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 |