Time-optimal uniform scattering in a grid

Poudel, Pavan; Sharma, Gokarna

doi:10.1145/3288599.3288622

Cited by 25 publications

(11 citation statements)

References 23 publications

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“…This problem has been studied for rings [9,20] and grids [3]. Recently, Poudel and Sharma [19] provided a Θ( √ n)-time algorithm for uniform scattering in a grid [7]. Furthermore, Dispersion is related to the load balancing problem, where a given load at the nodes has to be (re-)distributed among several processors (nodes).…”

Section: Algorithm Memory Per Robot Time (In Rounds) (In Bits)mentioning

confidence: 99%

Fast Dispersion of Mobile Robots on Arbitrary Graphs

Kshemkalyani

Molla

Sharma

2019

Algorithms for Sensor Systems

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The mobile robot dispersion problem on graphs asks k ≤ n robots placed initially arbitrarily on the nodes of an n-node anonymous graph to reposition autonomously to reach a configuration in which each robot is on a distinct node of the graph. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, and relocation of self-driven electric cars (robots) to recharge stations (nodes). In this paper, we provide two novel deterministic algorithms for dispersion, one for arbitrary graphs and another for grid graphs, in a synchronous setting where all robots perform their actions in every time step. Our algorithm for arbitrary graphs has O(min(m, k∆) · log k) steps runtime using O(log n) bits of memory at each robot, where m is the number of edges and ∆ is the maximum degree of the graph. This is an exponential improvement over the O(mk) steps best previously known algorithm. In particular, the runtime of our algorithm is optimal (up to a O(log k) factor) in constant-degree arbitrary graphs. Our algorithm for grid graphs has O(min(k, √ n)) steps runtime using Θ(log k) bits at each robot. This is the first algorithm for dispersion in grid graphs. Moreover, this algorithm is optimal for both memory and time when k = Ω(n). problem). This problem has many practical applications, for example, in relocating selfdriven electric cars (robots) to recharge stations (nodes), assuming that the cars have smart devices to communicate with each other to find a free/empty charging station [1,16]. This problem is also important due to its relationship to many other well-studied autonomous robot coordination problems, such as exploration, scattering, load balancing, covering, and self-deployment [1,16]. One of the key aspects of mobile-robot research is to understand how to use the resource-limited robots to accomplish some large task in a distributed manner [10,11]. In this paper, we study the trade-off between memory requirement of robots and the time to solve Dispersion on graphs.Augustine and Moses Jr.[1] studied Dispersion assuming k = n. They proved a memory lower bound of Ω(log n) bits at each robot and a time lower bound of Ω(D) (Ω(n) in arbitrary graphs) for any deterministic algorithm in any graph, where D is the diameter of the graph. They then provided deterministic algorithms using O(log n) bits at each robot to solve Dispersion on lines, rings, and trees in O(n) time. For arbitrary graphs, they provided two algorithms, one using O(log n) bits at each robot with O(mn) time and another using O(n log n) bits at each robot with O(m) time, where m is the number of edges in the graph. Recently, Kshemkalyani and Ali [16] provided an Ω(k) time lower bound for arbitrary graphs for k ≤ n. They then provided three deterministic algorithms for Dispersion in arbitrary graphs: (i) The first algorithm using O(k log ∆) bits at each robot with O(m) time, (ii) The second algorithm using O(D log ∆) bits at each robot with O(∆ D ) time, and ...

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Section: Algorithm Memory Per Robot Time (In Rounds) (In Bits)mentioning

confidence: 99%

Fast Dispersion of Mobile Robots on Arbitrary Graphs

Kshemkalyani

Molla

Sharma

2019

Algorithms for Sensor Systems

Self Cite

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“…is problem has been studied for rings [11,24] and grids [3]. Recently, Poudel and Sharma [23] provided a Θ( √ n)-time algorithm for uniform sca ering in a grid [8]. Furthermore, D is related to the load balancing problem, where a given load at the nodes has to be (re-)distributed among several processors (nodes).…”

Section: Algorithmmentioning

confidence: 99%

Dispersion of Mobile Robots in the Global Communication Model

Kshemkalyani

Molla

Sharma

2020

Proceedings of the 21st International Conference on Distributed Computing and Networking

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e dispersion problem on graphs asks k ≤ n robots placed initially arbitrarily on the nodes of an n-node anonymous graph to reposition autonomously to reach a con guration in which each robot is on a distinct node of the graph. is problem is of signi cant interest due to its relationship to other fundamental robot coordination problems, such as exploration, sca ering, load balancing, and relocation of self-driven electric cars (robots) to recharge stations (nodes). In this paper, we consider dispersion in the global communication model where a robot can communicate with any other robot in the graph (but the graph is unknown to robots). We provide three novel deterministic algorithms, two for arbitrary graphs and one for arbitrary trees, in a synchronous se ing where all robots perform their actions in every time step. For arbitrary graphs, our rst algorithm is based on a DFS traversal and guarantees O(min(m, k∆)) steps runtime using Θ(log(max(k, ∆))) bits at each robot, where m is the number of edges and ∆ is the maximum degree of the graph. e second algorithm for arbitrary graphs is based on a BFS traversal and guarantees O(max(D, k)∆(D + ∆)) steps runtime using O(max(D, ∆ log k)) bits at each robot, where D is the diameter of the graph. e algorithm for arbitrary trees is also based on a BFS travesal and guarantees O(D max(D, k)) steps runtime using O(max(D, ∆ log k)) bits at each robot. Our results are signi cant improvements compared to the existing results established in the local communication model where a robot can communication only with other robots present at the same node. Particularly, the DFS-based algorithm is optimal for both memory and time in constant-degree arbitrary graphs. e BFS-based algorithm for arbitrary trees is optimal with respect to runtime when k ≤ O(D).

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“…The dispersion problem of mobile robots on graphs is also discussed by Ajay D. Kshemkalyani et al, [17] where the robots are initially placed arbitrarily on the nodes of an n-node anonymous graph and they autonomously reposition themselves to reach a configuration in which each robot is on a distinct node of the graph. Also in [21] [16] [15] [12] the dispersion problem is discussed.…”

Section: Earlier Work and Our Contributionmentioning

confidence: 99%

Uniform Scattering of Robots on Alternate Nodes of a Grid

Mondal¹,

Chaudhuri²,

Chatterjee³

2020

Preprint

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In this paper, we propose a distributed algorithm to uniformly scatter the robots along a grid, with robots on alternate nodes of this grid distribution. These homogeneous, autonomous mobile robots place themselves equidistant apart on the grid, which can be required for guarding or covering a geographical area by the robots. The robots operate by executing cycles of the states "look-compute-move". In the look phase, it looks to see the position of the other robots; in the compute phase, it computes a destination to move to; and then in the move phase, it moves to that computed destination. They do not interact by message passing and can recollect neither the past actions nor the looked data from the previous cycle, i.e., oblivious. The robots are semi-synchronous, anonymous and have unlimited visibility. Eventually, the robots uniformly distribute themselves on alternate nodes of a grid, leaving the adjacent nodes of the grid vacant. The algorithm presented also assures no collision or deadlock among the robots.

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Time-optimal uniform scattering in a grid

Cited by 25 publications

References 23 publications

Fast Dispersion of Mobile Robots on Arbitrary Graphs

Fast Dispersion of Mobile Robots on Arbitrary Graphs

Dispersion of Mobile Robots in the Global Communication Model

Uniform Scattering of Robots on Alternate Nodes of a Grid

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