Pathological and Test Cases for Reeb Analysis

Abstract

After two decades of computational topology, it is clearly a computationally challenging area. Not only do we have the usual algorithmic and programming difficulties with establishing correctness, we also have a class of problems that are mathematically complex and notationally fragile. Effective development and deployment therefore requires an additional step -construction or selection of suitable test cases. Since we cannot test all possible inputs, our selection of test cases expresses our understanding of the task and of the problems involved. Moreover, the scale of the data sets we work with is such that, no matter how unlikely the behaviour mathematically, it is nearly guaranteed to occur at scale in every run. The test cases we choose are therefore tightly coupled with mathematically pathological cases, and need to be developed using the skills expressed most obviously in constructing mathematical counter-examples. This paper is therefore a first attempt at reporting, classifying and analyzing test cases previously used for algorithmic work in Reeb analysis (contour trees and Reeb graphs), and the expression of a philosophy of how to test topological code.

W-Structures in Contour Trees

“…W-structures were first visible in Carr et al [9], they have previously complicated proofs [11] and have caused issues with contour tree parallel algorithm design and analysis [12]. The contribution of this paper is to initiate a systematic study of these W-structures.…”

Distributed Hierarchical Contour Trees

“…Overall, therefore, Pascucci & Cole-McLaughlin defined a distributed approach for contour tree computation, but one that was hampered by the need to fan-in the entire contour tree to a single node. Subsequent approaches have primarily built a distributed merge tree, which is known to be easier than building the contour tree, especially in the presence of W-structures [9,15]. And, while subsequent work enabled some operations that usually require the contour treee [22], these operations are based on separate merge trees (join-and split-tree) that are not combined into a contour tree.…”

“…While search is logarithmic, hyperstructure construction is not, due to W-structures, which zig-zag horizontally through the tree [9,15], causing problems with parallel computation.…”

Computer Graphics and Imaging

“…Multivariate topological visualisation is a more recent research interest in the visualisation community in comparison to topological visualisations of a single field. Due to the relative infancy of the field, Carr et al [51] highlight the need for sufficiently complex data sets to extensively test the emerging algorithms in the area.…”