Optimal Randomized Comparison Based Algorithms for Collision

We consider the well known problem of finding two identical elements, called a collision, in a list of n numbers. Here, the (very fast) comparison based algorithms are randomized and will only report existing collisions, and do this with (small) probability p, the success probability. We find a trade-off between p and the running time t, and show that this trade-off is optimal up to a constant factor. For worst-case running time t, the optimal success probability is p = Θ (min{t/n, 1}t/(n log t)). For expected running time t, the success probability is p = Θ (t/(n logn)).