A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an ℝ-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an ℝ²-valued function, also satisfying an analogous weak equivalence. For a point cloud X ⊂ ℝ^d, our trifiltration has size O(|X|^{⌈(d+1)/2⌉+1}). We present an algorithm that computes this trifiltration in time O(|X|^{⌈d/2⌉+2}), together with an implementation. Our experiments demonstrate that the implementation can handle thousands of points in ℝ³, with memory growth that is nearly linear.

The Delaunay filtration D.(X) of a point cloud X ⊂ ℝd is a central tool of computational topology. Its use is justified by the topological equivalence of D. (X) and the offset (i.e., union-of-balls) filtration of X. Given a function γ : X → ℝ, we introduce a Delaunay bifiltration DC.(γ) that satisfies an analogous topological equivalence, ensuring that DC. (γ) topologically encodes the offset filtrations of all sublevel sets of γ, as well as the topological relations between them. DC.(γ) is of size , which for d odd matches the worst-case size of D. (X). Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute DC.(γ) in time Our implementation, based on CGAL, computes DC. (γ) with modest overhead compared to computing D. (X), and handles tens of thousands of points in ℝ3 within seconds.

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\in[1,\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\infty$ case, this gives a novel proof of universality for the interleaving distance on merge trees.