Element distinctness on one-tape Turing machines: a complete solution

Abstract

We give a complete characterization of the complexity of the element distinctness problem for n elements of $m\ge\log n$ bits each on deterministic and nondeterministic one-tape Turing machines. We present an algorithm running in time $O(n^2m(m+2-\log n))$ for deterministic machines and nondeterministic solutions that are of time complexity $O(nm(n + \log m))$ . For elements of logarithmic size $m = O(\log n)$ , on nondeterministic machines, these results close the gap between the known lower bound $\Omega(n^2\log n)$ and the previous upper bound $O(n^2(\log n)^{3/2}(\log\log n)^{1/2})$ . Additional lower bounds are given to show that the upper bounds are optimal for all other possible relations between m and n. The upper bounds employ hashing techniques, while the lower bounds make use of the communication complexity of set disjointness.