Some geometric lower bounds

Abstract

Dobkin and Lipton introduced the connected components argument to prove lower bounds in the linear decision tree model for membership problems, for example the element uniqueness problem. In this paper we apply the same idea to obtain lower bound statements for a variety of problems, each having the flavor of element uniqueness. In fact one of these problems is a parametric version of element uniqueness which asks, given n inputs a ₁,..., a _n and a query x≥ 0, whether there is a pair of inputs satisfying ¦a _i−a_j¦=x; the case x=0 IS element uniqueness. Then we apply some of these results to establish the fact that “search can be easier than uniqueness”; specifically we give two examples (one is the planar ham-sandwich cut) where finding or constructing a geometric object — known to exist — is less complex than answering the question about whether that object is unique. Finally we apply some of these results, along with a reduction argument, to get a nontrivial lower bound for the complexity of the least median of squares regression problem in the plane.

Research Supported by a Research Experiences for Undergraduates (REU) supplement to the NSF center grant to DIMACS.

Research Supported in Part by NSF grant CCR-9111491.

The author expresses gratitude to the NSF DIMACS Center at Rutgers.