Coreduction homology algorithm

Discrete and Computational Geometry

Volumn 41, Issue 1, 2009, Pages 96-118

  Mrozek, Marian ^a,b   Batko, Bogdan ^b,c  

^a JAGIELLONIAN UNIVERSITY   (Poland)

^b Graduate School of Business   (Poland)

^c PEDAGOGICAL UNIVERSITY   (Poland)

Author keywords

Cubical set; Homology algorithm; Reduction methods

Indexed keywords

EID: 57849165859     PISSN: 01795376     EISSN: 14320444     Source Type: Journal    
DOI: 10.1007/s00454-008-9073-y     Document Type: Article

References (30)

1. S. Basu 1999 On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets Discrete Comput. Geom. 22 1 18
2. Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré Conjecture. In: Saaty, T.L. (ed.) Lectures on Modern Mathematics, vol. II, pp. 93-128. Wiley, New York (1964)
3. C.J.A. Delfinado H. Edelsbrunner 1995 An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere Comput. Aided Geom. Des. 12 771 784
4. Donald, B.R., Chang, D.R.: On the complexity of computing the homology type of a triangulation. In: Proc. 32nd Ann. IEEE Sympos. Found. Comput. Sci., pp. 650-661 (1991)
5. Duff, I., Erisman, A., Reid, J.: Direct Methods for Sparse Matrices. Oxford University Press, London (1986)
6. J.-G. Dumas B.D. Saunders G. Villard 2001 On efficient sparse integer matrix Smith normal form computations J. Symb. Comput. 32 71 99
7. Dumas, J.-G., Heckenbach, F., Saunders, D., Velker, V.: Computing simplicial homology based on efficient smith normal form algorithms. In: Algebra, Geometry and Software Systems, pp. 177-207. Springer, Berlin (2003)
8. H. Edelsbrunner D. Letscher A. Zomorodian 2002 Topological persistence and simplification Discrete Comput. Geom. 28 511 533
9. Friedman, J.: Computing Betti numbers via combinatorial Laplacians. In: Proc. 28th Ann. ACM Sympos. Theory Comput., pp. 386-391 (1996)
10. M. Gameiro K. Mischaikow Th. Wanner 2005 Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation Acta Mater. 53 693 704
11. T. Kaczynski M. Mrozek M. Ślusarek 1998 Homology computation by reduction of chain complexes Comput. Math. Appl. 35 59 70
12. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Applied Mathematical Sciences, vol. 157. Springer, New York (2004)
13. Kalies, W.: Chom-A Cubical Homology Program (1999). http://www.math.fau. edu/kalies/chom.html
14. Kalies, W., Mischaikow, K., Watson, G.: Cubical approximation and computation of homology. In: Conley Index Theory, vol. 47, pp. 115-131. Banach Center Publications, Warsaw (1999)
15. S. Kirkpatrick B. Selman 1994 Critical behaviour in the satisfiability of random Boolean expressions Science 264 1297
16. Massey, W.S.: Homology and Cohomology Theory. Dekker, New York (1978)
17. K. Mischaikow M. Mrozek P. Pilarczyk 2005 Graph approach to the computation of the homology of continuous maps Found. Comput. Math. 5 199 229
18. Mrozek, M.: Homology Software (2006). http://www.ii.uj.edu.pl/~mrozek/ software/homology.html
19. M. Mrozek 2006 Index pairs algorithms Found. Comput. Math. 6 457 493
20. Mrozek, M., Pilarczyk, P., Zelazna, N.: Homology algorithm based on acyclic subspace. Comput. Math. Appl. (2008, accepted)
21. Mrozek, M., Batko, B.: Homology of representable sets (2008, submitted)
22. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984)
23. N.E. Steenrod 1940 Regular cycles of compact metric spaces Ann. Math. 41 833 851
24. Storjohann, A.: Near optimal algorithms for computing Smith normal form of integer matrices. In: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ISAAC 1996, pp. 267-274 (1996)
25. Urbańska, A.: Smith normal form algorithms for sparse matrices with applications to homology computations. M. Sc. Thesis, Jagiellonian University, Kraków (2007) (in Polish, with English Abstract)
26. M. Yannakakis 1981 Computing the minimum fill-in is NP-complete SIAM J. Matrix Anal. Appl. 2 77 79
27. A. Zomorodian G. Carlsson 2005 Computing persistent homology Discrete Comput. Geom. 33 249 274
28. Computational Homology Project: http://chomp.rutgers.edu
29. Computer Assisted Proofs in Dynamics: http://capd.wsb-nlu.edu.pl
30. Project LinBox: Exact computational linear algebra: http://www.linalg.org