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Ripser: efficient computation of Vietoris-Rips persistence barcodes

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Pith's one-line read Ripser computes exact Vietoris–Rips persistence barcodes without storing the filtration coboundary matrix, using apparent pairs to skip most column reductions.

desk verdict Ripser is a genuinely important methods paper: the apparent/emergent pair machinery is clean and proved, and the only real soft spots are the distinct-distances assumption behind the dim-1 shortcut and the single-run benchmarks. read the letter →

arxiv 1908.02518 v2 pith:H75WWCY2 submitted 2019-08-07 math.AT cs.CGcs.MS

Ulrich Bauer This is my paper · ORCID
classification math.ATcs.CGcs.MS MSC 55N31
keywords persistenthomologyVietoris-RipscomplexpersistencebarcodematrixreductioncohomologyapparentpairsdiscreteMorsetheoryimplicit
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper presents the algorithm behind Ripser, a program that computes Vietoris–Rips persistence barcodes exactly. The central claim is that the filtration coboundary matrix, which most persistence software builds and stores explicitly, can be represented implicitly: columns are recomputed on the fly, so memory use is low and most columns are never reduced at all. The key object is the apparent pair, a pair $(\sigma,\tau)$ in which $\sigma$ is the youngest facet of $\tau$ and $\tau$ is the oldest cofacet of $\sigma$; such pairs are already reduced, can be detected locally without matrix operations, and form a discrete gradient. The paper proves that, when all pairwise distances are distinct, every zero-persistence pair in dimension 1 is an apparent pair. Benchmarks against previous implementations show large reductions in both running time and memory.

What carries the argument

The load-bearing object is the apparent pair: in a simplexwise filtration, a pair $(\sigma,\tau)$ with $\dim\tau=\dim\sigma+1$ such that $\sigma$ is the youngest facet of $\tau$ and $\tau$ is the oldest cofacet of $\sigma$. Apparent pairs are persistence pairs, already reduced in the boundary or coboundary matrix, and together they form a discrete gradient. The algorithm wraps this in three supporting devices: the lexicographic refinement of the Vietoris-Rips filtration (order simplices by diameter, then dimension, then reverse colexicographic vertex order), the combinatorial number system (an order-preserving bijection from decreasing vertex tuples to natural numbers that makes facet and cofacet enumeration cheap), and implicit matrix reduction (compute the coboundary of a simplex when needed, keep only the reduction columns for death indices, and store pivots of already reduced columns instead of the reduced matrix). The emergent-pairs shortcut extends the same idea to pairs that become apparent during reduction, so that many coboundary columns are never fully constructed.

What would settle it

Run the same Vietoris-Rips computation on a finite metric space with many equal pairwise distances, such as equally spaced points on a line, and on a generic perturbation of it; if the tied case shows many zero-persistence dimension-1 pairs that are not apparent and a corresponding increase in reduced columns, the distinct-distances assumption is the load-bearing condition. A direct test of Theorem 3.10 is to search for any metric space with all pairwise distances distinct whose simplexwise refinement has a zero-persistence pair in dimension 1 that is not apparent; finding one would refute the theorem.

Watch

Extended reading notes

Core claim

The paper's own claim is that an exact persistence computation for a Vietoris-Rips filtration does not require an explicit filtration coboundary matrix, and that most columns of that matrix are already reduced. Ripser encodes the filtration order and the coboundary operator algorithmically, using the combinatorial number system to index simplices as integers and enumerating cofacets on demand, while storing only the reduction-matrix columns for death indices. Apparent pairs are the conceptual hinge: they are persistence pairs by Lemma 3.3, they constitute a discrete gradient by Lemma 3.5, and they can be read directly from facet and cofacet comparisons. Under the genericity assumption of distinct pairwise distances, Theorem 3.10 shows that in dimension 1 the zero-persistence pairs of the simplexwise refinement coincide exactly with the apparent pairs, so no column corresponding to a zero-persistence pair needs reduction. The combination of implicit reduction, cohomology, clearing, and apparent and emergent shortcuts yields an exact barcode with a far smaller column-reduction workload, which the experiments attribute to a large speedup and memory saving.

Load-bearing premise

The efficiency guarantee rests on the genericity assumption that all pairwise distances are distinct; with ties, zero-persistence pairs in dimension 1 need not be apparent pairs, so more columns must be reduced and the speedup can degrade.

Editorial extensions

If this is right

  • For any finite metric space with all pairwise distances distinct, the dimension-1 zero-persistence pairs of the lexicographic refinement are exactly the apparent pairs, so the shortcut detects every zero-persistence pair without column reduction.
  • Because apparent pairs are already reduced, their pivots need not be stored or looked up, so memory use scales with the number of non-apparent columns rather than with the full simplex count.
  • Computing persistent cohomology with clearing reduces the set of columns to reduce to the death columns plus one essential 0-dimensional column, making the reduction workload roughly proportional to the number of death simplices.
  • Apparent pairs give a canonical way to turn any total order on the simplices of a complex into a discrete Morse function, connecting the speedup directly to discrete Morse theory.
  • The same algorithmic representation of the coboundary applies to any filtration whose cofacets can be enumerated and whose filtration order can be compared, so the method is not tied to Vietoris-Rips complexes alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural stress test not performed in the paper is to measure how the shortcut degrades as pairwise distances become tied; Theorem 3.10's condition suggests that tie-breaking order could be engineered to restore the apparent-pairs property.
  • The apparent-pairs gradient may be useful beyond barcode computation, for example in simplifying Rips complexes by canceling critical cells or in constructing smaller discrete Morse complexes with predictable persistence.
  • The implicit-coboundary strategy transfers to other algebraic settings, such as cellular filtrations or algebraic discrete Morse theory, wherever an algebraic apparent pair with a unit coefficient can be detected locally; the gain depends on cofacet enumeration being much cheaper than memory access.
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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper presents the algorithm underlying the software Ripser for computing Vietoris–Rips persistence barcodes. The main ideas are threefold: the filtration coboundary matrix is represented implicitly and columns are recomputed on demand; the simplexwise lexicographic refinement of the Rips filtration is used with an implicit column-reduction scheme; and the notions of apparent pairs and emergent pairs are introduced so that many zero-persistence columns can be identified and skipped without performing the reduction. The theoretical part proves that apparent pairs form a discrete gradient, that apparent pairs are persistence pairs, and that, under distinct pairwise distances, every zero-persistence pair in dimension 1 is apparent. The experimental part compares Ripser with Dionysus, DIPHA, Gudhi, and Eirene on several datasets, reporting large speedups and memory reductions.

Significance. If the claims hold, this is a practically important contribution: exact Vietoris-Rips barcodes are computed with substantially less memory and time than previous software, and the apparent-pairs construction is of independent theoretical interest as a discrete gradient associated to a simplexwise filtration. The paper's mathematical core is sound: Lemmas 3.3, 3.5, 3.6, and 3.8 are proved carefully, Theorem 3.10 is a genuine structural result under its stated hypothesis, and the counting argument in Section 3.3 is parameter-free. The software and the benchmark harness are publicly available, which strengthens reproducibility. The principal caveat is that the most aggressive dimension-1 shortcut is proved only for metrics with distinct pairwise distances; the experimental claims are not scoped to that assumption, and no tied-distance benchmarks are reported.

major comments (2)
  1. [Section 3.5, Theorem 3.10] The dimension-1 zero-pair shortcut is proved only under the assumption of distinct pairwise distances, and the proof uses this assumption twice: once to make E the unique facet of F attaining diam(F), and once to ensure that every other edge of the cofacet tau has smaller diameter. Under ties, neither step is guaranteed, so a zero-persistence pair of the simplexwise refinement need not be apparent, and in principle it need not be initially emergent either. The sentence in Section 5 stating that "as predicted by Theorem 3.10, in dimension 1 every zero pair is an apparent pair" therefore overstates the scope, and Tables 1-3 contain no tied-distance input. Since the abstract's unqualified efficiency claim that "most columns are never reduced" rests substantially on this shortcut, please either prove a bound for tied inputs, add benchmarks with tied metrics, or explicitly scope the efficiency claim to the generic case.
  2. [Section 5, Tables 1 and 3] The experimental section reports no tied-distance datasets, even though the theoretical guarantee for the dimension-1 shortcut is restricted to distinct pairwise distances by Theorem 3.10. Integer-valued and grid-like distance matrices are common in applications of Vietoris-Rips persistence, so the central performance claim should be tested on at least one such input, or the paper should state clearly that the reported speedups are demonstrated only for generic inputs. In addition, the timings in Tables 1 and 3 appear to be single runs with no variance information; the difference between 15.3 s and 15.6 s in the sphere3 row of Table 3 is within typical run-to-run noise and should not be presented as a meaningful ordering.
minor comments (4)
  1. [Section 3.5, proof of Proposition 3.9] The last sentence of the proof, "Similarly, E must be the oldest cofacet of E in the filtration order," appears to contain a copy-and-paste error: it should say that E must be the youngest facet of F.
  2. [Section 4] The text says that facets are enumerated "implemented in the class simplex_coboundary_enumerator"; this is presumably a typo for the facet/boundary enumerator class, since the cofacet enumerator is described separately.
  3. [Section 5, Tables 1 and 3] The caption should state explicitly that each reported timing is a single run and that no confidence intervals or repeated runs are provided, so that small differences in the tables are not over-interpreted.
  4. [Section 5, Table 2] The column headers "non-zero pairs," "non-emergent," "non-shortcut," "non-apparent," and "total pairs" should be defined in the caption; the current layout makes it difficult to verify the statement that all dimension-1 zero pairs are shortcut pairs.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: algorithmic claims are proven from standard persistence theory and benchmarked against external packages.

full rationale

Ripser's derivation chain is self-contained against standard persistence theory. Correctness of the matrix reduction (Algorithm 1) is attributed to Cohen-Steiner et al. (Proposition 3.1), clearing to Chen and Kerber, cohomology duality to de Silva et al., and the lexicographic refinement with apparent/emergent pair shortcuts is established by Lemmas 3.3, 3.5, 3.6, 3.8 and Propositions 3.9, 3.12. Theorem 3.10 is an actual converse proof, not a restatement: it uses the distinct pairwise distances assumption to force the zero-persistence edge to be the youngest facet of the triangle and to make every other edge of the oldest cofacet older, then invokes Lemma 3.3 to conclude the cofacet coincides with the death simplex; no quantity is fitted and then called a prediction. The efficiency claims are benchmarked against four external packages (Dionysus, DIPHA, Gudhi, Eirene), and the same-author citations (DIPHA, PHAT) are baseline implementations rather than premises of the algorithm's correctness. The only caveat is the explicitly stated genericity assumption in Theorem 3.10, which narrows the scope of one efficiency guarantee under tied distances; this is a stated limitation, not a circular step. No equation or algorithmic shortcut reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters: the algorithm has no fitted constants and the benchmarks are external comparisons. The proof burden is carried by standard persistence theory (interval decomposition, matrix reduction) and prior algorithmic results (clearing, cohomology duality, enclosing-radius threshold), all cited. Apparent and emergent pairs are the paper's new constructs; they are defined formally, proved to be persistence pairs, and verified in experiments.

assumptions (6)
  • standard math Persistence modules over a field decompose into interval summands (Gabriel/Crawley-Boevey).
    Used in Section 2 to justify the barcode as a complete invariant; cited to [11, 53].
  • standard math Matrix reduction of the filtration boundary matrix yields persistence pairs and essential indices (Proposition 3.1, Cohen-Steiner et al.).
    Basis for all reduction algorithms in the paper; cited to [10].
  • domain assumption Persistence barcodes of homology and cohomology coincide, and the relative cohomology coboundary matrix is the reversed transpose of the boundary matrix.
    Adopted from de Silva et al. [13]; used in Section 3.3 to compute via cohomology.
  • domain assumption Clearing optimization remains valid when extended to the reduction matrix (Algorithm 2).
    The extension is argued in Section 3.2; it is central to implicit reduction.
  • domain assumption Above the minimum enclosing radius, the Vietoris-Rips complex is a simplicial cone and homology is trivial.
    Used in Section 4 as default threshold; attributed to Henselman-Petrusek [21].
  • standard math Combinatorial number system provides an order-preserving bijection between lexicographically ordered simplices and natural numbers.
    Used throughout Section 4 for simplex indexing; cited to [25, 41].
invented entities (2)
  • Apparent pair (Definition 3.2) independent evidence
    purpose: Identify persistence pairs and discrete gradient pairs directly from the filtration order without matrix reduction, and skip zero-persistence columns in Rips computations.
    Proved to be persistence pairs (Lemma 3.3) and to form a discrete gradient (Lemma 3.5); benchmarks (Tables 2-3) show most zero-persistence pairs are apparent, and independent software (Eirene, Ripser++) uses the same notion.
  • Emergent pair (Definition 3.11) independent evidence
    purpose: Detect persistence pairs that become apparent during column reduction, allowing the coboundary construction to be cut short.
    Defined algorithmically and used in Section 3.5; correctness follows from Proposition 3.1 and the reduction order, and Table 3 shows substantial speedups when enabled.

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Pith. "Pith review of Ripser: efficient computation of Vietoris-Rips persistence barcodes." pith.science (2026). https://pith.science/paper/H75WWCY2

@misc{pith2026190802518,
  author       = {Pith},
  title        = {Pith review of: Ripser: efficient computation of Vietoris-Rips persistence barcodes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H75WWCY2}},
  note         = {Machine review of arXiv:1908.02518}
}
read the original abstract

We present an algorithm for the computation of Vietoris-Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix. Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest. Our implementation shows substantial improvements over previous software both in time and memory usage.

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Reference graph

Works this paper leans on

54 extracted references · 51 canonical work pages · cited by 2 Pith papers

  1. [1]

    S. A. Barannikov. The framed Morse complex and its invariants. In Singularities and bifurcations, volume 21 of Adv. Soviet Math., pages 93–115. Amer. Math. Soc., Providence, RI, 1994

    work page 1994

  2. [2]

    U. Bauer. Ripser: a lean C++ code for the computation of Vi etoris–Rips persistence barcodes. http://ripser.org, 2015–2019

    work page 2015

  3. [3]

    Lifespan Functors and Natural Dualities in Persistent Homology

    U. Bauer and M. Schmahl. The structure of morphisms in per sistent homology, I. Functorial dualities. Preprint, 2020. arXiv:2012.12881

    work page Pith review arXiv 2020

  4. [4]

    Bauer, M

    U. Bauer, M. Kerber, and J. Reininghaus. Distributed computation of persistent homology . In Proceedings of the Sixteenth Workshop on Algorithm Enginee ring and Experiments (ALENEX) , pages 31–38. SIAM, 2014. Software available at https://github.com/DIPHA/dipha

    work page 2014

  5. [5]

    Bauer, M

    U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner. PHAT – Persistent Homology Algo- rithms Toolbox . Journal of Symbolic Computation , 78:76–90, 2017. Software available at https://bitbucket.org/phat-code/phat. 22 /u1D45B /u1D45D explicit implicit discarded emergent apparent sphere3 192 2 15.3 s, 4.1 GB 15.6 s, 4.1 GB 5.6 s, 196 MB 1.0 s, 196 MB 0.66 s, ...

    work page 2017

  6. [6]

    Binchi, E

    J. Binchi, E. Merelli, M. Rucco, G. Petri, and F. Vaccarin o. jHoles: A tool for understanding biological complex networks via clique weight rank persist ent homology . Electronic Notes in Theoretical Computer Science, 306:5–18, 2014. Software available at http://www.jholes.eu

    work page 2014

  7. [7]

    Chazal, V

    F. Chazal, V . de Silva, M. Glisse, and S. Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, 2016

    work page 2016

  8. [8]

    Chen and M

    C. Chen and M. Kerber. Persistent homology computation with a twist. In 27th European Workshop on Computational Geometry (EuroCG) , pages 197–200, 2011

    work page 2011

Show all 54 references
  1. [9]

    Chen and M

    C. Chen and M. Kerber. An output-sensitive algorithm for persistent homology . Comput. Geom., 46(4):435–447, 2013

    2013

  2. [10]

    Cohen-Steiner, H

    D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. Vines and vineyards by updating persistence in linear time. In SCG ’06: Proceedings of the twenty-second annual symposium on Computational geometry, pages 119–126, 2006

    2006

  3. [11]

    Crawley-Boevey.Decomposition of pointwise finite-dimensional persistence modules

    W . Crawley-Boevey.Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and Its Applications, 14(5):1550066+, 2015

    2015

  4. [12]

    de Silva, D

    V . de Silva, D. Morozov, and M. Vejdemo-Johansson. Persistent cohomology and circular coordi- nates. Discrete Comput. Geom., 45(4):737–759, 2011

    2011

  5. [13]

    de Silva, D

    V . de Silva, D. Morozov, and M. Vejdemo-Johansson.Dualities in persistent (co)homology. Inverse Problems, 27(12):124003, 17, 2011

    2011

  6. [14]

    Delgado-Friedrichs, V

    O. Delgado-Friedrichs, V . Robins, and A. Sheppard. Skeletonization and partitioning of digital im- ages using discrete Morse theory. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(3):654–666, 2015

    2015

  7. [15]

    Edelsbrunner and J

    H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, 2010

    2010

  8. [16]

    Edelsbrunner, D

    H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification . Discrete & Computational Geometry, 28(4):511–533, 2002

    2002

  9. [17]

    R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134(1):90–145, 1998

    1998

  10. [18]

    M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75–263. Springer, New Y ork, 1987. 23

    1987

  11. [19]

    Hausmann

    J.-C. Hausmann. On the Vietoris–Rips complexes and a co homology theory for metric spaces. In Prospects in topology (Princeton, NJ, 1994) , volume 138 of Ann. of Math. Stud. , pages 175–188. Princeton Univ. Press, Princeton, NJ, 1995

    1994

  12. [20]

    Henselman and R

    G. Henselman and R. Ghrist. Matroid filtrations and computational persistent ho- mology. arXiv preprint, 2016. arXiv:1606.00199. Software available at http://gregoryhenselman.org/eirene/

    2016 arXiv

  13. [21]

    Henselman-Petrusek

    G. Henselman-Petrusek. Matroids and Canonical Forms: Theory and Applications . PhD thesis, University of Pennsylvania, 2017

    2017

  14. [22]

    S. Huber. Libstick: a C++ library to compute persistent homology. https://www.sthu.org/code/libstick/, 2013–2014

    2013

  15. [23]

    Jöllenbeck and V

    M. Jöllenbeck and V . Welker.Minimal resolutions via algebraic discrete Morse theory. Mem. Amer. Math. Soc., 197(923):vi+74, 2009

    2009

  16. [24]

    M. Kahle. Random geometric complexes. Discrete & Computational Geometry , 45(3):553–573, 2011

    2011

  17. [25]

    D. E. Knuth. Generating all combinations , In The Art of Computer Programming , volume 4A: Combinatorial Algorithms, Part 1, chapter 7.2.1.3, pages 3 55–389. Addison-Wesley Professional, 2011

    2011

  18. [26]

    D. N. Kozlov. Discrete Morse theory for free chain complexes . C. R. Math. Acad. Sci. Paris , 340 (12):867–872, 2005

    2005

  19. [27]

    J. B. Kruskal, Jr. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc., 7:48–50, 1956

    1956

  20. [28]

    L. Lampret. Chain complex reduction via fast digraph traversal. Preprint, 2020. arXiv:1903.00783

    2020 arXiv

  21. [29]

    Latschev

    J. Latschev. Vietoris–Rips complexes of metric spaces near a closed Riem annian manifold. Arch. Math. (Basel), 77(6):522–528, 2001

    2001

  22. [30]

    R. H. Lewis. CTL: The Computational Topology Library. http://ctl.appliedtopology.org, 2013–2016

    2013

  23. [31]

    Mendoza-Smith and J

    R. Mendoza-Smith and J. Tanner. Parallel multi-scale reduction of persistent homol- ogy filtrations . arXiv preprint, 2017. arXiv:1708.04710. Software available at https://github.com/rodrgo/OpenPH

    2017 arXiv

  24. [32]

    Milosavljević, D

    N. Milosavljević, D. Morozov, and P. Škraba. Zigzag persistent homology in matrix multiplication time. In SoCG ’11: Proceedings of the twenty-seventh annual symposi um on Computational geometry, pages 216–225. ACM, New Y ork, 2011

    2011

  25. [33]

    D. Morozov. Dionysus: a C++ library for computing persi stent homology, 2006–2013. http://www.mrzv.org/software/dionysus

    2006

  26. [34]

    D. Morozov. Dionysus 2: a computational topology packa ge focused on persistent homology, 2014–2020. http://www.mrzv.org/software/dionysus2

    2014

  27. [35]

    Morozov and H

    D. Morozov and H. Edelsbrunner. Persistent homology. In J. E. Goodman, J. O’Rourke, and C. D. Tóth, editors, Handbook of Discrete and Computational Geometry , chapter 24. CRC Press, third edition, 2017. 24

    2017

  28. [36]

    Morozov and A

    D. Morozov and A. Nigmetov. Towards lockfree persistent homology . In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Archite ctures, SPAA ’20, page 555–557. Association for Computing Machinery, 2020

    2020

  29. [37]

    J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, 1984

    1984

  30. [38]

    V . Nanda. Perseus, the persistent homology software. http://people.maths.ox.ac.uk/nanda/perseus/index.html, 2010–2013

    2010

  31. [39]

    Olver and A

    S. Olver and A. Townsend. A practical framework for infinite-dimensional linear alge bra. In Proceedings of the 1st First Workshop for High Performance T echnical Computing in Dynamic Languages, HPTCDL ’14, page 57–62. IEEE Press, 2014

    2014

  32. [40]

    Otter, M

    N. Otter, M. A. Porter, U. Tillmann, P. Grindrod, and H. A . Harrington. A roadmap for the computation of persistent homology. EPJ Data Science, 6(1):17, 2017

    2017

  33. [41]

    E. Pascal. Sopra una formola numerica . Giornale di Matematiche, 25:45–49, 1887

  34. [42]

    Perry, V

    P. Perry, V . de Silva, L. Kettner, and A. Zomorodian. Ple x: Simplicial complexes in MATLAB. http://mii.stanford.edu/research/comptop/programs/, 2000–2006

    2000

  35. [43]

    Sexton and M

    H. Sexton and M. Vejdemo-Johansson. jPlex. http://www.math.colostate.edu/~adams/jplex/index.html, 2008

    2008

  36. [44]

    V . d. Silva and G. Carlsson. Topological estimation using witness complexes. In M. Gross, H. Pfister, M. Alexa, and S. Rusinkiewicz, editors, SPBG’04 Symposium on Point - Based Graphics 2004 . The Eurographics Association, 2004

    2004

  37. [45]

    Sköldberg

    E. Sköldberg. Morse theory from an algebraic viewpoint. Trans. Amer. Math. Soc., 358(1):115–129, 2006

    2006

  38. [46]

    R. E. Tarjan. A class of algorithms which require nonlinear time to mainta in disjoint sets . J. Comput. System Sci., 18(2):110–127, 1979

    1979

  39. [47]

    A. Tausz. pHom: Persistent Homology in R. https://cran.r-project.org/src/contrib/Archive/phom/, 2011-2014

    2011

  40. [48]

    Tausz, M

    A. Tausz, M. Vejdemo-Johansson, and H. Adams. JavaPlex : A research software pack- age for persistent (co)homology. In H. Hong and C. Y ap, edito rs, Proceedings of ICMS 2014, Lecture Notes in Computer Science 8592, pages 129–136, 201 4. Software available at http://appliedtop...

    2014

  41. [49]

    GUDHI User and Reference Manual

    The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial Board, 2015. Software available at http://gudhi.gforge.inria.fr

    2015

  42. [50]

    Vietoris

    L. Vietoris. Über den höheren Zusammenhang kompakter Räume und eine Klas se von zusammen- hangstreuen Abbildungen. Math. Ann., 97(1):454–472, 1927

    1927

  43. [51]

    Zhang, M

    S. Zhang, M. Xiao, C. Guo, L. Geng, H. Wang, and X. Zhang. H ypha: a framework based on separation of parallelisms to accelerate persistent homol ogy matrix reduction. In Proceedings of the ACM International Conference on Supercomputing , pages 69–81, 2019. Software available ...

    2019

  44. [52]

    Zhang, M

    S. Zhang, M. Xiao, and H. Wang. GPU-Accelerated Computation of Vietoris-Rips Persistenc e Barcodes. In S. Cabello and D. Z. Chen, editors, 36th International Symposium on Computational Geometry (SoCG 2020), volume 164 of Leibniz International Proceedings in Informatics (LIPIcs...

    2020

  45. [53]

    Zomorodian and G

    A. Zomorodian and G. Carlsson. Computing persistent homology . Discrete & Computational Geometry, 33(2):249–274, 2005

    2005

  46. [54]

    M. Čufar. Ripserer.jl: flexible and efficient persistent homology comp utation in julia . Journal of Open Source Software, 5(54):2614, 2020. 26

    2020

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