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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).