large shared memory. If simultaneous write access to a memory cell is forbidden, then a lower bound of $\Omega ( \log n )$ on the number of steps easily follows (from S. Cook, C. Dwork, and R. Reischuk, SIAM J. Comput., 15 (1986), pp. 87-97.) When several (different) values can be written simultaneously to any cell, then there is an simple algorithm requiring $O ( 1 )$ steps. We consider the intermediate model, in which simultaneous writes to a single cell are allowed only if all values written are equal. We prove a lower bound of $\Omega ( ( \log n )^{1 /2} )$ steps, improving the previous lower bound of $\Omega ( \log \log \log n )$ steps (F. E. Fich, F. Meyer auf der Heide, and A. Wigderson, Adv. in Comput., 4 (1987), pp. 1-15). The proof uses Ramsey-theoretic and combinatorial arguments. The result implies a separation between the powers of some variants of the PRAM model of parallel computation.