Clarksons Algorithm for Violator Spaces

TL;DR

This paper extends Clarkson's algorithm to violator spaces, simplifying its second stage and establishing its theoretical foundations, including the equivalence between violator spaces and hypercube partitions.

Contribution

It demonstrates the first simplification of Clarkson's second stage and confirms that previous first stage simplifications apply to violator spaces, also establishing their equivalence with hypercube partitions.

Findings

Clarkson's second stage can be simplified in violator spaces.

Previous simplifications of the first stage apply to violator spaces.

Violator spaces are equivalent to partitions of the hypercube by hypercubes.

Abstract

Clarksons algorithm is a two-staged randomized algorithm for solving linear programs. This algorithm has been simplified and adapted to fit the framework of LP-type problems. In this framework we can tackle a number of non-linear problems such as computing the smallest enclosing ball of a set of points in R^d . In 2006, it has been shown that the algorithm in its original form works for violator spaces too, which are a proper general- ization of LP-type problems. It was not clear, however, whether previous simplifications of the algorithm carry over to the new setting. In this paper we show the following theoretical results: (a) It is shown, for the first time, that Clarksons second stage can be simplified. (b) The previous simplifications of Clarksons first stage carry over to the violator space setting. (c) Furthermore, we show the equivalence of violator spaces and partitions of the hypercube by hypercubes.