Pub Date : 2022-12-01DOI: 10.1007/978-3-030-89240-1_6
Konstantinos Georgiou, S. Leizerovich, J. Lucier, S. Kundu
{"title":"Evacuating from ℓp Unit Disks in the Wireless Model - (Extended Abstract)","authors":"Konstantinos Georgiou, S. Leizerovich, J. Lucier, S. Kundu","doi":"10.1007/978-3-030-89240-1_6","DOIUrl":"https://doi.org/10.1007/978-3-030-89240-1_6","url":null,"abstract":"","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123856729","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 : 2022-09-18DOI: 10.48550/arXiv.2209.08544
Konstantinos Georgiou, W. Jang
. The input to the Triangle Evacuation problem is a triangle ABC . Given a starting point S on the perimeter of the triangle, a feasible solution to the problem consists of two unit-speed trajectories of mobile agents that eventually visit every point on the perimeter of ABC . The cost of a feasible solution (evacuation cost) is defined as the supremum over all points T of the time it takes that T is visited for the first time by an agent plus the distance of T to the other agent at that time. Similar evacuation type problems are well studied in the literature covering the unit circle, the ‘ p unit circle for p ≥ 1 , the square, and the equilateral triangle. We extend this line of research to arbitrary non-obtuse triangles. Motivated by the lack of symmetry of our search domain, we introduce 4 different algorithmic problems arising by letting the starting edge and/or the starting point S on that edge to be chosen either by the algorithm or the adversary. To that end, we provide a tight analysis for the algorithm that has been proved to be optimal for the previously studied search domains, as well as we provide lower bounds for each of the problems. Both our upper and lower bounds match and extend naturally the previously known results that were established only for equilateral triangles.
{"title":"Triangle Evacuation of 2 Agents in the Wireless Model","authors":"Konstantinos Georgiou, W. Jang","doi":"10.48550/arXiv.2209.08544","DOIUrl":"https://doi.org/10.48550/arXiv.2209.08544","url":null,"abstract":". The input to the Triangle Evacuation problem is a triangle ABC . Given a starting point S on the perimeter of the triangle, a feasible solution to the problem consists of two unit-speed trajectories of mobile agents that eventually visit every point on the perimeter of ABC . The cost of a feasible solution (evacuation cost) is defined as the supremum over all points T of the time it takes that T is visited for the first time by an agent plus the distance of T to the other agent at that time. Similar evacuation type problems are well studied in the literature covering the unit circle, the ‘ p unit circle for p ≥ 1 , the square, and the equilateral triangle. We extend this line of research to arbitrary non-obtuse triangles. Motivated by the lack of symmetry of our search domain, we introduce 4 different algorithmic problems arising by letting the starting edge and/or the starting point S on that edge to be chosen either by the algorithm or the adversary. To that end, we provide a tight analysis for the algorithm that has been proved to be optimal for the previously studied search domains, as well as we provide lower bounds for each of the problems. Both our upper and lower bounds match and extend naturally the previously known results that were established only for equilateral triangles.","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133511442","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 : 2022-07-07DOI: 10.48550/arXiv.2207.03275
Nada Almalki, O. Michail
In this work, we investigate novel algorithmic growth processes. In particular, we propose three growth operations, full doubling, RC doubling and doubling, and explore the algorithmic and structural properties of their resulting processes under a geometric setting. In terms of modeling, our system runs on a 2-dimensional grid and operates in discrete time-steps. The process begins with an initial shape $S_I=S_0$ and, in every time-step $t geq 1$, by applying (in parallel) one or more growth operations of a specific type to the current shape-instance $S_{t-1}$, generates the next instance $S_t$, always satisfying $|S_t|>|S_{t-1}|$. Our goal is to characterize the classes of shapes that can be constructed in $O(log n)$ or polylog $n$ time-steps and determine whether a final shape $S_F$ can be constructed from an initial shape $S_I$ using a finite sequence of growth operations of a given type, called a constructor of $S_F$. For full doubling, in which, in every time-step, every node generates a new node in a given direction, we completely characterize the structure of the class of shapes that can be constructed from a given initial shape. For RC doubling, in which complete columns or rows double, our main contribution is a linear-time centralized algorithm that for any pair of shapes $S_I$, $S_F$ decides if $S_F$ can be constructed from $S_I$ and, if the answer is yes, returns an $O(log n)$-time-step constructor of $S_F$ from $S_I$. For the most general doubling operation, where up to individual nodes can double, we show that some shapes cannot be constructed in sub-linear time-steps and give two universal constructors of any $S_F$ from a singleton $S_I$, which are efficient (i.e., up to polylogarithmic time-steps) for large classes of shapes. Both constructors can be computed by polynomial-time centralized algorithms for any shape $S_F$.
{"title":"On Geometric Shape Construction via Growth Operations","authors":"Nada Almalki, O. Michail","doi":"10.48550/arXiv.2207.03275","DOIUrl":"https://doi.org/10.48550/arXiv.2207.03275","url":null,"abstract":"In this work, we investigate novel algorithmic growth processes. In particular, we propose three growth operations, full doubling, RC doubling and doubling, and explore the algorithmic and structural properties of their resulting processes under a geometric setting. In terms of modeling, our system runs on a 2-dimensional grid and operates in discrete time-steps. The process begins with an initial shape $S_I=S_0$ and, in every time-step $t geq 1$, by applying (in parallel) one or more growth operations of a specific type to the current shape-instance $S_{t-1}$, generates the next instance $S_t$, always satisfying $|S_t|>|S_{t-1}|$. Our goal is to characterize the classes of shapes that can be constructed in $O(log n)$ or polylog $n$ time-steps and determine whether a final shape $S_F$ can be constructed from an initial shape $S_I$ using a finite sequence of growth operations of a given type, called a constructor of $S_F$. For full doubling, in which, in every time-step, every node generates a new node in a given direction, we completely characterize the structure of the class of shapes that can be constructed from a given initial shape. For RC doubling, in which complete columns or rows double, our main contribution is a linear-time centralized algorithm that for any pair of shapes $S_I$, $S_F$ decides if $S_F$ can be constructed from $S_I$ and, if the answer is yes, returns an $O(log n)$-time-step constructor of $S_F$ from $S_I$. For the most general doubling operation, where up to individual nodes can double, we show that some shapes cannot be constructed in sub-linear time-steps and give two universal constructors of any $S_F$ from a singleton $S_I$, which are efficient (i.e., up to polylogarithmic time-steps) for large classes of shapes. Both constructors can be computed by polynomial-time centralized algorithms for any shape $S_F$.","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121163645","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 : 2022-07-07DOI: 10.48550/arXiv.2207.03062
Matthew Connor, O. Michail
We study a model of programmable matter systems consisting of n devices lying on a 2-dimensional square grid, which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We are interested in characterising the class of shapes which can be transformed into each other in such a scenario, under the additional constraint of maintaining global connectivity at all times. This was one of the main problems left open by [Michail et al., JCSS’19]. Note that the considered question is about structural feasibility of transformations, which we exclusively deal with via centralised constructive proofs. Distributed solutions are left for future work and form an interesting research direction. Past work made some progress for the special class of nice shapes. We here consider the class of orthogonal convex shapes, where for any two nodes u, v in a horizontal or vertical line on the grid, there is no empty cell between u and v. We develop a generic centralised transformation and prove that, for any pair A, B of colour-consistent orthogonal convex shapes, it can transform A into B. In light of the existence of blocked shapes in the considered class, we use a minimal 3-node seed to trigger the transformation. The running time of our transformation is an optimal O(n) sequential moves, where n = |A| = |B|. We leave as an open problem the existence of a universal connectivity-preserving transformation with a small seed. Our belief is that the techniques developed in this paper might prove useful to answer this.
{"title":"Centralised Connectivity-Preserving Transformations by Rotation: 3 Musketeers for all Orthogonal Convex Shapes","authors":"Matthew Connor, O. Michail","doi":"10.48550/arXiv.2207.03062","DOIUrl":"https://doi.org/10.48550/arXiv.2207.03062","url":null,"abstract":"We study a model of programmable matter systems consisting of n devices lying on a 2-dimensional square grid, which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We are interested in characterising the class of shapes which can be transformed into each other in such a scenario, under the additional constraint of maintaining global connectivity at all times. This was one of the main problems left open by [Michail et al., JCSS’19]. Note that the considered question is about structural feasibility of transformations, which we exclusively deal with via centralised constructive proofs. Distributed solutions are left for future work and form an interesting research direction. Past work made some progress for the special class of nice shapes. We here consider the class of orthogonal convex shapes, where for any two nodes u, v in a horizontal or vertical line on the grid, there is no empty cell between u and v. We develop a generic centralised transformation and prove that, for any pair A, B of colour-consistent orthogonal convex shapes, it can transform A into B. In light of the existence of blocked shapes in the considered class, we use a minimal 3-node seed to trigger the transformation. The running time of our transformation is an optimal O(n) sequential moves, where n = |A| = |B|. We leave as an open problem the existence of a universal connectivity-preserving transformation with a small seed. Our belief is that the techniques developed in this paper might prove useful to answer this.","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132818794","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 : 2022-05-07DOI: 10.48550/arXiv.2205.03651
Vishwanath R. Singireddy, Manjanna Basappa
In this paper we consider the problem of locating $k$ obnoxious facilities (congruent disks of maximum radius) amidst $n$ demand points (existing repulsive facility sites) ordered from left to right in the plane so that none of the existing facility sites are affected (no demand point falls in the interior of the disks). We study this problem in two restricted settings: (i) the obnoxious facilities are constrained to be centered on along a predetermined horizontal line segment $bar{pq}$, and (ii) the obnoxious facilities are constrained to lie on the boundary arc of a predetermined disk $cal C$. An $(1-epsilon)$-approximation algorithm was given recently to solve the constrained problem in (i) in time $O((n+k)log{frac{||pq||}{2(k-1)epsilon}})$, where $epsilon>0$ cite{Sing2021}. Here, for the problem in (i), we first propose an exact polynomial-time algorithm based on a binary search on all candidate radii computed explicitly. This algorithm runs in $O((nk)^2log{(nk)}+(n+k)log{(nk)})$ time. We then show that using the parametric search technique of Megiddo cite{MG1983}; we can solve the problem exactly in $O((n+k)^2)$ time, which is faster than the latter. Continuing further, using the improved parametric technique we give an $O(nlog^2 n)$-time algorithm for $k=2$. We finally show that the above $(1-epsilon)$-approximation algorithm of cite{Sing2021} can be easily adapted to solve the circular constrained problem of (ii) with an extra multiplicative factor of $n$ in the running time.
{"title":"Dispersing Facilities on Planar Segment and Circle Amidst Repulsion","authors":"Vishwanath R. Singireddy, Manjanna Basappa","doi":"10.48550/arXiv.2205.03651","DOIUrl":"https://doi.org/10.48550/arXiv.2205.03651","url":null,"abstract":"In this paper we consider the problem of locating $k$ obnoxious facilities (congruent disks of maximum radius) amidst $n$ demand points (existing repulsive facility sites) ordered from left to right in the plane so that none of the existing facility sites are affected (no demand point falls in the interior of the disks). We study this problem in two restricted settings: (i) the obnoxious facilities are constrained to be centered on along a predetermined horizontal line segment $bar{pq}$, and (ii) the obnoxious facilities are constrained to lie on the boundary arc of a predetermined disk $cal C$. An $(1-epsilon)$-approximation algorithm was given recently to solve the constrained problem in (i) in time $O((n+k)log{frac{||pq||}{2(k-1)epsilon}})$, where $epsilon>0$ cite{Sing2021}. Here, for the problem in (i), we first propose an exact polynomial-time algorithm based on a binary search on all candidate radii computed explicitly. This algorithm runs in $O((nk)^2log{(nk)}+(n+k)log{(nk)})$ time. We then show that using the parametric search technique of Megiddo cite{MG1983}; we can solve the problem exactly in $O((n+k)^2)$ time, which is faster than the latter. Continuing further, using the improved parametric technique we give an $O(nlog^2 n)$-time algorithm for $k=2$. We finally show that the above $(1-epsilon)$-approximation algorithm of cite{Sing2021} can be easily adapted to solve the circular constrained problem of (ii) with an extra multiplicative factor of $n$ in the running time.","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133883492","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 : 2021-07-29DOI: 10.1007/978-3-031-22050-0_9
G. Mertzios, O. Michail, George Skretas, P. Spirakis, Michail Theofilatos
{"title":"The Complexity of Growing a Graph","authors":"G. Mertzios, O. Michail, George Skretas, P. Spirakis, Michail Theofilatos","doi":"10.1007/978-3-031-22050-0_9","DOIUrl":"https://doi.org/10.1007/978-3-031-22050-0_9","url":null,"abstract":"","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121858620","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 : 2020-10-09DOI: 10.1007/978-3-030-89240-1_3
Jannik Castenow, Jonas Harbig, Daniel Jung, Till Knollmann, F. Heide
{"title":"Gathering a Euclidean Closed Chain of Robots in Linear Time","authors":"Jannik Castenow, Jonas Harbig, Daniel Jung, Till Knollmann, F. Heide","doi":"10.1007/978-3-030-89240-1_3","DOIUrl":"https://doi.org/10.1007/978-3-030-89240-1_3","url":null,"abstract":"","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129111450","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 : 2020-09-09DOI: 10.1007/978-3-030-62401-9_9
Konstantinos Georgiou, J. Lucier
{"title":"Weighted Group Search on a Line - (Extended Abstract)","authors":"Konstantinos Georgiou, J. Lucier","doi":"10.1007/978-3-030-62401-9_9","DOIUrl":"https://doi.org/10.1007/978-3-030-62401-9_9","url":null,"abstract":"","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"19 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120972897","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 : 2020-05-17DOI: 10.1007/978-3-030-62401-9_6
Abdullah Almethen, O. Michail, I. Potapov
{"title":"On Efficient Connectivity-Preserving Transformations in a Grid","authors":"Abdullah Almethen, O. Michail, I. Potapov","doi":"10.1007/978-3-030-62401-9_6","DOIUrl":"https://doi.org/10.1007/978-3-030-62401-9_6","url":null,"abstract":"","PeriodicalId":159325,"journal":{"name":"Algorithmic Aspects of Wireless Sensor Networks","volume":"488 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120864602","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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