W-Structures in Contour Trees

Abstract

The contour tree is one of the principal tools in scientific visualisation. It captures the connectivity of level sets in scalar fields. In order to apply the contour tree to exascale data we need efficient shared memory and distributed algorithms. Recent work has revealed a parallel performance bottleneck caused by substructures of contour trees called W-structures. We report two novel algorithms that detect and extract the Wstructures. We also use the W-structures to show that extended persistence is not equivalent to branch decomposition and leafpruning.

Distributed Hierarchical Contour Trees

“…For a 3D mesh, the contour tree would scale with the cube of the linear dimension, but the boundary tree with the square. For W-structures, however, it is possible for N b = Ω(t), although we have never see this occur, and have reported typical statistics [15]. We assume that W-structures do not have a major impact on the size of the boundary tree.…”

“…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.…”

“…We test all boundary vertices for criticality, taking O(N 2 The relationship between t, the size of the contour tree, b, the size of the boundary, b ′ , the number of boundary critical points, and N b , the size of the boundary tree, is key. In the absence of W-structures [15], N b < 2b ′ < 2b, i.e. the complexity of the boundary tree is linear in the size of the boundary.…”

Pathological and Test Cases for Reeb Analysis

“…One of the particular values of this example is that the W structure ensures that only two branches are candidates at any given time, with the lower priority one chosen for removal. Here, our priority measure is the "height" of the branch, which is not in fact the persistence of the extremum [16]. The right-hand most superarc (200 − 60) in the illustration has a priority of 140, higher than the priority (110) left-hand superarc (230 − 120).…”