Shape Fitting on Point Sets with Probability Distributions

TL;DR

This paper addresses shape fitting problems on point sets with uncertain locations modeled by probability distributions, proposing efficient randomized and deterministic algorithms to approximate the distributions of various shape measures.

Contribution

It introduces novel data structures and algorithms for approximating shape fitting measures under probabilistic point locations, extending traditional geometric methods to uncertain data.

Findings

Efficient randomized algorithms for shape fitting with probabilistic data

Deterministic algorithms with polynomial runtime in data size and approximation factor

Experimental results demonstrating practicality of proposed methods

Abstract

A typical computational geometry problem begins: Consider a set P of n points in R^d. However, many applications today work with input that is not precisely known, for example when the data is sensed and has some known error model. What if we do not know the set P exactly, but rather we have a probability distribution mu_p governing the location of each point p in P? Consider a set of (non-fixed) points P, and let mu_P be the probability distribution of this set. We study several measures (e.g. the radius of the smallest enclosing ball, or the area of the smallest enclosing box) with respect to mu_P. The solutions to these problems do not, as in the traditional case, consist of a single answer, but rather a distribution of answers. We describe several data structures that approximate distributions of answers for shape fitting problems. We provide simple and efficient randomized algorithms for computing all of these data structures, which are easy to implement and practical. We provide some experimental results to assert this. We also provide more involved deterministic algorithms for some of these data structures that run in time polynomial in n and 1/eps, where eps is the approximation factor.