Computing Minimum Complexity 1D Curve Simplifications under the Fréchet Distance

We consider the problem of simplifying curves under the Fréchet distance. Let P be a curve and ε ≥ 0 be a distance threshold. An ε-simplification is a curve within Fréchet distance ε of P . We consider ε-simplifications of minimum complexity (i.e. minimum number of vertices). Parameterized by ε, we define a continuous family of minimum complexity ε-simplifications P ε of a curve P inone dimension. We present a data structure that after linear preprocessing time can report the ε-simplification in linear output-sensitive time. Moreover, for k ≥ 1, we show how this data structure can be used to report a simplification P ε with at most k vertices that is closest to P in O(k) time

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is is NP-complete when the shape is a vertical unit-segment

Computing Minimum Complexity 1D Curve Simplifications under the Fréchet Distance

We consider the problem of simplifying curves under the Fréchet distance. Let P be a curve and ε ≥ 0 be a distance threshold. An ε-simplification is a curve within Fréchet distance ε of P . We consider ε-simplifications of minimum complexity (i.e. minimum number of vertices). Parameterized by ε, we define a continuous family of minimum complexity ε-simplifications P ε of a curve P in one dimension. We present a data structure that after linear preprocessing time can report the ε-simplification in linear output-sensitive time. Moreover, for k ≥ 1, we show how this data structure can be used to report a simplification P ε with at most k vertices that is closest to P in O(k) time

The shape of things to come: Topological Data Analysis and biology, from molecules to organisms

Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of topological data analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in data sets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features—connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex data sets. We first provide a TDA primer for biologists before exploring the use of TDA across biological sub‐disciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data are shape and shape is data, will be relevant as biology transitions into a data‐driven era where the meaningful interpretation of large data sets is a limiting factor

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is is NP-complete when the shape is a vertical unit-segment

Geometry and topology of estuary and braided river channel networks automatically extracted from topographic data

Automatic extraction of channel networks from topography in systems with multiple interconnected channels, like braided rivers and estuaries, remains a major challenge in hydrology and geomorphology. Representing channelized systems as networks provides a mathematical framework for analyzing transport and geomorphology. In this paper, we introduce a mathematically rigorous methodology and software for extracting channel network topology and geometry from digital elevation models (DEMs) and analyze such channel networks in estuaries and braided rivers. Channels are represented as network links, while channel confluences and bifurcations are represented as network nodes. We analyze and compare DEMs from the field and those generated by numerical modeling. We use a metric called the volume parameter that characterizes the volume of deposited material separating channels to quantify the volume of reworkable sediment deposited between links, which is a measure for the spatial scale associated with each network link. Scale asymmetry is observed in most links downstream of bifurcations, indicating geometric asymmetry and bifurcation stability. The length of links relative to system size scales with volume parameter value to the power of 0.24–0.35, while the number of links decreases and does not exhibit power law behavior. Link depth distributions indicate that the estuaries studied tend to organize around a deep main channel that exists at the largest scale while braided rivers have channel depths that are more evenly distributed across scales. The methods and results presented establish a benchmark for quantifying the topology and geometry of multichannel networks from DEMs with a new automatic extraction tool