Better lower bounds on detecting affine and spherical degeneracies

Abstract

We show that in the worst case, (n d ) sidedness queries are required to determine whether a set of n points in IR d is a nely degenerate, i.e., whether it contains d + 1 p o i n ts on a common hyperplane. This matches known upper bounds. We g i v e a straightforward adversary argument, based on the explicit construction of a point set containing (n d ) \collapsible" simplices, any o n e o f which can be made degenerate without changing the orientation of any o t h e r simplex. As an immediate corollary, w e h a ve a n ( n d ) l o wer bound on the number of sidedness queries required to determine the order type of a set of n points in IR d . Using similar techniques, we a l s o s h o w t h a t ( n d+1 ) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in IR d .

On the least median square problem

“…If this conjecture is true, the Ω(n d ) lower bound is tight; the problem can be solved in O(n d ) time by constructing the dual hyperplane arrangement. Erickson and Seidel [13,12] proved an Ω(n d ) lower bound on the number of sidedness queries required to solve this problem; however, the model of computation in which their lower bound holds is not strong enough to solve the LMS problem, since it does not allow us to compare widths of different slabs. The strongest lower bound known in any general model of computation is Ω(n log n), for any fixed dimension, in the algebraic decision and computation tree models [2,25], although the problem is known to be NP-complete when d is not fixed [17,12].…”

Lower bounds for linear degeneracy testing

“…Decision trees have often shown to be realistic and effective models for proving lower bounds on the complexity of fundamental geometric problems [Ben-Or 1983;Björner et al 1992;Dobkin and Lipton 1979;Erickson 1999aErickson , 1999bErickson and Seidel 1995;Grigoriev et al 1996Grigoriev et al , 1997Steele and Yao 1982;Yao 1997Yao , 1995. Testing degeneracy is one such example.…”

Sub-linear Time Hybrid Approximations for Least Trimmed Squares Estimator and Related Problems

“…LTS exhibits high time complexity when solved even approximately. Mount et al [25] showed that there is an Ω((n−m) d−1 ) lower bound on the running time of any residual or quantile approximation, provided that the conjectured Ω(n d )-time lower bound for the affine degeneracy problem holds [9,11]. This means that both residual and quantile approximations likely require Ω(n d−1 ) time to solve when the ratio of m/n is a positive constant c < 1, which is often the case in applications.…”