An $\bm{O(n\log n)}$ Shortest Path Algorithm Based on Delaunay Triangulation

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-hard to find a cheapest biconnected spanning graph, where edge costs are the Euclidean distances of the respective points. We propose a different type of subgraph, based on forbidding (due to failure) the use of a vertex. A minimum spanning multi-tree is a spanning graph that contains for each possible forbidden vertex a spanning tree that is minimum among the spanning trees that do not use the forbidden vertex. We propose a worst-case time optimal algorithm for computing a minimum spanning multi-tree for a planar Euclidean point set. A minimum spanning multi-tree is cheap, even though it embeds a linear number of MSTs: Its cost is more than the MST cost only by a constant factor. Furthermore, we propose a linear time algorithm for computing a cheap vertex failure tolerant graph, given the Delaunay triangulation. This graph bounds the cost of the minimum spanning multi-tree from above.

How to collect balls moving in the Euclidean plane

N points are given in the Euclidean plane, and the minimum spanning tree problem seeks for a minimum spanning tree interconnecting the n points so that there is only one path between any two points. One of the classic and frequently-used algorithms for minimum spanning tree problem is Prim's algorithm, but it consumes large time and space complexity for the plane minimum spanning tree problem is of O(n2) numbers of edges. Luckily, it was proved that the plane minimum spanning tree is a sub-graph of Delaunay triangulation for the given points in the plane, and the number of edges in the triangulation is O(n). This motivates us to efficiently compute the Delaunay triangulation of the given points and then find the minimum spanning tree in the triangulation. This paper presents an algorithm based on the divide and conquer for Delaunay triangulation together with the Prim's algorithm to produce an O(nlogn) algorithm for minimum spanning tree problem in the plane, implements the visual graphic interface with various selected algorithms for plane minimum spanning tree and compares their running time.

Mobile robots are mostly used in many industrial and military applications. Researches in the path planning is one of the most important aspects in mobile robot researches. Path planning for a mobile robot need to find a collision free path through the robot's environment with obstacles from the specified start location to a desired(target) goal location while satisfying certain optimization conditions. Existing path planning methods like graphical methods such as visibility graph, the potential field and the cell decomposition are designed for static environments, in which there are stationary obstacles. In practical systems such as Marine Science Research, Robots in Industry, and military combat applications, robots usually face dynamic environments where both moving and stationary obstacles exist. This project proposes a heuristic based method to search the feasible initial path efficiently. The heuristic based method is then combined into the simulated annealing algorithm based approach for dynamic robot path planning. Therefore the quality of the solution is characterised by the length of the planned path and it is improved with the combined heuristic method in the simulated annealing based approach for both runtime and offline path planning.

The performance of robot arm motion generated via neural network are presented in this paper. The robot arm motion for obstacle avoidance simultaneously optimizing three functions; minimum distance, minimum time and minimum energy are generated. Four different initial and goal position had been chosen to test and analyze the performance of generated neural controller. The same neural controllers can be employed to a different range of initial and goal position. The motion generated yield good results in the simulator. In this research a new approach for intelligent robot arm path and motion generation are successfully implemented.

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

Alternative pathfinding requires finding a set of k alternative paths (including the shortest path) between a given source s and a target t. Intuitively, these paths should be significantly different from each other and meaningful/natural (e.g., must not contain loops or unnecessary detours). While finding alternative paths in road networks has been extensively studied, to the best of our knowledge, we are the first to formally study alternative pathfinding in game maps which are typically represented as Euclidean planes containing polygonal obstacles. First, we adapt the existing techniques designed for road networks to find alternative paths in the game maps. Then, based on our web-based system that visualises alternative paths generated by different approaches, we conduct a user study that shows that the existing road network approaches generate high-quality alternative paths when adapted for the game maps. However, these existing approaches are computationally inefficient especially when compared to the state-of-the-art shortest path algorithms. Motivated by this, we propose novel data structures and exploit these to develop an efficient algorithm to compute high-quality alternative paths. that shows that the existing road network approaches generate high-quality alternative paths in game maps. Our extensive experimental study demonstrates that our proposed algorithm is more than an order of magnitude faster than the existing approaches and returns alternative paths of comparable quality. Furthermore, our algorithm is comparable to a state-of-the-art shortest path algorithm in terms of running time.

The Euclidean Shortest Path Problem (ESPP) is a classic problem which requires finding the shortest path in a Euclidean plane with polygonal obstacles. The state-of-the-art solution, Euclidean Hub Labeling (EHL), offers ultra-fast query performance but comes with significant memory overhead, requiring up to tens of gigabytes of storage on large maps, limiting its use in memory-constrained environments like mobile phones. Additionally, EHL's memory usage can only be determined after index construction, and while it provides a memory-runtime tradeoff, it does not fully optimize memory utilization. In this work, we introduce an improved version of EHL, called EHL*, which overcomes these limitations. A key contribution of EHL* is its ability to create an index that adheres to a specified memory budget while optimizing query runtime performance. Moreover, EHL* can leverage pre-known query distributions, a common scenario in many real-world applications, to further enhance runtime efficiency. Our results show that EHL* can reduce memory usage by up to 10-20 times without much impact on query runtime performance compared to EHL, making it a highly effective solution for optimal pathfinding in memory-constrained environments. We also present a theoretical analysis comparing EHL* with EHL, providing insights into their indexing and query processing cost.

We consider a compound central field in the Euclidean plane which is governed by a finite number of bell-shaped potential functions with finite range. The study of the qualitative behavior of the Hamilton flow in such a potential field can be reduced to that of the so-called perturbed billiards system. The main result in this paper is the construction of a symbolic dynamics of the Hamilton flow by using the perturbed billiards system provided that the energy E > 0 is small enough. We also try to show an analogue of the prime number theorem for the closed orbits of the flow following ideas presented by Morita and Parry and Pollicott.

<strong class="journal-contentHeaderColor">Abstract.</strong> Barnes interpolation is a method that is widely used in geospatial sciences like meteorology to remodel data values recorded at irregularly distributed points into a representative analytical field. When implemented naively, the computational complexity of Barnes interpolation depends directly on both the number of sample points and the number of grid points. In the era of highly resolved grids and overwhelming numbers of sample points, which originate e.g. from the Internet of Things or from crowd-sourced data, this computation can be quite demanding even on high-performance machines. This paper presents new approaches how very good approximations of Barnes interpolation can be implemented using fast algorithms. Two use cases are in particular considered, namely (1) where the used grid is embedded in the Euclidean plane and (2) where the grid is located on the unit sphere.

Geometric planar networks on bichromatic collinear points

We study a scenario in which n mobile robots with a limited viewing range are distributed in the Euclidean plane and have to solve a formation problem. The formation problems we consider are the G athering problem and the C hain-Formation problem. In the G athering problem, the robots have to gather in one (not predefined) point, while in the C hain -F ormation problem they have to form a connected communication chain of minimal length between two stationary base stations. Each robot may base its decisions where to move only on the current relative positions of neighboring robots (that are within its viewing range); that is, besides having a limited viewing range, the robots are oblivious (they do not use information from the past), have none or only very limited identities, and they do not have a common sense of direction. Variants of these problems (especially for the G athering problem) have been studied extensively in different discrete time models. In contrast, our work focuses on a continuous time model; that is, the robots continuously sense the positions of other robots within their viewing range and continuously adapt their speed and direction according to some simple, local rules. Hereby, we assume that the robots have a maximum movement speed of one. We show that this idealized idea of continuous sensing allows us to solve the mentioned formation problems in linear time O( n ) (which, given the maximum speed of one, immediately yields a maximum traveled distance of O( n )). Note that in the more classical discrete time models, the best known strategies need at least Θ( n 2 ) or even Θ( n 2 log n ) timesteps to solve these problems. For the G athering problem, our analysis solves a problem left open by Gordon et al. [2004], where the authors could prove that gathering in a continuous model is possible in finite time, but were not able to give runtime bounds. Apart from these linear bounds, we also provide runtime bounds for both formation problems that relate the runtime of our strategies to the runtime of an optimal, global algorithm. Specifically, we show that our strategy for the G athering problem is log OPT-competitive and the strategy for the C hain -F ormation problem is log n -competitive. Here, by c -competitive, we mean that our (local) strategy is asymptotically by at most a factor of c slower than an optimal, global strategy.

Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O( log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n2). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O(( log n)2) time. In this dynamic data structure, a query with output size k can be solved in O( log n+k( log n)2) time. The second dynamic data structure, which allows only semi-online updates, has O(( log n)2) amortized update time and O( log n+k) query time.

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem that has numerous valuable practical applications. It is known, that CVRP is strongly NP-hard even on the Euclidean plane and APX-hard in its metric setting even for any fixed capacity \(q\ge 3\). For the Euclidean setting, there are known several approximation schemes. But, to the best of our knowledge, polynomial bounds for their time complexity were proved either for a fixed capacity q or under the restriction \(q\ll n\). Moreover, most of these schemes were developed for the simplest case, where each customer has a unit demand, and cannot be extended to the general case of a non-uniform demand (both splittable or not) directly.

The Large Millimeter Telescope observatory is extending its night time operation to the day time. A sun avoidance strategy was therefore implemented in the control system in real-time to avoid excessive heating and damage to the secondary mirror and the prime focus. The LMT uses an ”on-the-fly” trajectory generator that receives as input the target location of the telescope and in turn outputs a commanded position to the servo system. The sun avoidance strategy is also implemented ”on-the-fly” where it intercepts the input to the trajectory generator and alters that input to avoid the sun. Two sun avoidance strategies were explored. The first strategy uses a potential field approach where the sun is represented as a high-potential obstacle in the telescope’s workspace and the target location is represented as a low-potential goal. The potential field is repeatedly calculated as the sun and the telescope move and the telescope follows the induced force by this field. The second strategy is based on path planning using visibility graphs where the sun is represented as a polygonal obstacle and the telescope follows the shortest path from its actual position to the target location via the vertices of the sun’s polygon. The visibility graph approach was chosen as the favorable strategy due to the efficiency of its algorithm and the simplicity of its computation.