264 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Oriented Spanners

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    Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in ?(n?) time for n points, and a greedy algorithm that computes a 5-spanner in ?(nlog n) time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented ?(1)-spanner

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Algorithms for Triangles, Cones & Peaks

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    Three different geometric objects are at the center of this dissertation: triangles, cones and peaks. In computational geometry, triangles are the most basic shape for planar subdivisions. Particularly, Delaunay triangulations are a widely used for manifold applications in engineering, geographic information systems, telecommunication networks, etc. We present two novel parallel algorithms to construct the Delaunay triangulation of a given point set. Yao graphs are geometric spanners that connect each point of a given set to its nearest neighbor in each of kk cones drawn around it. They are used to aid the construction of Euclidean minimum spanning trees or in wireless networks for topology control and routing. We present the first implementation of an optimal O(nlogn)\mathcal{O}(n \log n)-time sweepline algorithm to construct Yao graphs. One metric to quantify the importance of a mountain peak is its isolation. Isolation measures the distance between a peak and the closest point of higher elevation. Computing this metric from high-resolution digital elevation models (DEMs) requires efficient algorithms. We present a novel sweep-plane algorithm that can calculate the isolation of all peaks on Earth in mere minutes

    Coresets for Clustering in Geometric Intersection Graphs

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    Algorithms for Unit-Disk Graphs and Related Problems

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    In this dissertation, we study algorithms for several problems on unit-disk graphs and related problems. The unit-disk graph can be viewed as an intersection graph of a set of congruent disks. Unit-disk graphs have been extensively studied due to many of their applications, e.g., modeling the topology of wireless sensor networks. Lots of problems on unit-disk graphs have been considered in the literature, such as shortest paths, clique, independent set, distance oracle, diameter, etc. Specifically, we study the following problems in this dissertation: L1 shortest paths in unit-disk graphs, reverse shortest paths in unit-disk graphs, minimum bottleneck moving spanning tree, unit-disk range reporting, distance selection, etc. We develop efficient algorithms for these problems and our results are either first-known solutions or somehow improve the previous work. Given a set P of n points in the plane and a parameter r \u3e 0, a unit-disk graph G(P) can be defined using P as its vertex set and two points of P are connected by an edge if the distance between these two points is at most r. The weight of an edge is one in the unweighted case and is equal to the distance between the two endpoints in the weighted case. Note that the distance between two points can be measured by different metrics, e.g., L1 or L2 metric. In the first problem of L1 shortest paths in unit-disk graphs, we are given a point set P and a source point s ∈ P, the problem is to find all shortest paths from s to all other vertices in the L1 weighted unit-disk graph defined on set P. We present an O(n log n) time algorithm, which matches the Ω(n log n)-time lower bound. In the second problem, we are given a set P of n points, parameters r, λ \u3e 0, and two points s and t of P, the goal is to compute the smallest r such that the shortest path length between s and t in the unit-disk graph with respect to set P and parameter r is at most λ. This problem can be defined in both unweighted and weighted cases. We propose an algorithm of O(⌊λ⌋ · n log n) time and another algorithm of O(n5/4 log7/4 n) time for the unweighted case. We also given an O(n5/4 log5/2 n) time algorithm for the weighted case. In the third problem, we are given a set P of n points that are moving in the plane, the problem is to compute a spanning tree for these moving points that does not change its combinatorial structure during the point movement such that the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement is minimized. We present an algorithm that runs in O(n4/3 log3 n) time. The fourth problem is unit-disk range reporting in which we are given a set P of n points in the plane and a value r, we need to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We build a data structure of O(n) space in O(n log n) time that can answer each query in O(k + log n) time, where k is the output size. The time complexity of our algorithm is the same as the previous result but our approach is much simpler. Finally, for the problem of distance selection, we are given a set P of n points in the plane and an integer 1 ≤ k ≤ (n2), the distance selection problem is to find the k-th smallest interpoint distance among all pairs of points of p. We propose an algorithm that runs in O(n4/3 log n) time. Our techniques yield two algorithmic frameworks for solving geometric optimization problems. Many algorithms and techniques developed in this dissertation are quite general and fundamental, and we believe they will find other applications in future

    Computational Intelligence for Cooperative Swarm Control

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    Over the last few decades, swarm intelligence (SI) has shown significant benefits in many practical applications. Real-world applications of swarm intelligence include disaster response and wildlife conservation. Swarm robots can collaborate to search for survivors, locate victims, and assess damage in hazardous environments during an earthquake or natural disaster. They can coordinate their movements and share data in real-time to increase their efficiency and effectiveness while guiding the survivors. In addition to tracking animal movements and behaviour, robots can guide animals to or away from specific areas. Sheep herding is a significant source of income in Australia that could be significantly enhanced if the human shepherd could be supported by single or multiple robots. Although the shepherding framework has become a popular SI mechanism, where a leading agent (sheepdog) controls a swarm of agents (sheep) to complete a task, controlling a swarm of agents is still not a trivial task, especially in the presence of some practical constraints. For example, most of the existing shepherding literature assumes that each swarm member has an unlimited sensing range to recognise all other members’ locations. However, this is not practical for physical systems. In addition, current approaches do not consider shepherding as a distributed system where an agent, namely a central unit, may observe the environment and commu- nicate with the shepherd to guide the swarm. However, this brings another hurdle when noisy communication channels between the central unit and the shepherd af- fect the success of the mission. Also, the literature lacks shepherding models that can cope with dynamic communication systems. Therefore, this thesis aims to design a multi-agent learning system for effective shepherding control systems in a partially observable environment under communication constraints. To achieve this goal, the thesis first introduces a new methodology to guide agents whose sensing range is limited. In this thesis, the sheep are modelled as an induced network to represent the sheep’s sensing range and propose a geometric method for finding a shepherd-impacted subset of sheep. The proposed swarm optimal herding point uses a particle swarm optimiser and a clustering mechanism to find the sheepdog’s near-optimal herding location while considering flock cohesion. Then, an improved version of the algorithm (named swarm optimal modified centroid push) is proposed to estimate the sheepdog’s intermediate waypoints to the herding point considering the sheep cohesion. The approaches outperform existing shepherding methods in reducing task time and increasing the success rate for herding. Next, to improve shepherding in noisy communication channels, this thesis pro- poses a collaborative learning-based method to enhance communication between the central unit and the herding agent. The proposed independent pre-training collab- orative learning technique decreases the transmission mean square error by half in 10% of the training time compared to existing approaches. The algorithm is then ex- tended so that the sheepdog can read the modulated herding points from the central unit. The results demonstrate the efficiency of the new technique in time-varying noisy channels. Finally, the central unit is modelled as a mobile agent to lower the time-varying noise caused by the sheepdog’s motion during the task. So, I propose a Q-learning- based incremental search to increase transmission success between the shepherd and the central unit. In addition, two unique reward functions are presented to ensure swarm guidance success with minimal energy consumption. The results demonstrate an increase in the success rate for shepherding

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Applications of Centrality Measures and Extremal Combinatorics

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    Centrality measures assign numbers or rankings to network nodes that reflect their importance. There are many types of centrality measures, each suitable for different types of networks and applications. In Chapter 2, we consider a model of astronaut health during a space mission. Katz centrality is commonly used to measure the influence of nodes in social and biological networks. We motivate its use in this application to estimate the expected quality time lost due to the progression of medical conditions. In Chapter 3, we find dominating sets in satellite networks. To do this, we use the Shapley value, a centrality measure that originates in game theory and is computed based only on local network information. We demonstrate that the Shapley value is an effective and time-efficient tool for finding small dominating sets in various random graph families and a set of real-world networks. In Chapter 4, we introduce a novel algorithm for categorizing which nodes are nearest the boundary, called boundary nodes, in a network that uses Chvátal’s definition of a line in a graph. We test this algorithm on random geometric graphs and compare its effectiveness to other known methods for boundary node detection. In Chapter 5, for certain sets S and equations eq, we look for the smallest number of colors rb(S, eq) such that for every surjective rb(S, eq)-coloring of S, there exists a solution to eq where every element of the solution set is assigned a distinct color. We show that rb([n], x_1 + x_2 = x_3) = ⌊log_2(m) + 2⌋ and rb([m] × [n], x_1 + x_2 = x_3) = m + n + 1 for m, n \u3e 1. In Chapter 6, a graph G is H-semi-saturated if adding an edge e to G that is not currently in G causes H to appear as a subgraph in G that contains e. We say that G is H-saturated if G does not contain H as a subgraph before adding e. The smallest number of edges in an H-semi-saturated (H-saturated) graph is called the semi-saturation number of H (saturation number of H). We show that the saturation and semi-saturation numbers differ by at least 1 for a disjoint union of paths called a linear forest. Additionally, we find graph families for which the saturation number is monotonic with respect to the subgraph relation
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