Pub Date : 2020-10-22DOI: 10.1007/978-3-030-83508-8_10
Stav Ashur, M. J. Katz
{"title":"A 4-Approximation of the $frac{2pi }{3}$-MST","authors":"Stav Ashur, M. J. Katz","doi":"10.1007/978-3-030-83508-8_10","DOIUrl":"https://doi.org/10.1007/978-3-030-83508-8_10","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"29 1","pages":"101914"},"PeriodicalIF":0.0,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86048531","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 : 2020-10-01DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.2
Sayan Bandyapadhyay, Neeraj Kumar, S. Suri, Kasturi R. Varadarajan
Abstract In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is an approximation framework that gives an O ( n α ( n ) ) -approximation for polygonal obstacles, where α ( n ) denotes the inverse Ackermann's function. For pseudodisks and rectilinear polygons, the same technique achieves an O ( n ) -approximation. The technique also gives O ( n ) -approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.
{"title":"Improved Approximation Bounds for the Minimum Constraint Removal Problem","authors":"Sayan Bandyapadhyay, Neeraj Kumar, S. Suri, Kasturi R. Varadarajan","doi":"10.4230/LIPIcs.APPROX-RANDOM.2018.2","DOIUrl":"https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.2","url":null,"abstract":"Abstract In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is an approximation framework that gives an O ( n α ( n ) ) -approximation for polygonal obstacles, where α ( n ) denotes the inverse Ackermann's function. For pseudodisks and rectilinear polygons, the same technique achieves an O ( n ) -approximation. The technique also gives O ( n ) -approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"29 1","pages":"101650"},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87801387","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 : 2020-10-01DOI: 10.1016/j.comgeo.2020.101656
I. Aldana-Galván, Carlos Alegría-Galicia, J. L. Álvarez-Rebollar, N. Marín-Nevárez, E. Solís-Villarreal, J. Urrutia, C. Velarde
{"title":"Finding Minimum Witness Sets in Orthogonal Polygons","authors":"I. Aldana-Galván, Carlos Alegría-Galicia, J. L. Álvarez-Rebollar, N. Marín-Nevárez, E. Solís-Villarreal, J. Urrutia, C. Velarde","doi":"10.1016/j.comgeo.2020.101656","DOIUrl":"https://doi.org/10.1016/j.comgeo.2020.101656","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"4 1","pages":"101656"},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86396259","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 : 2020-09-30DOI: 10.4230/LIPIcs.ISAAC.2020.13
M. V. Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, J. Vermeulen
Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.
{"title":"Between Shapes, Using the Hausdorff Distance","authors":"M. V. Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, J. Vermeulen","doi":"10.4230/LIPIcs.ISAAC.2020.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2020.13","url":null,"abstract":"Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"9 1","pages":"101817"},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84216355","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 : 2020-09-16DOI: 10.4230/LIPIcs.ISAAC.2020.9
Joachim Gudmundsson, Y. Sha, Sampson Wong
In 2012 Driemel et al. cite{DBLP:journals/dcg/DriemelHW12} introduced the concept of $c$-packed curves as a realistic input model. In the case when $c$ is a constant they gave a near linear time $(1+varepsilon)$-approximation algorithm for computing the Frechet distance between two $c$-packed polygonal curves. Since then a number of papers have used the model. In this paper we consider the problem of computing the smallest $c$ for which a given polygonal curve in $mathbb{R}^d$ is $c$-packed. We present two approximation algorithms. The first algorithm is a $2$-approximation algorithm and runs in $O(dn^2 log n)$ time. In the case $d=2$ we develop a faster algorithm that returns a $(6+varepsilon)$-approximation and runs in $O((n/varepsilon^3)^{4/3} polylog (n/varepsilon)))$ time. We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of $c$-packedness is a useful realistic input model for many curves and trajectories.
{"title":"Approximating the packedness of polygonal curves","authors":"Joachim Gudmundsson, Y. Sha, Sampson Wong","doi":"10.4230/LIPIcs.ISAAC.2020.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2020.9","url":null,"abstract":"In 2012 Driemel et al. cite{DBLP:journals/dcg/DriemelHW12} introduced the concept of $c$-packed curves as a realistic input model. In the case when $c$ is a constant they gave a near linear time $(1+varepsilon)$-approximation algorithm for computing the Frechet distance between two $c$-packed polygonal curves. Since then a number of papers have used the model. \u0000In this paper we consider the problem of computing the smallest $c$ for which a given polygonal curve in $mathbb{R}^d$ is $c$-packed. We present two approximation algorithms. The first algorithm is a $2$-approximation algorithm and runs in $O(dn^2 log n)$ time. In the case $d=2$ we develop a faster algorithm that returns a $(6+varepsilon)$-approximation and runs in $O((n/varepsilon^3)^{4/3} polylog (n/varepsilon)))$ time. \u0000We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of $c$-packedness is a useful realistic input model for many curves and trajectories.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"19 1","pages":"101920"},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88100493","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 : 2020-05-27DOI: 10.1007/s00454-020-00210-2
M. Huicochea
{"title":"On the Number of Monochromatic Lines in $pmb {mathbb {R}}^d$","authors":"M. Huicochea","doi":"10.1007/s00454-020-00210-2","DOIUrl":"https://doi.org/10.1007/s00454-020-00210-2","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"70 1","pages":"1061-1086"},"PeriodicalIF":0.0,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73659132","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 : 2020-04-01DOI: 10.4230/LIPIcs.ESA.2017.65
Thomas Schibler, S. Suri
We study the problem of k-dominance in a set of d-dimensional vectors, prove bounds on the number of maxima (skyline vectors), under both worst-case and average-case models, perform experimental evaluation using synthetic and real-world data, and explore an application of k-dominant skyline for extracting a small set of top-ranked vectors in high dimensions where the full skylines can be unmanageably large.
{"title":"K-Dominance in Multidimensional Data: Theory and Applications","authors":"Thomas Schibler, S. Suri","doi":"10.4230/LIPIcs.ESA.2017.65","DOIUrl":"https://doi.org/10.4230/LIPIcs.ESA.2017.65","url":null,"abstract":"We study the problem of k-dominance in a set of d-dimensional vectors, prove bounds on the number of maxima (skyline vectors), under both worst-case and average-case models, perform experimental evaluation using synthetic and real-world data, and explore an application of k-dominant skyline for extracting a small set of top-ranked vectors in high dimensions where the full skylines can be unmanageably large.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"09 1","pages":"101594"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85964131","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 : 2020-02-18DOI: 10.4230/LIPIcs.ISAAC.2020.57
A. Dumitrescu, Anirban Ghosh, Csaba D. T'oth
A unit disk graph $G$ on a given set of points $P$ in the plane is a geometric graph where an edge exists between two points $p,q in P$ if and only if $|pq| leq 1$. A subgraph $G'$ of $G$ is a $k$-hop spanner if and only if for every edge $pqin G$, the topological shortest path between $p,q$ in $G'$ has at most $k$ edges. We obtain the following results for unit disk graphs. (i) Every $n$-vertex unit disk graph has a $5$-hop spanner with at most $5.5n$ edges. We analyze the family of spanners constructed by Biniaz (WADS 2019) and improve the upper bound on the number of edges from $9n$ to $5.5n$. (ii) Using a new construction, we show that every $n$-vertex unit disk graph has a $3$-hop spanner with at most $11n$ edges. (iii) Every $n$-vertex unit disk graph has a $2$-hop spanner with $O(n^{3/2})$ edges. This is the first construction of a $2$-hop spanner with a subquadratic number of edges. (iv) For every sufficiently large $n$, there exists a set $P$ of $n$ points such that every plane hop spanner on $P$ has hop stretch factor at least $4$. Previously, no lower bound greater than $2$ was known. (v) For every point set on a circle, there exists a plane $4$-hop spanner. As such, this provides a tight bound for points on a circle. (vi) The maximum degree of $k$-hop spanners cannot be bounded above by a function of $k$.
{"title":"Sparse Hop Spanners for Unit Disk Graphs","authors":"A. Dumitrescu, Anirban Ghosh, Csaba D. T'oth","doi":"10.4230/LIPIcs.ISAAC.2020.57","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2020.57","url":null,"abstract":"A unit disk graph $G$ on a given set of points $P$ in the plane is a geometric graph where an edge exists between two points $p,q in P$ if and only if $|pq| leq 1$. A subgraph $G'$ of $G$ is a $k$-hop spanner if and only if for every edge $pqin G$, the topological shortest path between $p,q$ in $G'$ has at most $k$ edges. We obtain the following results for unit disk graphs. \u0000(i) Every $n$-vertex unit disk graph has a $5$-hop spanner with at most $5.5n$ edges. We analyze the family of spanners constructed by Biniaz (WADS 2019) and improve the upper bound on the number of edges from $9n$ to $5.5n$. \u0000(ii) Using a new construction, we show that every $n$-vertex unit disk graph has a $3$-hop spanner with at most $11n$ edges. \u0000(iii) Every $n$-vertex unit disk graph has a $2$-hop spanner with $O(n^{3/2})$ edges. This is the first construction of a $2$-hop spanner with a subquadratic number of edges. \u0000(iv) For every sufficiently large $n$, there exists a set $P$ of $n$ points such that every plane hop spanner on $P$ has hop stretch factor at least $4$. Previously, no lower bound greater than $2$ was known. \u0000(v) For every point set on a circle, there exists a plane $4$-hop spanner. As such, this provides a tight bound for points on a circle. \u0000(vi) The maximum degree of $k$-hop spanners cannot be bounded above by a function of $k$.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"62 1","pages":"101808"},"PeriodicalIF":0.0,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90412363","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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