Pub Date : 2022-02-24DOI: 10.1007/978-3-031-20624-5_44
Arunjana Das, Sandip Das, G. D. D. Fonseca, Y. Gérard, B. Rivier
{"title":"Complexity Results on Untangling Red-Blue Matchings","authors":"Arunjana Das, Sandip Das, G. D. D. Fonseca, Y. Gérard, B. Rivier","doi":"10.1007/978-3-031-20624-5_44","DOIUrl":"https://doi.org/10.1007/978-3-031-20624-5_44","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"5 1","pages":"101974"},"PeriodicalIF":0.0,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81389472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1016/j.comgeo.2022.101863
M. Huicochea, J. Leaños, Luis Manuel Rivera
{"title":"A note on the minimum number of red lines needed to pierce the intersections of blue lines","authors":"M. Huicochea, J. Leaños, Luis Manuel Rivera","doi":"10.1016/j.comgeo.2022.101863","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101863","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"54 1","pages":"101863"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79818168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4230/LIPIcs.ISAAC.2022.57
Shangqi Lu, Yufei Tao
Let P be a set of n points in R d where each point p ∈ P carries a weight drawn from a commutative monoid ( M , + , 0). Given a d -rectangle r upd (i.e., an orthogonal rectangle in R d ) and a value ∆ ∈ M , a range update adds ∆ to the weight of every point p ∈ P ∩ r upd ; given a d -rectangle r qry , a range sum query returns the total weight of the points in P ∩ r qry . The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of ˜ O ( n ) space that handles an update in ˜ O ( T upd ) time and a query in ˜ O ( T qry ) time for arbitrary functions T upd ( n ) and T qry ( n ) satisfying T upd · T qry = n . The result holds for any fixed dimensionality d ≥ 2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture. 2012 ACM Subject Classification Theory of computation → Data structures design and analysis
设P是R d中n个点的集合，其中每个点P∈P都有一个从交换单形(M， +， 0)中得到的权值。给定一个d -矩形R upd(即R d中的一个正交矩形)和一个值∆∈M，范围更新将∆加到每个点P∈P∩R upd的权值上;给定一个矩形r查询，一个范围和查询返回P∩r查询中所有点的总权重。我们的目标是将P存储在一个结构中，以支持更新和查询，并提供有吸引力的性能保证。我们描述了一个~ O (n)空间结构，它处理满足T upd·T qry = n的任意函数T upd (n)和T qry (n)在~ O (T upd)时间内的更新和在~ O (T qry)时间内的查询。这个结果对任何固定维数d≥2都成立。我们的查询更新权衡严格到受omv猜想约束的多对数因子。2012 ACM学科分类:计算理论→数据结构设计与分析
{"title":"Range Updates and Range Sum Queries on Multidimensional Points with Monoid Weights","authors":"Shangqi Lu, Yufei Tao","doi":"10.4230/LIPIcs.ISAAC.2022.57","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2022.57","url":null,"abstract":"Let P be a set of n points in R d where each point p ∈ P carries a weight drawn from a commutative monoid ( M , + , 0). Given a d -rectangle r upd (i.e., an orthogonal rectangle in R d ) and a value ∆ ∈ M , a range update adds ∆ to the weight of every point p ∈ P ∩ r upd ; given a d -rectangle r qry , a range sum query returns the total weight of the points in P ∩ r qry . The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of ˜ O ( n ) space that handles an update in ˜ O ( T upd ) time and a query in ˜ O ( T qry ) time for arbitrary functions T upd ( n ) and T qry ( n ) satisfying T upd · T qry = n . The result holds for any fixed dimensionality d ≥ 2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture. 2012 ACM Subject Classification Theory of computation → Data structures design and analysis","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"19 1","pages":"102030"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84786991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1007/978-3-030-61792-9_22
Mincheol Kim, S. Yoon, Hee-Kap Ahn
{"title":"Shortest Rectilinear Path Queries to Rectangles in a Rectangular Domain","authors":"Mincheol Kim, S. Yoon, Hee-Kap Ahn","doi":"10.1007/978-3-030-61792-9_22","DOIUrl":"https://doi.org/10.1007/978-3-030-61792-9_22","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"77 1","pages":"101796"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79283512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-24DOI: 10.4230/LIPIcs.ISAAC.2021.14
Sarita de Berg, F. Staals
Our aim is to develop dynamic data structures that support $k$-nearest neighbors ($k$-NN) queries for a set of $n$ point sites in the plane in $O(f(n) + k)$ time, where $f(n)$ is some polylogarithmic function of $n$. The key component is a general query algorithm that allows us to find the $k$-NN spread over $t$ substructures simultaneously, thus reducing an $O(tk)$ term in the query time to $O(k)$. Combining this technique with the logarithmic method allows us to turn any static $k$-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, $O(log^2n/loglog n +k)$ query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic emph{geodesic} $k$-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only $k$-NN data structure. More generally, we obtain a dynamic planar $k$-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.
{"title":"Dynamic Data Structures for k-Nearest Neighbor Queries","authors":"Sarita de Berg, F. Staals","doi":"10.4230/LIPIcs.ISAAC.2021.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.14","url":null,"abstract":"Our aim is to develop dynamic data structures that support $k$-nearest neighbors ($k$-NN) queries for a set of $n$ point sites in the plane in $O(f(n) + k)$ time, where $f(n)$ is some polylogarithmic function of $n$. The key component is a general query algorithm that allows us to find the $k$-NN spread over $t$ substructures simultaneously, thus reducing an $O(tk)$ term in the query time to $O(k)$. Combining this technique with the logarithmic method allows us to turn any static $k$-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, $O(log^2n/loglog n +k)$ query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic emph{geodesic} $k$-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only $k$-NN data structure. More generally, we obtain a dynamic planar $k$-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"24 1","pages":"101976"},"PeriodicalIF":0.0,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76141256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-15DOI: 10.4230/LIPIcs.ISAAC.2021.3
B. Aronov, M. D. Berg, J. Cardinal, Esther Ezra, J. Iacono, M. Sharir
We present subquadratic algorithms in the algebraic decision-tree model for several textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the plane, we want to count, for each triangle $Deltain C$, the number of intersection points between the segments of $A$ and those of $B$ that lie in $Delta$. The problems considered in this paper have been studied by Chan~(2020), who gave algorithms that solve them, in the standard real-RAM model, in $O((n^2/log^2n)log^{O(1)}log n)$ time. We present solutions in the algebraic decision-tree model whose cost is $O(n^{60/31+varepsilon})$, for any $varepsilon>0$. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl~(2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the emph{order type} of the lines, a"handicap"that turns out to be beneficial for speeding up our algorithm.
{"title":"Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model","authors":"B. Aronov, M. D. Berg, J. Cardinal, Esther Ezra, J. Iacono, M. Sharir","doi":"10.4230/LIPIcs.ISAAC.2021.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.3","url":null,"abstract":"We present subquadratic algorithms in the algebraic decision-tree model for several textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the plane, we want to count, for each triangle $Deltain C$, the number of intersection points between the segments of $A$ and those of $B$ that lie in $Delta$. The problems considered in this paper have been studied by Chan~(2020), who gave algorithms that solve them, in the standard real-RAM model, in $O((n^2/log^2n)log^{O(1)}log n)$ time. We present solutions in the algebraic decision-tree model whose cost is $O(n^{60/31+varepsilon})$, for any $varepsilon>0$. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl~(2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the emph{order type} of the lines, a\"handicap\"that turns out to be beneficial for speeding up our algorithm.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"42 1","pages":"101945"},"PeriodicalIF":0.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84862050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-20DOI: 10.1016/j.comgeo.2022.101862
H. Alpert, R. Barnes, S. Bell, A. Mauro, N. Nevo, N. Tucker, H. Yang
{"title":"Routing by matching on convex pieces of grid graphs","authors":"H. Alpert, R. Barnes, S. Bell, A. Mauro, N. Nevo, N. Tucker, H. Yang","doi":"10.1016/j.comgeo.2022.101862","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101862","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"11 1","pages":"101862"},"PeriodicalIF":0.0,"publicationDate":"2021-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75789458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.1016/j.comgeo.2020.101709
David Kübel, E. Langetepe
{"title":"On the approximation of shortest escape paths","authors":"David Kübel, E. Langetepe","doi":"10.1016/j.comgeo.2020.101709","DOIUrl":"https://doi.org/10.1016/j.comgeo.2020.101709","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"44 1","pages":"101709"},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79766421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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