Computational Geometry-Theory and Applications最新文献

{"title":"Engineering an algorithm for constructing low-stretch geometric graphs with near-greedy average degrees","authors":"FNU Shariful ,&nbsp;Justin Weathers ,&nbsp;Anirban Ghosh ,&nbsp;Giri Narasimhan","doi":"10.1016/j.comgeo.2025.102201","DOIUrl":"10.1016/j.comgeo.2025.102201","url":null,"abstract":"<div><div>We design and engineer <span>Fast-Sparse-Spanner</span>, a simple and practical (fast and memory-efficient) algorithm for constructing sparse low stretch factor geometric graphs on large pointsets in the plane. To our knowledge, this is the first practical algorithm to construct fast low stretch factor graphs on large pointsets with average degrees (hence, the number of edges) competitive with that of greedy spanners, the sparsest known class of Euclidean geometric spanners. Although theoretically not guaranteed to produce <em>t</em>-spanners, we always found in our rigorous experiments that <span>Fast-Sparse-Spanner</span> generated near-greedy size <em>t</em>-spanners.</div><div>To evaluate our implementation in terms of computation speed, memory usage, and quality of output, we performed extensive experiments with synthetic and real-world pointsets, and by comparing it to our closest competitor <span>Bucketing</span>, the fastest known greedy spanner algorithm for pointsets in the plane, devised by Alewijnse et al. (2017) <span><span>[5]</span></span>. Our experiment with constructing a 1.1-spanner on a large synthetic pointset with 128<em>K</em> points uniformly distributed within a square shows more than a 41-fold speedup with roughly a third of the memory usage of that of <span>Bucketing</span>, but with only a 3% increase in the average degree of the resulting graph. When ran on a pointset with a million points drawn from the same distribution, we observed a 130-fold speedup, with roughly a fourth of the memory usage of that of <span>Bucketing</span>, and just a 6% increase in the average degree. In terms of diameter, the graphs generated by <span>Fast-Sparse-Spanner</span> beat greedy spanners in most cases (have substantially lower diameter) while maintaining near-greedy average degree. Further, our algorithm can be easily parallelized to take advantage of parallel environments.</div><div>We share the implementations via <span>GitHub</span> for broader uses and future research.</div><div><strong>GitHub repository.</strong> <span><span>https://github.com/ghoshanirban/FSS</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102201"},"PeriodicalIF":0.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Realizable dimension of periodic frameworks","authors":"Ryoshun Oba,&nbsp;Shin-ichi Tanigawa","doi":"10.1016/j.comgeo.2025.102200","DOIUrl":"10.1016/j.comgeo.2025.102200","url":null,"abstract":"<div><div>Belk and Connelly introduced the realizable dimension <span><math><mi>rd</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a finite graph <em>G</em>, which is the minimum nonnegative integer <em>d</em> such that every framework <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> in any dimension admits a framework in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with the same edge lengths. They characterized finite graphs with realizable dimension at most 1, 2, or 3 in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to <span><math><mi>Z</mi></math></span>-symmetric graphs. We give a forbidden minor characterization of <span><math><mi>Z</mi></math></span>-symmetric graphs with realizable dimension at most 1 or 2, and show that the characterization can be checked in linear time when a graph is given as a quotient <span><math><mi>Z</mi></math></span>-labeled graph.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102200"},"PeriodicalIF":0.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144071810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parallel line centers with guaranteed separation","authors":"Chaeyoon Chung ,&nbsp;Taehoon Ahn ,&nbsp;Sang Won Bae ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102185","DOIUrl":"10.1016/j.comgeo.2025.102185","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points in the plane and an integer <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>k</em>-line-center problem asks <em>k</em> slabs whose union covers <em>P</em> that minimizes the maximum width of the <em>k</em> slabs. In this paper, we introduce a new variant of the <em>k</em>-line-center problem for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, in which the resulting <em>k</em> lines are parallel and a prescribed separation between two line centers is guaranteed. More precisely, we define a measure of separation, namely the gap-ratio of <em>k</em> parallel slabs, to be the minimum distance between any two slabs, divided by the width of the smallest slab enclosing the <em>k</em> slabs. We present efficient algorithms for the following problems: (1) Given a real <span><math><mn>0</mn><mo>&lt;</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn></math></span>, compute <em>k</em> parallel slabs of minimum width that cover <em>P</em> with gap-ratio at least <em>ρ</em>. (2) Compute <em>k</em> parallel slabs that cover <em>P</em> with maximum possible gap-ratio. Our algorithms run in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>k</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msubsup><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>k</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time, respectively, using <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></math></span> space, where <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span> denotes the maximum possible gap-ratio of any <em>k</em> parallel slabs that cover <em>P</em>. Using linear space, the running times only slightly increase to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>⋅</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msubsup><mo>⋅</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102185"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constrained boundary labeling","authors":"Thomas Depian ,&nbsp;Martin Nöllenburg ,&nbsp;Soeren Terziadis ,&nbsp;Markus Wallinger","doi":"10.1016/j.comgeo.2025.102191","DOIUrl":"10.1016/j.comgeo.2025.102191","url":null,"abstract":"<div><div>Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce <em>grouping</em> and <em>ordering constraints</em> in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is <span>NP</span>-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is <span>NP</span>-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102191"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounds on the edge-length ratio of 2-outerplanar graphs","authors":"Emilio Di Giacomo ,&nbsp;Walter Didimo ,&nbsp;Giuseppe Liotta ,&nbsp;Henk Meijer ,&nbsp;Fabrizio Montecchiani ,&nbsp;Stephen Wismath","doi":"10.1016/j.comgeo.2025.102192","DOIUrl":"10.1016/j.comgeo.2025.102192","url":null,"abstract":"<div><div>The edge-length ratio of a planar straight-line drawing Γ of a graph <em>G</em> is the largest ratio between the lengths of every pair of edges of Γ. If the ratio is measured by considering only pairs of edges that are incident to a common vertex, we talk about local edge-length ratio. The (local) edge-length ratio of a planar graph is the infimum over all (local) edge-length ratios of its planar straight-line drawings. It is known that the edge-length ratio of outerplanar graphs is upper bounded by a constant, while there exist graph families with non-constant outerplanarity that have non-constant lower bounds on their edge-length ratios. In this paper we prove an <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> lower bound on the local edge-length ratio (and hence on the edge-length ratio) of the <em>n</em>-vertex 2-outerplanar graphs. We also prove a constant upper bound on the edge-length ratio of Halin graphs, pseudo-Halin graphs, and their generalizations.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102192"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Connected matchings","authors":"Oswin Aichholzer ,&nbsp;Sergio Cabello ,&nbsp;Viola Mészáros ,&nbsp;Patrick Schnider ,&nbsp;Jan Soukup","doi":"10.1016/j.comgeo.2025.102174","DOIUrl":"10.1016/j.comgeo.2025.102174","url":null,"abstract":"<div><div>We show that each set of <span><math><mi>n</mi><mo>⩾</mo><mn>2</mn></math></span> points in the plane in general position has a straight-line matching with at least <span><math><mo>(</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>27</mn></math></span> edges whose segments form a connected set, and such a matching can be computed in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. As an upper bound, we show that for some planar point sets in general position the largest matching whose segments form a connected set has <span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌉</mo></math></span> edges. We also consider a colored version, where each edge of the matching should connect points with different colors.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102174"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143550937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On exact covering with unit disks","authors":"Ji Hoon Chun,&nbsp;Christian Kipp,&nbsp;Sandro Roch","doi":"10.1016/j.comgeo.2025.102193","DOIUrl":"10.1016/j.comgeo.2025.102193","url":null,"abstract":"<div><div>We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an exact cover is not possible.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102193"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Flips in odd matchings","authors":"Oswin Aichholzer ,&nbsp;Anna Brötzner ,&nbsp;Daniel Perz ,&nbsp;Patrick Schnider","doi":"10.1016/j.comgeo.2025.102184","DOIUrl":"10.1016/j.comgeo.2025.102184","url":null,"abstract":"<div><div>Let <span><math><mi>P</mi></math></span> be a set of <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></math></span> points in the plane in general position. We define the graph <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> whose vertex set is the set of all plane matchings on <span><math><mi>P</mi></math></span> with exactly <em>m</em> edges. Two vertices in <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> are connected if the two corresponding matchings have <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span> edges in common. In this work we show that <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> is connected and give an upper bound of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> on its diameter. Moreover, we present a lower bound of <span><math><mi>n</mi><mo>−</mo><mn>2</mn></math></span> and an upper bound of <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for the diameter of <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> for <span><math><mi>P</mi></math></span> in convex position.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102184"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Revisiting the Fréchet distance between piecewise smooth curves","authors":"Jacobus Conradi ,&nbsp;Anne Driemel ,&nbsp;Benedikt Kolbe","doi":"10.1016/j.comgeo.2025.102194","DOIUrl":"10.1016/j.comgeo.2025.102194","url":null,"abstract":"<div><div>Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div><div>We assume that the algorithm is given two continuous curves, each consisting of a sequence of <em>m</em>, resp. <em>n</em>, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mo>)</mo></math></span> time, leading to a computation of the Fréchet distance that runs in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time.</div><div>Furthermore, we study approximation algorithms for piecewise smooth curves that are also <em>c</em>-packed for some fixed value <em>c</em>. We adapt the existing framework for polygonal curves that leads to a near-linear <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation to the Fréchet distance to the setting of piecewise smooth curves.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102194"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pattern formation for fat robots with memory","authors":"Rusul J. Alsaedi,&nbsp;Joachim Gudmundsson,&nbsp;André van Renssen","doi":"10.1016/j.comgeo.2025.102189","DOIUrl":"10.1016/j.comgeo.2025.102189","url":null,"abstract":"<div><div>Given a set of <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> rounds with probability at least <span><math><mn>1</mn><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>q</mi></mrow></msup></math></span>. The algorithms are collision-free and do not require the knowledge of the number of robots.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102189"},"PeriodicalIF":0.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}