[Paper Review] Geometric Computations on Indecisive and Uncertain Points

Allan Grønlund Jørgensen, Maarten Löffler+1 more|arXiv (Cornell University)|May 1, 2012

Data Management and Algorithms50 references10 citations

TL;DR

This paper presents efficient algorithms for computing geometric shape-fitting distributions on uncertain or indecisive points, where each point has a finite set of possible locations with associated probabilities. It introduces a randomized approximation algorithm for general uncertain data and a deterministic, exact polynomial-time algorithm for LP-type problems on indecisive points using Shape Inclusion Probability (SIP) functions.

ABSTRACT

We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a finite number of locations, the points are called indecisive points. In particular, we focus on geometric shape-fitting problems and on building compact distributions to describe how the solutions to these problems vary with respect to the uncertainty in the points. Our main results are: (1) a simple and efficient randomized approximation algorithm for calculating the distribution of any statistic on uncertain data sets; (2) a polynomial, deterministic and exact algorithm for computing the distribution of answers for any LP-type problem on an indecisive point set; and (3) the development of shape inclusion probability (SIP) functions which captures the ambient distribution of shapes fit to uncertain or indecisive point sets and are admissible to the two algorithmic constructions.

Motivation & Objective

• To address the challenge of computing complete probability distributions over geometric outputs when input points are uncertain or indecisive.
• To develop practical and efficient methods for representing and querying the distribution of solutions to geometric problems under uncertainty.
• To identify conditions under which exact and approximate computations on uncertain geometric data are tractable.
• To introduce Shape Inclusion Probability (SIP) functions as a compact, query-efficient representation of solution distributions.

Proposed method

• Proposes a randomized approximation algorithm to estimate the distribution of any statistic on uncertain point sets by sampling from the finite set of possible configurations.
• Introduces a deterministic, exact algorithm for LP-type problems on indecisive points by enumerating all potential bases and using a full violation test to compute weights.
• Employs SIP functions to represent the probability that a query shape is included in the solution shape for a random configuration of uncertain points.
• Uses planar subdivisions and arrangement structures to build data structures that support efficient SIP queries in logarithmic time.
• Applies the framework to problems like smallest enclosing disk and other LP-type problems with constant combinatorial dimension.
• Extends the approach to $$-SIP functions for continuous uncertain point sets through specific invocations of theoretical results.

Experimental results

Research questions

• RQ1 Can we compute the full probability distribution of geometric outputs (e.g., shape parameters) when input points are uncertain or indecisive?
• RQ2 What classes of geometric problems allow for efficient, exact computation of solution distributions under uncertainty?
• RQ3 How can we represent and query the distribution of solution shapes compactly and efficiently?
• RQ4 What is the trade-off between approximation and exactness in computing distributions for uncertain geometric data?
• RQ5 Are there polynomial-time algorithms for computing distributions of complex geometric shapes on indecisive points?

Key findings

• A randomized approximation algorithm is proposed for computing the distribution of any statistic on uncertain data sets, providing practical scalability.
• A deterministic, exact polynomial-time algorithm is developed for LP-type problems on indecisive points, with time complexity $O((nk)^{eta+1})$ and space $O((nk)^{eta})$, where $n$ is the number of points, $k$ the number of locations per point, and $eta$ the combinatorial dimension.
• The Shape Inclusion Probability (SIP) function enables compact representation of solution distributions and supports exact queries in $O( ext{poly}(nk))$ time.
• An optimized data structure for SIP queries achieves $O( ext{poly}(nk))$ preprocessing and $O( ext{polylog}(nk))$ query time using planar subdivisions.
• The diameter problem is shown to be #P-hard even for computing a single value of the output distribution, indicating inherent intractability for some problems.
• The framework extends to $$-SIP functions for continuous uncertain point sets, enabling deterministic approximation under specific uncertainty models.