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Table 1 We summarize several types of complexes that are used for PH

From: A roadmap for the computation of persistent homology

Complex K Size of K Theoretical guarantee
Čech \(2^{\mathcal {O}(N)}\) Nerve theorem
Vietoris–Rips (VR) \(2^{\mathcal {O}(N)}\) Approximates Čech complex
Alpha \(N^{\mathcal {O}(\lceil d/2 \rceil)}\) (N points in \(\mathbb {R}^{d}\)) Nerve theorem
Witness \(2^{\mathcal {O}(|L|)}\) For curves and surfaces in Euclidean space
Graph-induced complex \(2^{\mathcal {O}(|Q|)}\) Approximates VR complex
Sparsified Čech \(\mathcal {O}(N)\) Approximates Čech complex
Sparsified VR \(\mathcal {O}(N)\) Approximates VR complex
  1. We indicate the theoretical guarantees and the worst-case sizes of the complexes as functions of the cardinality N of the vertex set. For the witness complexes (see Section 5.2.4), L denotes the set of landmark points, while Q denotes the subsample set for the graph-induced complex (see Section 5.2.5).