A tail estimate for Mulmuley’s segment intersection algorithm

Abstract

We analyze the distribution of the running time of Mulmuley’s randomized algorithm for computing the intersections of n given segments in the plane. Its expectation has been known to be O(n log n + k), where k is the number of intersections. We show that for values of k not too close to n (k ≥ Cnlog15n for a large enough constant C), the running time is sharply concentrated around the expected value; e.g., the probability that the expected value is exceeded more than twice is O(n-c), where c can be an arbitrarily large constant (its choice determines the value of C needed in the assumption). Our proof uses an isoperimetric inequality for permutations.