. Secondly, T ) in P, with q s dividing Q T . Consequently, during all the computation of the while loop in line 3, q s | Q S . When exiting this while loop, low(j, Q S ) is undefined, implying in particular that low(j, q s ) is undefined and column j of R(Z qs ) is zero, and hence reduced

·. and Z. Qr, The persistent homology algorithm described in Section 8.1, applied on K with coefficients in a field k, requires O(n 3 ) operations in k. For a field Z q these operations take constant time and the algorithm has complexity O(n 3 ) The output of the algorithm is the persistence diagram, which has size O(n) for any field. For a set of prime numbers {q 1 , · · · , q r }, let n ? be the total number of distinct pairs in all persistence diagrams for the persistent homology of K with coefficient fields Z q 1, We express the complexity of the modular References [1] CGAL

S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Y. Wu, An optimal algorithm for approximate nearest neighbor searching fixed dimensions, Journal of the ACM, vol.45, issue.6, pp.891-923, 1998.
DOI : 10.1145/293347.293348

M. Atiyah and I. Macdonald, Introduction To Commutative Algebra . Addison-Wesley series in mathematics, 1994.

D. Attali, A. Lieutier, and D. Salinas, Efficient data structure for representing and simplifying simplicial complexes in high dimensions, Proceedings of the 27th annual ACM symposium on Computational geometry, SoCG '11, pp.279-304, 2012.
DOI : 10.1145/1998196.1998277
URL : https://hal.archives-ouvertes.fr/hal-00785082

D. Attali, A. Lieutier, and D. Salinas, Efficient data structure for representing and simplifying simplicial complexes in high dimensions, Proceedings of the 27th annual ACM symposium on Computational geometry, SoCG '11, pp.279-304, 2012.
DOI : 10.1145/1998196.1998277
URL : https://hal.archives-ouvertes.fr/hal-00785082

D. Attali, A. Lieutier, and D. Salinas, Vietoris???Rips complexes also provide topologically correct reconstructions of sampled shapes, Computational Geometry, vol.46, issue.4, pp.448-465, 2013.
DOI : 10.1016/j.comgeo.2012.02.009
URL : https://hal.archives-ouvertes.fr/hal-00579864

U. Bauer, M. Kerber, and J. Reininghaus, Clear and Compress: Computing Persistent Homology in Chunks, Topological Methods in Data Analysis and Visualization III, pp.103-117
DOI : 10.1007/978-3-319-04099-8_7

U. Bauer, M. Kerber, and J. Reininghaus, Distributed Computation of Persistent Homology, ALENEX, pp.31-38, 2014.
DOI : 10.1137/1.9781611973198.4

J. L. , B. , and R. Sedgewick, Fast algorithms for sorting and searching strings, Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.5-7, 1997.

I. N. Bernstein, I. M. Gelfand, and V. A. Ponomarev, COXETER FUNCTORS AND GABRIEL'S THEOREM, Russian Mathematical Surveys, vol.28, issue.2, pp.17-32, 1973.
DOI : 10.1070/RM1973v028n02ABEH001526

J. Boissonnat, O. Devillers, S. Pion, M. Teillaud, and M. Yvinec, Triangulations in CGAL, Computational Geometry, vol.22, issue.1-3, pp.5-19, 2002.
DOI : 10.1016/S0925-7721(01)00054-2
URL : https://hal.archives-ouvertes.fr/hal-01179408

J. Boissonnat, T. K. Dey, and C. Maria, The compressed annotation matrix: An efficient data structure for computing persistent cohomology, ESA, pp.695-706, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00761468

J. Boissonnat, T. K. Dey, and C. Maria, The compressed annotation matrix: An efficient data structure for computing persistent cohomology, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00761468

J. Boissonnat and C. Maria, The simplex tree: An efficient data structure for general simplicial complexes, ESA, pp.731-742, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01108416

J. Boissonnat and C. Maria, Computing Persistent Homology with Various Coefficient Fields in a Single Pass
DOI : 10.1007/978-3-662-44777-2_16
URL : https://hal.archives-ouvertes.fr/hal-00922572

J. Boissonnat and C. Maria, The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes, Algorithmica, vol.132, issue.23, pp.406-427, 2014.
DOI : 10.1007/s00453-014-9887-3
URL : https://hal.archives-ouvertes.fr/hal-01108416

E. Brisson, Representing geometric structures ind dimensions: Topology and order, Discrete & Computational Geometry, vol.5, issue.1, pp.387-426, 1993.
DOI : 10.1007/BF02189330

P. Bubenik and J. A. Scott, Categorification of Persistent Homology, Discrete & Computational Geometry, vol.33, issue.2, pp.600-627, 2014.
DOI : 10.1007/s00454-014-9573-x

A. Benjamin, T. Burton, J. Lewiner, J. Paixão, and . Spreer, Parameterized complexity of discrete Morse theory, Symposium on Computational Geometry, pp.127-136, 2013.

E. Gunnar, V. Carlsson, and S. De, Zigzag persistence, Foundations of Computational Mathematics, vol.10, issue.4, pp.367-405, 2010.

E. Gunnar, . Carlsson, D. Vin-de-silva, and . Morozov, Zigzag persistent homology and real-valued functions, Symposium on Computational Geometry, pp.247-256, 2009.

E. Gunnar, T. Carlsson, . Ishkhanov, A. Vin-de-silva, and . Zomorodian, On the local behavior of spaces of natural images, International Journal of Computer Vision, vol.76, issue.1, pp.1-12, 2008.

F. Chazal, D. Cohen-steiner, and A. Lieutier, A Sampling Theory for Compact Sets in Euclidean Space, Discrete & Computational Geometry, vol.18, issue.3, pp.461-479, 2009.
DOI : 10.1007/s00454-009-9144-8
URL : https://hal.archives-ouvertes.fr/hal-00864493

C. Chen and M. Kerber, Persistent homology computation with a twist, Proceedings 27th European Workshop on Computational Geometry, 2011.

C. Chen and M. Kerber, An output-sensitive algorithm for persistent homology, Computational Geometry, vol.46, issue.4, pp.435-447, 2013.
DOI : 10.1016/j.comgeo.2012.02.010
URL : http://hdl.handle.net/11858/00-001M-0000-0024-476F-A

D. Cohen-steiner, H. Edelsbrunner, and J. Harer, Stability of Persistence Diagrams, Discrete & Computational Geometry, vol.37, issue.1, pp.103-120, 2007.
DOI : 10.1007/s00454-006-1276-5

D. Cohen-steiner, H. Edelsbrunner, and D. Morozov, Vines and vineyards by updating persistence in linear time, Proceedings of the twenty-second annual symposium on Computational geometry , SCG '06, pp.119-126, 2006.
DOI : 10.1145/1137856.1137877

H. Thomas, C. Cormen, R. L. Stein, C. E. Rivest, and . Leiserson, Introduction to Algorithms, 2001.

S. Vin-de, A weak characterisation of the delaunay triangulation, Geometriae Dedicata, vol.135, issue.1, pp.39-64, 2008.

D. Vin-de-silva, M. Morozov, and . Vejdemo-johansson, Dualities in persistent (co)homology. CoRR, abs/1107, 2011.

D. Vin-de-silva, M. Morozov, and . Vejdemo-johansson, Persistent Cohomology and Circular Coordinates, Discrete & Computational Geometry, vol.33, issue.2, pp.737-759, 2011.
DOI : 10.1007/s00454-011-9344-x

C. Jose, A. Delfinado, and H. Edelsbrunner, An incremental algorithm for betti numbers of simplicial complexes, Symposium on Computational Geometry, pp.232-239, 1993.

C. Jose, A. Delfinado, and H. Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere, Computer Aided Geometric Design, vol.12, issue.7, pp.771-784, 1995.

H. Derksen and J. Weyman, Quiver representations, Notices of the AMS, vol.52, issue.2, pp.200-206, 2005.

K. Tamal, H. Dey, S. Edelsbrunner, D. V. Guha, and . Nekhayev, Topology preserving edge contraction, Publ. Inst. Math, pp.23-45, 1998.

K. Tamal, F. Dey, Y. Fan, and . Wang, Computing topological persistence for simplicial maps, Symposium on Computational Geometry, p.345, 2014.

J. Dumas, B. D. Saunders, and G. Villard, On Efficient Sparse Integer Matrix Smith Normal Form Computations, Journal of Symbolic Computation, vol.32, issue.1-2, pp.71-99, 2001.
DOI : 10.1006/jsco.2001.0451

H. Edelsbrunner and J. Harer, Computational Topology -an Introduction, 2010.

H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological Persistence and Simplification, Discrete & Computational Geometry, vol.28, issue.4, pp.511-533, 2002.
DOI : 10.1007/s00454-002-2885-2

R. Engelking, General topology. Monografie matematyczne, 1977.

M. Fürer, Faster Integer Multiplication, SIAM Journal on Computing, vol.39, issue.3, pp.979-1005, 2009.
DOI : 10.1137/070711761

J. Von, Z. Gathen, and J. Gerhard, Modern Computer Algebra, 2003.

A. Hatcher, Algebraic Topology, 2001.

A. B. Lee, K. S. Pedersen, and D. Mumford, The nonlinear statistics of high-contrast patches in natural images, International Journal of Computer Vision, vol.54, issue.1/2, pp.83-103, 2003.
DOI : 10.1023/A:1023705401078

P. Lienhardt, N-DIMENSIONAL GENERALIZED COMBINATORIAL MAPS AND CELLULAR QUASI-MANIFOLDS, International Journal of Computational Geometry & Applications, vol.04, issue.03, pp.275-324, 1994.
DOI : 10.1142/S0218195994000173

C. Maria and . Gudhi, simplicial complexes and persistent homology packages. https://project.inria.fr/gudhi/software

C. Maria, J. Boissonnat, M. Glisse, and M. Yvinec, The Gudhi Library: Simplicial Complexes and Persistent Homology, International Congress on Mathematical Software, pp.167-174, 2014.
DOI : 10.1007/978-3-662-44199-2_28
URL : https://hal.archives-ouvertes.fr/hal-01005601

C. Maria and S. Y. Oudot, Zigzag Persistence via Reflections and Transpositions, SODA, 2015.
DOI : 10.1137/1.9781611973730.14
URL : https://hal.archives-ouvertes.fr/hal-01091949

S. Martin, A. Thompson, E. A. Coutsias, and J. Watson, Topology of cyclo-octane energy landscape, The Journal of Chemical Physics, vol.132, issue.23, p.234115, 2010.
DOI : 10.1063/1.3445267

N. Milosavljevic, D. Morozov, and P. Skraba, Zigzag persistent homology in matrix multiplication time, Proceedings of the 27th annual ACM symposium on Computational geometry, SoCG '11, 2011.
DOI : 10.1145/1998196.1998229
URL : https://hal.archives-ouvertes.fr/inria-00520171

K. Mischaikow and V. Nanda, Morse Theory for Filtrations and Efficient Computation of Persistent Homology, Discrete & Computational Geometry, vol.37, issue.10, pp.330-353, 2013.
DOI : 10.1007/s00454-013-9529-6

D. Morozov, Persistence algorithm takes cubic time in worst case, BioGeometry News, Dept. Comput. Sci, 2005.

M. David, S. Mount, . Arya, and . Ann, Approximate Nearest Neighbors Library

J. R. Munkres, Elements of algebraic topology, 1984.

S. Y. Oudot, Persistence theory: from quiver representations to data analysis. Book in preparation, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01247501

Y. Steve, D. R. Oudot, and . Sheehy, Zigzag Zoology: Rips Zigzags for Homology Inference, J. Foundations of Computational Mathematics, 2014.

S. Roman, Fundamentals of Group Theory: An Advanced Approach . SpringerLink : Bücher, 2011.
DOI : 10.1007/978-0-8176-8301-6

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. ijm, pp.64-94, 1962.

D. Sheehy, Linear-Size Approximations to the Vietoris???Rips Filtration, Discrete & Computational Geometry, vol.49, issue.4, pp.778-796, 2013.
DOI : 10.1007/s00454-013-9513-1
URL : https://hal.archives-ouvertes.fr/hal-01111878

C. Webb, Decomposition of Graded Modules Proceedings of the, pp.565-571, 1985.

E. Welzl, Smallest enclosing disks (balls and ellipsoids) New Results and New Trends in Computer Science, pp.359-370, 1991.

A. Zomorodian, The tidy set, Proceedings of the 2010 annual symposium on Computational geometry, SoCG '10, pp.257-266, 2010.
DOI : 10.1145/1810959.1811004

A. Zomorodian and G. E. Carlsson, Computing Persistent Homology, Discrete & Computational Geometry, vol.33, issue.2, pp.249-274, 2005.
DOI : 10.1007/s00454-004-1146-y
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.5064