Discrete & Computational Geometry最新文献

A Faithful Discretization of Verbose Directional Transforms. 详细方向变换的忠实离散化。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-25 DOI: 10.1007/s00454-025-00791-w

Brittany Terese Fasy, Samuel Micka, David L Millman, Anna Schenfisch, Lucia Williams

引用次数: 0

Morse Theory for the k-NN Distance Function. k-NN距离函数的Morse理论。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-19 DOI: 10.1007/s00454-025-00795-6

Yohai Reani, Omer Bobrowski

引用次数: 0

Maximum Betti Numbers of Čech Complexes. Čech配合物的最大贝蒂数。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-10 DOI: 10.1007/s00454-025-00796-5

Herbert Edelsbrunner, János Pach

{"title":"Maximum Betti Numbers of Čech Complexes.","authors":"Herbert Edelsbrunner, János Pach","doi":"10.1007/s00454-025-00796-5","DOIUrl":"https://doi.org/10.1007/s00454-025-00796-5","url":null,"abstract":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> and any radius satisfies <math> <mrow><msub><mi>β</mi> <mi>p</mi></msub> <mrow></mrow> <mo>=</mo> <mi>O</mi> <mrow><mo>(</mo> <msup><mi>N</mi> <mi>m</mi></msup> <mo>)</mo></mrow> </mrow> </math> , with <math><mrow><mi>m</mi> <mo>=</mo> <mo>min</mo> <mo>{</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mrow><mrow><mo>⌈</mo></mrow> <mi>d</mi> <mo>/</mo> <mn>2</mn> <mrow><mo>⌉</mo></mrow> </mrow> <mo>}</mo></mrow> </math> . We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of <math><mrow><mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> </math> points in <math> <msup><mrow><mi>R</mi></mrow> <mn>3</mn></msup> </math> and two radii such that the first Betti number of the Čech complex at one radius is <math> <mrow> <msup><mrow><mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> <mo>-</mo> <mn>1</mn></mrow> </math> , and the second Betti number of the Čech complex at the other radius is <math><msup><mi>n</mi> <mn>2</mn></msup> </math> .","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 2","pages":"597-624"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12953301/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147357507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the Size of Chromatic Delaunay Mosaics. 论彩色德劳内马赛克的尺寸。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-09-30 DOI: 10.1007/s00454-025-00778-7

Ranita Biswas, Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, Morteza Saghafian

{"title":"On the Size of Chromatic Delaunay Mosaics.","authors":"Ranita Biswas, Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, Morteza Saghafian","doi":"10.1007/s00454-025-00778-7","DOIUrl":"10.1007/s00454-025-00778-7","url":null,"abstract":"Given a locally finite set <math><mrow><mi>A</mi> <mo>⊆</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> and a coloring <math><mrow><mi>χ</mi> <mo>:</mo> <mi>A</mi> <mo>→</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>s</mi> <mo>}</mo></mrow> </math> , we introduce the chromatic Delaunay mosaic of <math><mi>χ</mi></math> , which is a Delaunay mosaic in <math> <msup><mrow><mi>R</mi></mrow> <mrow><mi>d</mi> <mo>+</mo> <mi>s</mi></mrow> </msup> </math> 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 <math><mrow><mi>n</mi> <mo>=</mo> <mrow><mo>#</mo> <mi>A</mi></mrow> </mrow> </math> , and the coloring is random, then the chromatic Delaunay mosaic has <math><mrow><mi>O</mi> <mo>(</mo> <msup><mi>n</mi> <mrow><mo>⌈</mo> <mi>d</mi> <mo>/</mo> <mn>2</mn> <mo>⌉</mo></mrow> </msup> <mo>)</mo></mrow> </math> cells in expectation. In contrast, for Delone sets and Poisson point processes in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> 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.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 1","pages":"24-47"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12748319/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145879476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Stability and Inference of the Euler Characteristic Transform. 欧拉特征变换的稳定性与推理。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-02-09 DOI: 10.1007/s00454-025-00763-0

Lewis Marsh, David Beers

引用次数: 0

Error Resilient Space Partitioning. 错误弹性空间分区。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-28 DOI: 10.1007/s00454-025-00804-8

Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, Ran J Tessler

{"title":"Error Resilient Space Partitioning.","authors":"Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, Ran J Tessler","doi":"10.1007/s00454-025-00804-8","DOIUrl":"https://doi.org/10.1007/s00454-025-00804-8","url":null,"abstract":"A major research area in discrete geometry is to consider the best way to partition the d-dimensional Euclidean space <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> to a discrete subset of representative values. Specifically, we study partitions of <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> into bounded-size tiles colored by one of k colors, such that tiles of the same color have a distance of at least t from each other. Such tilings allow for error-resilient rounding, as two points of the same color and distance less than t from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors k and the distance t, for various dimensions d. On the qualitative side, we show that in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , using <math><mrow><mi>k</mi> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </math> colors is both sufficient and necessary to achieve <math><mrow><mi>t</mi> <mo>></mo> <mn>0</mn></mrow> </math> . On the quantitative side, we achieve numerous upper and lower bounds on t as a function of k. In particular, for <math><mrow><mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>24</mn></mrow> </math> , we obtain sharp asymptotic bounds on t, as <math><mrow><mi>k</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> . We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat's connector-free lemma, and Čech cohomology.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 3","pages":"839-870"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13050337/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147628884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Subword Complexes and Kalai's Conjecture on Reconstruction of Spheres. 子词复合体与卡莱关于球的重构猜想。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2025-05-13 DOI: 10.1007/s00454-025-00733-6

Cesar Ceballos, Joseph Doolittle

引用次数: 0

The χ -Binding Function of d-Directional Segment Graphs. d向线段图的χ -绑定函数。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2025-05-17 DOI: 10.1007/s00454-025-00737-2

Lech Duraj, Ross J Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The <ns0:math><ns0:mi>χ</ns0:mi></ns0:math> -Binding Function of d-Directional Segment Graphs.","authors":"Lech Duraj, Ross J Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald","doi":"10.1007/s00454-025-00737-2","DOIUrl":"https://doi.org/10.1007/s00454-025-00737-2","url":null,"abstract":"Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> having at most d slopes. Since each slope induces an interval graph, it easily follows for every G in d-DIR with clique number at most <math><mi>ω</mi></math> that the chromatic number <math><mrow><mi>χ</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </math> of G is at most <math><mrow><mi>d</mi> <mi>ω</mi></mrow> </math> . We show for every even value of <math><mi>ω</mi></math> how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the <math><mi>χ</mi></math> -binding function of d-DIR is <math><mrow><mi>ω</mi> <mo>↦</mo> <mi>d</mi> <mi>ω</mi></mrow> </math> for <math><mi>ω</mi></math> even and <math><mrow><mi>ω</mi> <mo>↦</mo> <mi>d</mi> <mo>(</mo> <mi>ω</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> <mo>+</mo> <mn>1</mn></mrow> </math> for <math><mi>ω</mi></math> odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case <math><mrow><mi>d</mi> <mo>=</mo> <mn>2</mn></mrow> </math> .","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 3","pages":"758-770"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12484362/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145214309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Compact Metric Spaces with Infinite Cop Number. 具有无限Cop数的紧度量空间。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2024-10-14 DOI: 10.1007/s00454-024-00696-0

Agelos Georgakopoulos

引用次数: 0

Monochromatic Infinite Sets in Minkowski Planes. 闵可夫斯基平面上的单色无限集。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2024-11-12 DOI: 10.1007/s00454-024-00702-5

Nóra Frankl, Panna Gehér, Arsenii Sagdeev, Géza Tóth

引用次数: 0