Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data – An Algorithm and A Benchmark

by   Pierre Guillou, et al.

This paper introduces an efficient algorithm for persistence diagram computation, given an input piecewise linear scalar field f defined on a d-dimensional simplicial complex K, with d ≤ 3. Our work revisits the seminal algorithm "PairSimplices" [31], [103] with discrete Morse theory (DMT) [34], [80], which greatly reduces the number of input simplices to consider. Further, we also extend to DMT and accelerate the stratification strategy described in "PairSimplices" for the fast computation of the 0^th and (d - 1)^th diagrams, noted D_0(f) and D_d-1(f). Minima-saddle persistence pairs (D_0(f)) and saddle-maximum persistence pairs (D_d-1(f)) are efficiently computed by processing, with a Union-Find, the unstable sets of 1-saddles and the stable sets of (d - 1)-saddles. This fast pre-computation for the dimensions 0 and (d - 1) enables an aggressive specialization of [4] to the 3D case, which results in a drastic reduction of the number of input simplices for the computation of D_1(f), the intermediate layer of the sandwich. Finally, we document several performance improvements via shared-memory parallelism. We provide an open-source implementation of our algorithm for reproducibility purposes. We also contribute a reproducible benchmark package, which exploits three-dimensional data from a public repository and compares our algorithm to a variety of publicly available implementations. Extensive experiments indicate that our algorithm improves by two orders of magnitude the time performance of the seminal "PairSimplices" algorithm it extends. Moreover, it also improves memory footprint and time performance over a selection of 14 competing approaches, with a substantial gain over the fastest available approaches, while producing a strictly identical output.


page 2

page 4

page 6

page 7

page 11

page 12

page 18


Progressive Wasserstein Barycenters of Persistence Diagrams

This paper presents an efficient algorithm for the progressive approxima...

Fast Approximation of Persistence Diagrams with Guarantees

This paper presents an algorithm for the efficient approximation of the ...

Ripser: efficient computation of Vietoris-Rips persistence barcodes

We present an algorithm for the computation of Vietoris-Rips persistence...

Wasserstein Dictionaries of Persistence Diagrams

This paper presents a computational framework for the concise encoding o...

Fast persistent homology computation for functions on ℝ

0-dimensional persistent homology is known, from a computational point o...

TACHYON: Efficient Shared Memory Parallel Computation of Extremum Graphs

The extremum graph is a succinct representation of the Morse decompositi...

A Progressive Approach to Scalar Field Topology

This paper introduces progressive algorithms for the topological analysi...

Please sign up or login with your details

Forgot password? Click here to reset