Algorithms for ham-sandwich cuts

Abstract

Given disjoint sets PI, P2 ..... Pd in R a with n points in total, a hamsandwich 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.

Orthogonal Ham-Sandwich Theorem in ℝ3

“…Interestingly, our algorithm is faster than the current best algorithm [10] for computing (standard) ham-sandwich cuts in ℝ 3 which running time depends on the complexity of -sets in the plane. An open question is the complexity of the problem of computing an orthogonal ham-sandwich cut in ℝ 3 .…”

“…Cole, Sharir and Yap [6] described an algorithm that has the same complexity (as pointed out in [10]). Lo, Matoušek and Steiger [10] found a linear time algorithm for computing a ham-sandwich cut in the plane. They also designed algorithms for computing ham-sandwich cuts in higher dimensions.…”

LDRD report : parallel repartitioning for optimal solver performance.

“…A linear time algorithm exists for n=2, and some efficient algorithms exist for other low dimensions [38].…”

Generalizing Ham Sandwich Cuts to Equitable Subdivisions

“…The recursive procedure has input g, n, m and two sets Rand B of points. If g is even we use the linear time algorithm of Lo et al for the Ham Sandwich Problem [13]. Otherwise provided g > 1 we find a triple (gI' g2, g3) using Theorem 9, and find an equitable 3-cutting by Theorem 5 (in fact, the algorithm may find 2-cutting instead of 3-cutting).…”

“…The Ham Sandwich problem is well studied from an algorithmic point of view [2], [5]- [7], [12]- [14], [17], [19]. An optimal algorithm ofLo et al [13] finds a Ham Sandwich cut in linear time. Kaneko and Kano [11] considered balanced partitions of two sets in the plane.…”