Table 1.

We summarize several types of complexes that are used for PH

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

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).