Boris Aronov, Tsuri Farhana, Matthew J. Katz, Indu Ramesh
It is unlikely that the discrete Fr'echet distance between two curves of length $n$ can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, $P$, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to $P$ in sublinear time. Since there is evidence that this is impossible for query curves of length $Theta(n^alpha)$, for any $alpha > 0$, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds. We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, $t$-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph $G$ in the family, so that, given a query segment and a pair $u,v$ of vertices in $G$, one can quickly compute the smallest discrete Fr'echet distance between the segment and any $(u,v)$-path in $G$. The answer is exact, if $t=1$, and approximate if $t>1$.
{"title":"Discrete Fréchet Distance Oracles","authors":"Boris Aronov, Tsuri Farhana, Matthew J. Katz, Indu Ramesh","doi":"arxiv-2404.04065","DOIUrl":"https://doi.org/arxiv-2404.04065","url":null,"abstract":"It is unlikely that the discrete Fr'echet distance between two curves of\u0000length $n$ can be computed in strictly subquadratic time. We thus consider the\u0000setting where one of the curves, $P$, is known in advance. In particular, we\u0000wish to construct data structures (distance oracles) of near-linear size that\u0000support efficient distance queries with respect to $P$ in sublinear time. Since\u0000there is evidence that this is impossible for query curves of length\u0000$Theta(n^alpha)$, for any $alpha > 0$, we focus on query curves of (small)\u0000constant length, for which we are able to devise distance oracles with the\u0000desired bounds. We extend our tools to handle subcurves of the given curve, and even\u0000arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we\u0000construct an oracle that can quickly compute the distance between a short\u0000polygonal path (the query) and a path in the preprocessed tree between two\u0000query-specified vertices. Moreover, we define a new family of geometric graphs,\u0000$t$-local graphs (which strictly contains the family of geometric spanners with\u0000constant stretch), for which a similar oracle exists: we can preprocess a graph\u0000$G$ in the family, so that, given a query segment and a pair $u,v$ of vertices\u0000in $G$, one can quickly compute the smallest discrete Fr'echet distance\u0000between the segment and any $(u,v)$-path in $G$. The answer is exact, if $t=1$,\u0000and approximate if $t>1$.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587003","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}
Gr"unbaum's equipartition problem asked if for any measure on $mathbb{R}^d$ there are always $d$ hyperplanes which divide $mathbb{R}^d$ into $2^d$ $mu$-equal parts. This problem is known to have a positive answer for $dle 3$ and a negative one for $dge 5$. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for $dle 2$ and there is reason to expect it to have a negative answer for $dge 3$. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of $8n$ in $mathbb{R}^3$ can be split evenly by $3$ mutually orthogonal planes. To our surprise, it seems the probability that a random set of $8$ points chosen uniformly and independently in the unit cube does not admit such a partition is less than $0.001$.
{"title":"On the orthogonal Grünbaum partition problem in dimension three","authors":"Gerardo L. Maldonado, Edgardo Roldán-Pensado","doi":"arxiv-2404.01504","DOIUrl":"https://doi.org/arxiv-2404.01504","url":null,"abstract":"Gr\"unbaum's equipartition problem asked if for any measure on $mathbb{R}^d$\u0000there are always $d$ hyperplanes which divide $mathbb{R}^d$ into $2^d$\u0000$mu$-equal parts. This problem is known to have a positive answer for $dle 3$\u0000and a negative one for $dge 5$. A variant of this question is to require the\u0000hyperplanes to be mutually orthogonal. This variant is known to have a positive\u0000answer for $dle 2$ and there is reason to expect it to have a negative answer\u0000for $dge 3$. In this note we exhibit measures that prove this. Additionally,\u0000we describe an algorithm that checks if a set of $8n$ in $mathbb{R}^3$ can be\u0000split evenly by $3$ mutually orthogonal planes. To our surprise, it seems the\u0000probability that a random set of $8$ points chosen uniformly and independently\u0000in the unit cube does not admit such a partition is less than $0.001$.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587027","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}
We describe the heuristics used by the Shadoks team in the CG:SHOP 2024 Challenge. Each instance consists of a convex polygon called container and a multiset of items, where each item is a simple polygon and has an associated value. The goal is to pack some of the items inside the container using translations, in order to maximize the sum of their values. Our strategy consists of obtaining good initial solutions and improving them with local search. To obtain the initial solutions we used integer programming and a carefully designed greedy approach.
{"title":"Shadoks Approach to Knapsack Polygonal Packing","authors":"Guilherme D. da Fonseca, Yan Gerard","doi":"arxiv-2403.20123","DOIUrl":"https://doi.org/arxiv-2403.20123","url":null,"abstract":"We describe the heuristics used by the Shadoks team in the CG:SHOP 2024\u0000Challenge. Each instance consists of a convex polygon called container and a\u0000multiset of items, where each item is a simple polygon and has an associated\u0000value. The goal is to pack some of the items inside the container using\u0000translations, in order to maximize the sum of their values. Our strategy\u0000consists of obtaining good initial solutions and improving them with local\u0000search. To obtain the initial solutions we used integer programming and a\u0000carefully designed greedy approach.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587024","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}
Annika Bonerath, Martin Nöllenburg, Soeren Terziadis, Markus Wallinger, Jules Wulms
Schematic depictions in text books and maps often need to label specific point features with a text label. We investigate one variant of such a labeling, where the image contour is a circle and the labels are placed as circular arcs along the circumference of this circle. To map the labels to the feature points, we use orbital-radial leaders, which consist of a circular arc concentric with the image contour circle and a radial line to the contour. In this paper, we provide a framework, which captures various dimensions of the problem space as well as several polynomial time algorithms and complexity results for some problem variants.
{"title":"On Orbital Labeling with Circular Contours","authors":"Annika Bonerath, Martin Nöllenburg, Soeren Terziadis, Markus Wallinger, Jules Wulms","doi":"arxiv-2403.19052","DOIUrl":"https://doi.org/arxiv-2403.19052","url":null,"abstract":"Schematic depictions in text books and maps often need to label specific\u0000point features with a text label. We investigate one variant of such a\u0000labeling, where the image contour is a circle and the labels are placed as\u0000circular arcs along the circumference of this circle. To map the labels to the\u0000feature points, we use orbital-radial leaders, which consist of a circular arc\u0000concentric with the image contour circle and a radial line to the contour. In\u0000this paper, we provide a framework, which captures various dimensions of the\u0000problem space as well as several polynomial time algorithms and complexity\u0000results for some problem variants.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140326003","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}
This paper addresses two problems needed to support four-dimensional ($3d + t$) spacetime numerical simulations. The first contribution is a general algorithm for producing conforming spacetime meshes of moving geometries. Here, the surface points of the geometry are embedded in a four-dimensional space as the geometry moves in time. The geometry is first tessellated at prescribed time steps and then these tessellations are connected in the parameter space of each geometry entity to form tetrahedra. In contrast to previous work, this approach allows the resolution of the geometry to be controlled at each time step. The only restriction on the algorithm is the requirement that no topological changes to the geometry are made (i.e. the hierarchical relations between all geometry entities are maintained) as the geometry moves in time. The validity of the final mesh topology is verified by ensuring the tetrahedralizations represent a closed 3-manifold. For some analytic problems, the $4d$ volume of the tetrahedralization is also verified. The second problem addressed in this paper is the design of a system to interactively visualize four-dimensional meshes, including tetrahedra (embedded in $4d$) and pentatopes. Algorithms that either include or exclude a geometry shader are described, and the efficiency of each approach is then compared. Overall, the results suggest that visualizing tetrahedra (either those bounding the domain, or extracted from a pentatopal mesh) using a geometry shader achieves the highest frame rate, in the range of $20-30$ frames per second for meshes with about $50$ million tetrahedra.
{"title":"Tessellation and interactive visualization of four-dimensional spacetime geometries","authors":"Philip Claude Caplan","doi":"arxiv-2403.19036","DOIUrl":"https://doi.org/arxiv-2403.19036","url":null,"abstract":"This paper addresses two problems needed to support four-dimensional ($3d +\u0000t$) spacetime numerical simulations. The first contribution is a general\u0000algorithm for producing conforming spacetime meshes of moving geometries. Here,\u0000the surface points of the geometry are embedded in a four-dimensional space as\u0000the geometry moves in time. The geometry is first tessellated at prescribed\u0000time steps and then these tessellations are connected in the parameter space of\u0000each geometry entity to form tetrahedra. In contrast to previous work, this\u0000approach allows the resolution of the geometry to be controlled at each time\u0000step. The only restriction on the algorithm is the requirement that no\u0000topological changes to the geometry are made (i.e. the hierarchical relations\u0000between all geometry entities are maintained) as the geometry moves in time.\u0000The validity of the final mesh topology is verified by ensuring the\u0000tetrahedralizations represent a closed 3-manifold. For some analytic problems,\u0000the $4d$ volume of the tetrahedralization is also verified. The second problem\u0000addressed in this paper is the design of a system to interactively visualize\u0000four-dimensional meshes, including tetrahedra (embedded in $4d$) and\u0000pentatopes. Algorithms that either include or exclude a geometry shader are\u0000described, and the efficiency of each approach is then compared. Overall, the\u0000results suggest that visualizing tetrahedra (either those bounding the domain,\u0000or extracted from a pentatopal mesh) using a geometry shader achieves the\u0000highest frame rate, in the range of $20-30$ frames per second for meshes with\u0000about $50$ million tetrahedra.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140324334","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}
A subset $S$ of the unit sphere $mathbb{S}^2$ is called orthogonal-pair-free if and only if there do not exist two distinct points $u, v in S$ at distance $frac{pi}{2}$ from each other. Witsenhausen cite{witsenhausen} asked the following question: {it What is the least upper bound $alpha_3$ on the Lesbegue measure of any measurable orthogonal-pair-free subset of $mathbb{S}^2$?} We prove the following result in this paper: Let $mathcal{A}$ be the collection of all orthogonal-pair-free sets $S$ such that $S$ consists of a finite number of mutually disjoint convex sets. Then, $alpha_3 = limsup_{S in mathcal{A}} mu(S)$. Thus, if the double cap conjecture cite{kalai1} is not true, there is a set in $mathcal{A}$ with measure strictly greater than the measure of the double cap.
{"title":"Convexity of near-optimal orthogonal-pair-free sets on the unit sphere","authors":"Apurva Mudgal","doi":"arxiv-2403.18404","DOIUrl":"https://doi.org/arxiv-2403.18404","url":null,"abstract":"A subset $S$ of the unit sphere $mathbb{S}^2$ is called orthogonal-pair-free\u0000if and only if there do not exist two distinct points $u, v in S$ at distance\u0000$frac{pi}{2}$ from each other. Witsenhausen cite{witsenhausen} asked the\u0000following question: {it What is the least upper bound $alpha_3$ on the\u0000Lesbegue measure of any measurable orthogonal-pair-free subset of\u0000$mathbb{S}^2$?} We prove the following result in this paper: Let $mathcal{A}$\u0000be the collection of all orthogonal-pair-free sets $S$ such that $S$ consists\u0000of a finite number of mutually disjoint convex sets. Then, $alpha_3 =\u0000limsup_{S in mathcal{A}} mu(S)$. Thus, if the double cap conjecture\u0000cite{kalai1} is not true, there is a set in $mathcal{A}$ with measure\u0000strictly greater than the measure of the double cap.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311567","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}
Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovic, Shay Solomon, Cuong Than
A $(1+varepsilon)textit{-stretch tree cover}$ of a metric space is a collection of trees, where every pair of points has a $(1+varepsilon)$-stretch path in one of the trees. The celebrated $textit{Dumbbell Theorem}$ [Arya et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+varepsilon)$-stretch tree cover with $O_d(varepsilon^{-d} cdot log(1/varepsilon))$ trees, where the $O_d$ notation suppresses terms that depend solely on the dimension~$d$. The running time of their construction is $O_d(n log n cdot frac{log(1/varepsilon)}{varepsilon^{d}} + n cdot varepsilon^{-2d})$. Since the same point may occur in multiple levels of the tree, the $textit{maximum degree}$ of a point in the tree cover may be as large as $Omega(log Phi)$, where $Phi$ is the aspect ratio of the input point set. In this work we present a $(1+varepsilon)$-stretch tree cover with $O_d(varepsilon^{-d+1} cdot log(1/varepsilon))$ trees, which is optimal (up to the $log(1/varepsilon)$ factor). Moreover, the maximum degree of points in any tree is an $textit{absolute constant}$ for any $d$. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a $(1+varepsilon)$-stretch $textit{Steiner}$ tree cover (that may use Steiner points) with $O_d(varepsilon^{(-d+1)/{2}} cdot log(1/varepsilon))$ trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive $O_d(n log n)$ term; this improves over the running time underlying the Dumbbell Theorem.
{"title":"Optimal Euclidean Tree Covers","authors":"Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovic, Shay Solomon, Cuong Than","doi":"arxiv-2403.17754","DOIUrl":"https://doi.org/arxiv-2403.17754","url":null,"abstract":"A $(1+varepsilon)textit{-stretch tree cover}$ of a metric space is a\u0000collection of trees, where every pair of points has a $(1+varepsilon)$-stretch\u0000path in one of the trees. The celebrated $textit{Dumbbell Theorem}$ [Arya\u0000et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean\u0000space admits a $(1+varepsilon)$-stretch tree cover with $O_d(varepsilon^{-d}\u0000cdot log(1/varepsilon))$ trees, where the $O_d$ notation suppresses terms\u0000that depend solely on the dimension~$d$. The running time of their construction\u0000is $O_d(n log n cdot frac{log(1/varepsilon)}{varepsilon^{d}} + n cdot\u0000varepsilon^{-2d})$. Since the same point may occur in multiple levels of the\u0000tree, the $textit{maximum degree}$ of a point in the tree cover may be as\u0000large as $Omega(log Phi)$, where $Phi$ is the aspect ratio of the input\u0000point set. In this work we present a $(1+varepsilon)$-stretch tree cover with\u0000$O_d(varepsilon^{-d+1} cdot log(1/varepsilon))$ trees, which is optimal (up\u0000to the $log(1/varepsilon)$ factor). Moreover, the maximum degree of points in\u0000any tree is an $textit{absolute constant}$ for any $d$. As a direct corollary,\u0000we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We\u0000also present a $(1+varepsilon)$-stretch $textit{Steiner}$ tree cover (that\u0000may use Steiner points) with $O_d(varepsilon^{(-d+1)/{2}} cdot\u0000log(1/varepsilon))$ trees, which too is optimal. The running time of our two\u0000constructions is linear in the number of edges in the respective tree covers,\u0000ignoring an additive $O_d(n log n)$ term; this improves over the running time\u0000underlying the Dumbbell Theorem.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311415","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}
Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, Saurabh Ray
Let $Gamma$ be a finite set of Jordan curves in the plane. For any curve $gamma in Gamma$, we denote the bounded region enclosed by $gamma$ as $tilde{gamma}$. We say that $Gamma$ is a non-piercing family if for any two curves $alpha , beta in Gamma$, $tilde{alpha} setminus tilde{beta}$ is a connected region. A non-piercing family of curves generalizes a family of $2$-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG '89) proved that if we are given a family $mathcal{C}$ of $2$-intersecting curves and a fixed curve $Cinmathcal{C}$, then the arrangement can be emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p in tilde{C}$ in such a way that the we have a family of $2$-intersecting curves throughout the process. In this paper, we generalize the result of Snoeyink and Hershberger to the setting of non-piercing curves. We show that given an arrangement of non-piercing curves $Gamma$, and a fixed curve $gammain Gamma$, the arrangement can be swept by $gamma$ so that the arrangement remains non-piercing throughout the process. We also give a shorter and simpler proof of the result of Snoeyink and Hershberger and cite applications of their result, where our result leads to a generalization.
{"title":"Sweeping Arrangements of Non-Piercing Curves in Plane","authors":"Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, Saurabh Ray","doi":"arxiv-2403.16474","DOIUrl":"https://doi.org/arxiv-2403.16474","url":null,"abstract":"Let $Gamma$ be a finite set of Jordan curves in the plane. For any curve\u0000$gamma in Gamma$, we denote the bounded region enclosed by $gamma$ as\u0000$tilde{gamma}$. We say that $Gamma$ is a non-piercing family if for any two\u0000curves $alpha , beta in Gamma$, $tilde{alpha} setminus tilde{beta}$ is\u0000a connected region. A non-piercing family of curves generalizes a family of\u0000$2$-intersecting curves in which each pair of curves intersect in at most two\u0000points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG\u0000'89) proved that if we are given a family $mathcal{C}$ of $2$-intersecting\u0000curves and a fixed curve $Cinmathcal{C}$, then the arrangement can be\u0000emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p in\u0000tilde{C}$ in such a way that the we have a family of $2$-intersecting curves\u0000throughout the process. In this paper, we generalize the result of Snoeyink and\u0000Hershberger to the setting of non-piercing curves. We show that given an\u0000arrangement of non-piercing curves $Gamma$, and a fixed curve $gammain\u0000Gamma$, the arrangement can be swept by $gamma$ so that the arrangement\u0000remains non-piercing throughout the process. We also give a shorter and simpler\u0000proof of the result of Snoeyink and Hershberger and cite applications of their\u0000result, where our result leads to a generalization.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140298212","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}
The use of drones or Unmanned Aerial Vehicles (UAVs) for aerial photography and cinematography is becoming widespread. The following optimization problem has been recently considered. Let us imagine a sporting event where a group of runners are competing and a team of drones with cameras are used to cover the event. The media emph{director} selects a set of emph{filming scenes} (determined by locations and time intervals) and the goal is to maximize the total emph{filming time} (the sum of recordings) achieved by the aerial cinematographers. Recently, it has been showed that this problem can be solved in polynomial time assuming the drones have unlimited battery endurance. In this paper, we prove that the problem is NP-hard for the more realistic case in which the battery endurance of the drones is limited.
{"title":"Filming runners with drones is hard","authors":"José-Miguel Díaz-Báñez, Ruy Fabila-Monroy","doi":"arxiv-2403.14033","DOIUrl":"https://doi.org/arxiv-2403.14033","url":null,"abstract":"The use of drones or Unmanned Aerial Vehicles (UAVs) for aerial photography\u0000and cinematography is becoming widespread. The following optimization problem\u0000has been recently considered. Let us imagine a sporting event where a group of\u0000runners are competing and a team of drones with cameras are used to cover the\u0000event. The media emph{director} selects a set of emph{filming scenes}\u0000(determined by locations and time intervals) and the goal is to maximize the\u0000total emph{filming time} (the sum of recordings) achieved by the aerial\u0000cinematographers. Recently, it has been showed that this problem can be solved\u0000in polynomial time assuming the drones have unlimited battery endurance. In\u0000this paper, we prove that the problem is NP-hard for the more realistic case in\u0000which the battery endurance of the drones is limited.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201084","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}
We revisit a standard polygon containment problem: given a convex $k$-gon $P$ and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under translation and rotation (if it exists), or more generally, find the largest copy of $P$ inside $Q$ under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required $Omega(n^2)$ time, even in the simplest $k=3$ case. We present a significantly faster new algorithm for $k=3$ achieving $O(n$polylog $n)$ running time. Moreover, we extend the result for general $k$, achieving $O(k^{O(1/varepsilon)}n^{1+varepsilon})$ running time for any $varepsilon>0$. Along the way, we also prove a new $O(k^{O(1)}n$polylog $n)$ bound on the number of similar copies of $P$ inside $Q$ that have 4 vertices of $P$ in contact with the boundary of $Q$ (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).
{"title":"Convex Polygon Containment: Improving Quadratic to Near Linear Time","authors":"Timothy M. Chan, Isaac M. Hair","doi":"arxiv-2403.13292","DOIUrl":"https://doi.org/arxiv-2403.13292","url":null,"abstract":"We revisit a standard polygon containment problem: given a convex $k$-gon $P$\u0000and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under\u0000translation and rotation (if it exists), or more generally, find the largest\u0000copy of $P$ inside $Q$ under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and\u0000Agarwal, Amenta, and Sharir (1998) all required $Omega(n^2)$ time, even in the\u0000simplest $k=3$ case. We present a significantly faster new algorithm for $k=3$\u0000achieving $O(n$polylog $n)$ running time. Moreover, we extend the result for\u0000general $k$, achieving $O(k^{O(1/varepsilon)}n^{1+varepsilon})$ running time\u0000for any $varepsilon>0$. Along the way, we also prove a new $O(k^{O(1)}n$polylog $n)$ bound on the\u0000number of similar copies of $P$ inside $Q$ that have 4 vertices of $P$ in\u0000contact with the boundary of $Q$ (assuming general position input), disproving\u0000a conjecture by Agarwal, Amenta, and Sharir (1998).","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201091","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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