# Building Efficient and Compact Data Structures for Simplicial Complexes

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## Abstract

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the ST while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree and the Simplex Array List. We analyze the compressed ST, the Maximal Simplex Tree, and the Simplex Array List under various settings.

### Keywords

Simplicial complex Compact data structures Automaton NP-hard## Notes

### Acknowledgments

We would like to thank Eylon Yogev for helping us with some of the experiments. We would like to thank the anaonymous reviewers for pointing out a mistake in the previous version of the proof of Lemma 8, and for helping us improve the presentation of the paper. We would like to thank Rajesh Chitnis for pointing out a short proof of Lemma 8. We would like to thank François Godi for pointing out a mistake in the analysis of the cost of the edge contraction operation for the Simplex Array List as it appeared in [8]. We would like to thank Marc Glisse for several comments on earlier versions of this paper and also for pointing out a tightening of the cost of the edge contraction operation for the Simplex Array List. We would like to thank Marc Glisse and Sivaprasad S. for implementing SAL and sharing their results with us. Finally, we would like to thank Dorian Mazauric for pointing out Theorem 2.

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