Geometric Separators and the Parabolic Lift

A geometric separator for a set U of n geometric objects (usually balls) is a small (sublinear in n) subset whose removal disconnects the intersection graph of U into roughly equal sized parts. These separators provide a natural way to do divide and conquer in geometric settings. A particularly nice geometric separator algorithm originally introduced by Miller and Thurston has three steps: compute a centerpoint in a space of one dimension higher than the input, compute a conformal transformation that “centers” the centerpoint, and finally, use the computed transformation to sample a sphere in the original space. The output separator is the subset of S intersecting this sphere. It is both simple and elegant. We show that a change of perspective (literally) can make this algorithm even simpler by eliminating the entire middle step. By computing the centerpoint of the points lifted onto a paraboloid rather than using the stereographic map as in the original method, one can sample the desired sphere directly, without computing the conformal transformation. 1 Geometric Separators A spherical geometric separator of a collection of n balls in R is a sphere S that has at least n d+2 balls centered inside, at least n d+2 centered outside, and intersects at most O(n1− 1 d ) of them (see Section 2 for a formal definition). The existence of such separators in two and three dimensions was established by Miller and Thurston, though their method was quickly adapted to higher dimensions. Across a series of papers, Miller, Thurston, Teng, and Vavasis laid out the theory of geometric separators and their applications to scientific computing [14, 15, 13, 10, 12]. This line of work is a treasure trove for computational geometers as it hinges on a novel trick that combines projective and combinatorial geometry to solve an important algorithmic problem, solving linear systems arising in finite element analysis. More generally, geometric separators give a natural way to do divide and conquer for geometric problems. They have been applied to various nearest neighbor search problems [11] as well as to mesh compression [1]. Other variations of geometric separators have been used ∗Department of Computer Science and Engineering, University of Connecticut, [email protected] for the Traveling Salesman and Minimum Steiner Tree problems in geometric settings [18], or for packing and piercing problems [2]. The Miller-Thurston algorithm for computing a geometric separator maps the n points (the centers of the balls) to a unit d-sphere in R via a stereographic map. It then computes a centerpoint, which is a geometric generalization of a median (a formal definition is given in Section 2). There exists a conformal transformation of the points in R so that this centerpoint will lie exactly at the origin. To output a separator, one samples a random unit vector in R. The hyperplane through the origin normal to this vector intersects the unit d-sphere at a (d − 1)-sphere. The output is just the stereographic projection of this (d− 1)-sphere back to R. With high probability, such a sphere will be a geometric separator. The one aspect of this algorithm that was left to the reader, was the linear algebra required to compute the desired conformal transformation. In the original paper, it was simply asserted that it exists. Later papers explained that it can be computed via Householder transformations and cited a textbook on matrix computations. In this paper, we show that this phase of the algorithm is entirely unnecessary. By working initially with the parabolic lifting