Preprocessing Imprecise Points for the Pareto Front

TL;DR

This paper develops efficient algorithms for constructing the Pareto front from uncertain data modeled by regions, introducing a new optimality notion based on entropy, and analyzing the limits of preprocessing and reconstruction phases.

Contribution

It presents a novel preprocessing approach for uncertain data regions to efficiently reconstruct Pareto fronts, with a new optimality measure related to entropy and bounds on algorithmic performance.

Findings

Preprocessing can efficiently reconstruct Pareto fronts for disjoint axis-aligned rectangles.

Introduces a new notion of uncertainty-region optimality based on entropy.

Proves that instance optimality is impossible when the problem reduces to sorting.

Abstract

In the preprocessing model for uncertain data we are given a set of regions R which model the uncertainty associated with an unknown set of points P. In this model there are two phases: a preprocessing phase, in which we have access only to R, followed by a reconstruction phase, in which we have access to points in P at a certain retrieval cost C per point. We study the following algorithmic question: how fast can we construct the pareto front of P in the preprocessing model? We show that if R is a set of pairwise-disjoint axis-aligned rectangles, then we can preprocess R to reconstruct the Pareto front of P efficiently. To refine our algorithmic analysis, we introduce a new notion of algorithmic optimality which relates to the entropy of the uncertainty regions. Our proposed uncertainty-region optimality falls on the spectrum between worst-case optimality and instance optimality. We prove that instance optimality is unobtainable in the preprocessing model, whenever the classic algorithmic problem reduces to sorting. Our results are worst-case optimal in the preprocessing phase; in the reconstruction phase, our results are uncertainty-region optimal with respect to real RAM instructions, and instance optimal with respect to point retrievals.