# Building Efficient and Compact Data Structures for Simplicial Complexes

- 175 Downloads

## 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.

## References

- 1.Acharya, A., Zhu, H., Shen, K.: Adaptive algorithms for cache-efficient trie search. In: Workshop on Algorithm Engineering and Experimentation ALENEX 99, Baltimore (1999)Google Scholar
- 2.Andersson, A., Nilsson, S.: Improved behaviour of tries by adaptive branching. Inf. Process. Lett.
**46**, 295–300 (1993)MathSciNetCrossRefMATHGoogle Scholar - 3.Appel, A.W., Jacobson, G.J.: The world’s fastest scrabble program. In: Communications of the ACM
**31**(1988)Google Scholar - 4.Attali, D., Lieutier, A., Salinas, D.: Efficient data structure for representing and simplifying simplicial complexes in high dimensions. Int. J. Comput. Geom. Appl.
**22**(4), 279–303 (2012)MathSciNetCrossRefMATHGoogle Scholar - 5.Badr, A., Geffert, V., Shipman, I.: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl.
**43**(1), 69–94 (2009)MathSciNetCrossRefMATHGoogle Scholar - 6.Bentley, J.L., Sedgewick, R.: Fast algorithms for sorting and searching strings. In: Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 360–369 (1997)Google Scholar
- 7.Billera, L.J., Björner, A.: Face numbers of polytopes on complexes. In: Handbook of Discrete and Computational Geometry. CRC Press, pp. 291–310 (1997)Google Scholar
- 8.Boissonnat, J-D., Karthik C.S., Tavenas, S.: Building efficient and compact data structures for simplicial complexes. In: Symposium on Computational Geometry, pp. 642–656 (2015)Google Scholar
- 9.Boissonnat, J.-D., Maria, C.: The simplex tree: an efficient data structure for general simplicial complexes. Algorithmica
**70**(3), 406–427 (2014)MathSciNetCrossRefMATHGoogle Scholar - 10.Boissonnat, J.-D., Mazauric, D.: On the complexity of the representation of simplicial complexes by trees. Theor. Comput. Sci.
**617**, 28–44 (2016)MathSciNetCrossRefMATHGoogle Scholar - 11.Câmpeanu, C., Sântean, N., Yu, S.: Minimal cover-automata for finite languages. Theor. Comput. Sci.
**267**(1–2), 3–16 (2001)MathSciNetCrossRefMATHGoogle Scholar - 12.Carrasco, R., Forcada, M.: Incremental construction and maintenanceof minimal finite-state automata. In: Computational Linguistics, vol. 28 (2002)Google Scholar
- 13.Comer, D., Sethi, R.: Complexity of trie index construction. In: Proceedings of Foundations of Computer Science, pp. 197-207 (1976)Google Scholar
- 14.Daciuk, J., Mihov, S., Watson, B., Watson, R.: Incremental construction of minimal acyclic finite-state automata. Comput. Linguist.
**26**, 3–16 (2000)MathSciNetCrossRefMATHGoogle Scholar - 15.Eppstein, D., Löffler, M., Strash, D.: Listing all maximal cliques in sparse graphs in near-optimal time. In: International Symposium on Algorithms and Computation, vol. (1), pp. 403-414 (2010)Google Scholar
- 16.Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman publishers, Newyork (1979)MATHGoogle Scholar
- 17.Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, Cambridge (2004)MATHGoogle Scholar
- 18.Grohe, M., Kreutzer, S., Siebertz, S.: Characterisations of nowhere dense graphs. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 21-40 (2013)Google Scholar
- 19.Hopcroft, J.: An n log n algorithm for minimizing states in a finite automaton. In: Theory of Machines and Computations, pp. 189–196 (1971)Google Scholar
- 20.Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of a Symposium on the Complexity of Computer Computations, held March 20–22, 1972, pp. 85–103. IBM, New York (1972)Google Scholar
- 21.Maia, E., Moreira, N., Reis, R.: Incomplete transition complexity of some basic operations. In: International Conference on Current Trends in Theory and Practice of Computer Science, pp. 319–331 (2013)Google Scholar
- 22.Maletti, A.: Notes on hyper-minimization. In: Proceedings 13th International Conference Automata and Formal Languages, pp. 34–49 (2011)Google Scholar
- 23.Nerode, A.: Linear automaton transformations. Proc. Am. Math. Soc.
**9**, 541–544 (1958)MathSciNetCrossRefMATHGoogle Scholar - 24.Revuz, D.: Minimisation of acyclic deterministic automata in linear time. Theor. Comput. Sci.
**92**(1), 181–189 (1992)MathSciNetCrossRefMATHGoogle Scholar - 25.Sgarbas, K., Fakotakis, N., Kokkinakis, G.: Optimal insertion in deterministic DAWGs. In: Theoretical Computer Science, pp. 103–117 (2003)Google Scholar
- 26.Teuhola, J., Raita, T.: Text compression using prediction. In: Proceedings of ACM Conference on Research and Development in Information Retrieval, pp. 97–102 (1986)Google Scholar
- 27.Yellin, D.: Algorithms for subset testing and finding maximal sets. In: SODA, pp. 386–392 (1992)Google Scholar