Abstract Given a locally finite set $$A \subseteq {{\mathbb R}}^d$$ A ⊆ R d and a coloring $$\chi :A \rightarrow \{0,1,\ldots ,s\}$$ χ : A → { 0 , 1 , … , s } , we introduce the chromatic Delaunay mosaic of $$\chi $$ χ , which is a Delaunay mosaic in $${{\mathbb R}}^{d+s}$$ R d + s that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with $$n = {{\#}{A}}$$ n = # A , and the coloring is random, then the chromatic Delaunay mosaic has $$O(n^{{\lceil d/2 \rceil }})$$ O ( n ⌈ d / 2 ⌉ ) cells in expectation. In contrast, for Delone sets and Poisson point processes in $${{\mathbb R}}^d$$ R d , the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in $${{\mathbb R}}^2$$ R 2 all colorings of a well spread set of n points have chromatic Delaunay mosaics of size O ( n ). This encourages the use of chromatic Delaunay mosaics in applications.

In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain persistent homology -- a multi-scale topological descriptor -- for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.

Given a finite set, $A \subseteq \mathbb{R}^2$, and a subset, $B \subseteq A$, the \emph{MST-ratio} is the combined length of the minimum spanning trees of $B$ and $A \setminus B$ divided by the length of the minimum spanning tree of $A$. The question of the supremum, over all sets $A$, of the maximum, over all subsets $B$, is related to the Steiner ratio, and we prove this sup-max is between $2.154$ and $2.427$. Restricting ourselves to $2$-dimensional lattices, we prove that the sup-max is $2.0$, while the inf-max is $1.25$. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than $1.25$.

We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.

Abstract The Voronoi tessellation in $${{{\mathbb {R}}}}^d$$ R d is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.