article published yes Chao Chen author 3E92416E-F248-11E8-B48F-1D18A9856A87 Michael Kerber author 36E4574A-F248-11E8-B48F-1D18A9856A870000-0002-8030-9299 HeEd department In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ &gt; 0, it returns only those homology classes with persistence at least Γ. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant δ ∈ (0, 1), the running time is O (C (1 - δ) Γ R d (n) log n), where C (1 - δ) Γ is the number of homology classes with persistence at least (1 - δ) Γ, n is the total number of simplices in the complex, d its dimension, and R d (n) is the complexity of computing the rank of an n × n matrix with O (d n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O (C (1 - δ) Γ n 2.376) algorithm, an O (C (1 - δ) Γ n 2.28) Las-Vegas algorithm, or an O (C (1 - δ) Γ n 2 + ε{lunate}) Monte-Carlo algorithm for an arbitrary ε{lunate} &gt; 0. The space complexity of the Monte-Carlo version is bounded by O (d n) = O (n log n). Elsevier2013 eng 00031443700000410.1016/j.comgeo.2012.02.010 464435 - 447 https://research-explorer.ista.ac.at/record/3367 Chen C, Kerber M. An output sensitive algorithm for persistent homology. <i>Computational Geometry: Theory and Applications</i>. 2013;46(4):435-447. doi:<a href="https://doi.org/10.1016/j.comgeo.2012.02.010">10.1016/j.comgeo.2012.02.010</a> Chen, Chao, and Michael Kerber. “An Output Sensitive Algorithm for Persistent Homology.” <i>Computational Geometry: Theory and Applications</i>, vol. 46, no. 4, Elsevier, 2013, pp. 435–47, doi:<a href="https://doi.org/10.1016/j.comgeo.2012.02.010">10.1016/j.comgeo.2012.02.010</a>. C. Chen and M. Kerber, “An output sensitive algorithm for persistent homology,” <i>Computational Geometry: Theory and Applications</i>, vol. 46, no. 4. Elsevier, pp. 435–447, 2013. Chen, C., &#38; Kerber, M. (2013). An output sensitive algorithm for persistent homology. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href="https://doi.org/10.1016/j.comgeo.2012.02.010">https://doi.org/10.1016/j.comgeo.2012.02.010</a> Chen, Chao, and Michael Kerber. “An Output Sensitive Algorithm for Persistent Homology.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2013. <a href="https://doi.org/10.1016/j.comgeo.2012.02.010">https://doi.org/10.1016/j.comgeo.2012.02.010</a>. C. Chen, M. Kerber, Computational Geometry: Theory and Applications 46 (2013) 435–447. Chen C, Kerber M. 2013. An output sensitive algorithm for persistent homology. Computational Geometry: Theory and Applications. 46(4), 435–447. 29392018-12-11T12:00:27Z2025-09-29T13:26:21Z