Parallel Enumeration of Triangulations

Abstract

A b s t r ac t . We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before.2010 Mathematics Subject Classification. 52B55 (68U05).

Finding hexahedrizations for small quadrangulations of the sphere

“…One technique commonly used to deal with this type of issue is to define a canonical representation for objects under constructions, so that all those that belong to a given isomorphism class are transformed into the same representative element [Brinkmann and McKay 2007;Burton 2011]. A significant portion of the execution time is then spent computing the canonical representations of partial solutions, which may completely change after every operation [Jordan et al 2018]. The symmetry breaking method used within our algorithm instead compares partial solutions directly, and exploits the tree-shaped structure of the search in order to reuse results from previous computations.…”

mts: a light framework for parallelizing tree search codes

“…The previous examples are from the tutorial on reverse search; a more substantial application of mts is the parallel enumeration of triangulations implemented in mptopcom [29]. This was the first parallel code for the problem, and has already found applications [30,43].…”

New Counts for the Number of Triangulations of Cyclic Polytopes

“…In [8] Imai et al described an algorithm for computing all (regular) triangulations of a given point configurations, which is based on the reverse search enumeration scheme of Avis and Fukuda [2]. That algorithm, which we call down-flip reverse search, was improved and implemented by Skip Jordan with the authors of this extended abstract [9]. The basic idea is to orient each flip according to lexicographic ordering of the GKZ-vectors.…”

“…The open source software MPTOPCOM is designed for computing triangulations in a massively parallel setup. Its algorithm is the down-flip reverse search method of Imai et al [8] with several improvements as described in [9]. As its key feature reverse search is output sensitive, and this makes it attractive for extremely large enumeration problems.…”