Visual Complexity of Point Set Mappings

TL;DR

This paper investigates the complexity of animated point set transitions by focusing on group translations as a measure of visual complexity, providing algorithms and hardness results for various problem variants.

Contribution

It introduces a new framework for measuring visual complexity via group translations and analyzes the computational complexity of related problems, offering algorithms and hardness proofs.

Findings

Polynomial time algorithms for certain problem variants

NP-hardness results for other variants

Approximation algorithms for some open problems

Abstract

We study the visual complexity of animated transitions between point sets. Although there exist many metrics for point set similarity, these metrics are not adequate for this purpose, as they typically treat each point separately. Instead, we propose to look at translations of entire subsets/groups of points to measure the visual complexity of a transition between two point sets. Specifically, given two labeled point sets A and B in R^d, the goal is to compute the cheapest transformation that maps all points in A to their corresponding point in B, where the translation of a group of points counts as a single operation in terms of complexity. In this paper we identify several problem dimensions involving group translations that may be relevant to various applications, and study the algorithmic complexity of the resulting problems. Specifically, we consider different restrictions on the groups that can be translated, and different optimization functions. For most of the resulting problem variants we are able to provide polynomial time algorithms, or establish that they are NP-hard. For the remaining open problems we either provide an approximation algorithm or establish the NP-hardness of a restricted version of the problem. Furthermore, our problem classification can easily be extended with additional problem dimensions giving rise to new problem variants that can be studied in future work.