Near-Optimal Time-Space Tradeoff for Element Distinctness

It was conjectured in Borodin et al. [J. Comput. System Sci., 22 (1981), pp. 351--364] that to solve the element distinctness problem requires $TS = \Omega(n^2)$ on a comparison-based branching program using space $S$ and time $T$, which, if true, would be close to optimal since $TS = O((n \log n)^2)$ is achievable. Recently, Borodin et al. [SIAM J. Comput., 16 (1987), pp. 97--99] showed that $TS = \Omega (n^{3/2}(\log n)^{1/2})$. This paper presents a near-optimal tradeoff $TS = \Omega(n^{2-\epsilon(n)})$, where $\epsilon(n) = O(1/(\log n)^{1/2})$.