  Multi-parameter persistent homology is a recent branch of topological data
analysis. In this area, data sets are investigated through the lens of homology
with respect to two or more scale parameters. The high computational cost of
many algorithms calls for a preprocessing step to reduce the input size. In
general, a minimal presentation is the smallest possible representation of a
persistence module. Lesnick and Wright proposed recently an algorithm (the
LW-algorithm) for computing minimal presentations based on matrix reduction. In
this work, we propose, implement and benchmark several improvements over the
LW-algorithm. Most notably, we propose the use of priority queues to avoid
extensive scanning of the matrix columns, which constitutes the computational
bottleneck in the LW-algorithm, and we combine their algorithm with ideas from
the multi-parameter chunk algorithm by Fugacci and Kerber. Our extensive
experiments show that our algorithm outperforms the LW-algorithm and computes
the minimal presentation for data sets with millions of simplices within a few
seconds. Our software is publicly available.
