Computing wave impact in self-organised mussel beds

We model the effects of byssal connections made by mussels within patterned mussel beds on bed stability as a disk graph, and propose a formula for assessing which mussels, if any, would get dislodged from the\u3cbr/\u3ebed under the impact of a wave. We formulate the computation as a flow problem, giving access to efficient algorithms to evaluate the formula. We then analyse the geometry of the graph, and show that we only need to compute a maximum flow in a restricted part of the graph, giving rise to a near-linear solution in practise

Computing the similarity between moving curves

\u3cp\u3eIn this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fréchet distance between surfaces. While the Fréchet distance between surfaces is generally NP-hard, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on a relation between obstacles.\u3c/p\u3

Computing the Fréchet distance between real-valued surfaces

\u3cp\u3eThe Fréchet distance is a well-studied measure for the similarity of shapes. While efficient algorithms for computing the Fréchet distance between curves exist, there are only few results on the Fréchet distance between surfaces. Recent work has shown that the Fréchet distance is computable between piecewise linear functions f and g : M → Rk with M a triangulated surface of genus zero. We focus on the case k = 1 and M being a topological sphere or disk with constant boundary. Intuitively, we measure the distance between terrains based solely on the height function. Our main result is that in this case computing the Fréchet distance between f and g is in NP. We additionally show that already for k = 1, computing a factor 2 - ϵ approximation of the Fréchet distance is NP-hard, showing that this problem is in fact NP-complete. We also define an intermediate distance, between contour trees, which we also show to be NP-complete to compute. Finally, we discuss how our and other distance measures between contour trees relate to each other.\u3c/p\u3

Seth says:weak Fréchet distance is faster, but only if it is continuous and in one dimension

\u3cp\u3eWe show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.\u3c/p\u3

Data structures for Fréchet queries in trajectory data

Let π be a trajectory in the plane, represented as a polyline with n edges. We show how to preprocess π into a data structure such that for any horizontal query segment σ in the plane and a subtrajectory between two vertices of π, one can quickly determine the Fréchet distance between σ and that subtrajectory. We provide data structures for these queries that need O(n2 log2 n) preprocessing time, O(n2 log2 n) space, and O(log2 n) query time. If we are interested only in the Fréchet distance between the complete trajectory π and a horizontal query segment σ, we can answer these queries in O(log2 n) time using only O(n2) space. Compilation copyright © 2017 Michiel Smid Copyright of individual papers retained by authors.All right reserved

A KDS for discrete Morse-Smale complexes

The Morse-Smale complex of a terrain is a topological complex that provides information about the features of the terrain. It consists of the critical points (minima, saddles and maxima), together with steepest-descent paths from saddles to minima and steepest-ascent paths from saddles to maxima. We describe a kinetic data structure to maintain the Morse-Smale-complex for a triangulated terrain whose vertex heights change continuously. This can be used to efficiently analyze time-varying data

Kinetic volume-based persistence for 1D terrains

Persistence is the method of choice to simplify terrains by removing insignificant features while retaining topologically important ones. Motivated by applications in geomorphology, we study volume-persistence, a variant of persistence which is based on the volume underneath the terrain (instead of the usual vertex heights). Specifically, we want to kinetically maintain a volume-simplified time-varying terrain. In this paper we describe a kinetic data structure (KDS) that maintains the pruned split tree of an area-persistent 1D terrain under linear vertex motion. The main ingredient of this KDS is an algorithm that detects those combinatorial events when a pruned part of the terrain attains a certain threshold volume

Monotone contractions of the boundary of the disc

In this paper, we study contractions of the boundary of a Riemannian 2-disc where the maximal length of the intermediate curves is minimized. We prove that with an arbitrarily small overhead in the lengths of the intermediate curves, there exists such an optimal contraction that is monotone, i.e., where the intermediate curves are simple closed curves which are pairwise disjoint. This proves a conjecture of Chambers and Rotman