# The Simplex Tree: an Efficient Data Structure for General Simplicial Complexes

arXiv: https://arxiv.org/abs/2001.02581 · HTML: https://mlsanity.com/paper/2001.02581/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2020.

## Built on by (most cited first, 14 of 14)

- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)
- [Topological Measurement of Deep Neural Networks Using Persistent Homology](https://mlsanity.com/paper/2106.03016/) (2021)
- [What makes math problems hard for reinforcement learning: a case study](https://mlsanity.com/paper/2408.15332/) (2024)
- [Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems](https://mlsanity.com/paper/2212.01666/) (2022)
- [Classification based on Topological Data Analysis](https://mlsanity.com/paper/2102.03709/) (2021)
- [A Topological Data Analysis Based Classifier](https://mlsanity.com/paper/2111.05214/) (2021)
- [Dory: Overcoming Barriers to Computing Persistent Homology](https://mlsanity.com/paper/2103.05608/) (2021)
- [TopoPrune: Robust Data Pruning via Unified Latent Space Topology](https://mlsanity.com/paper/2602.02739/) (2026)
- [Geometric Model Checking of Continuous Space](https://mlsanity.com/paper/2105.06194/) (2021)
- [Persistent Homology via Ellipsoids](https://mlsanity.com/paper/2408.11450/) (2024)
- [A DNN Biophysics Model with Topological and Electrostatic Features](https://mlsanity.com/paper/2409.03658/) (2024)
- [Classification of Histopathology Slides with Persistent Homology Convolutions](https://mlsanity.com/paper/2507.14378/) (2025)
- [Frequent subgraph-based persistent homology for graph classification](https://mlsanity.com/paper/2512.24917/) (2025)

## Abstract (arXiv)

This paper introduces a data structure, called simplex tree, to represent abstract simplicial complexes of any dimension. All faces of the simplicial complex are explicitly stored in a trie whose nodes are in bijection with the faces of the complex. This data structure allows to efficiently implement a large range of basic operations on simplicial complexes. We provide theoretical complexity analysis as well as detailed experimental results. We more specifically study Rips and witness complexes.