@inproceedings{fa1d5263da984c8a8911e4dd538d7abc,
title = "Grid peeling and the affine curve-shortening flow",
abstract = "In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call grid peeling) is the convex-layer decomposition of subsets G ? Z2 of the integer grid, previously studied for the particular case G = {1, . . ., m}2 by Har-Peled and Lidick{\'y} (2013). The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where G = N2 is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times t > 0. We prove that, in the grid peeling of N2, (1) the number of grid points removed up to iteration n is ?(n3/2 log n); and (2) the boundary at iteration n is sandwiched between two hyperbolas that are separated from each other by a constant factor.",
keywords = "Computational geometry, Constant factors;, Differential geometry, Iterative methods, L-shaped, Number of Grids, Theoretical arguments",
author = "David Eppstein and Sariel Har-Peled and Gabriel Nivasch",
note = "Publisher Copyright: Copyright {\textcopyright} 2018 by SIAM.; 20th Workshop on Algorithm Engineering and Experiments, ALENEX 2018 ; Conference date: 07-01-2018 Through 08-01-2018",
year = "2018",
doi = "https://doi.org/10.1137/1.9781611975055.10",
language = "الإنجليزيّة",
series = "Proceedings of the Workshop on Algorithm Engineering and Experiments",
publisher = "Society for Industrial and Applied Mathematics Publications",
pages = "109--116",
editor = "Rasmus Pagh and Suresh Venkatasubramanian",
booktitle = "2018 Proceedings of the 20th Workshop on Algorithm Engineering and Experiments, ALENEX 2018",
address = "الولايات المتّحدة",
}