Algorithms for Ham Sandwich Cuts

Given disjoint sets PI, P2 ..... Pd in R a with n points in total, a ham-sandwich cut is a hyperplane that simultaneously bisects the Pi. We present algorithms for finding ham-sandwich cuts in every dimension d > 1. When d = 2, the algorithm is optimal, having complexity O(n). For dimension d > 2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement of n hyperplanes in dimension d-1. This, in turn, is related to the number of k-sets in R d-~. With the current estimates, we get complexity close to O(n 3/2) for d = 3, roughly O(n s/3) for d = 4, and O(n d-1-atd~) for some a(d) > 0 (going to zero as d increases) for larger d. We also give a linear-time algorithm for ham-sandwich cuts in R 3 when the three sets are suitably separated.