The Parallel Complexity of Element Distinctness is Omega (sqrt(log n))

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 imp...