Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location

In this paper we introduce an optimization problem which involves maximization of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems consisting of both mobile and stationary facilities. Let ψ be a circular path through which mobile service stations are plying, and S be a set of n stationary facilities (points) inside ψ. A demand point p is served from a mobile facility plying along ψ if the distance of p from the boundary of ψ is less than that from any member in S. On the other hand, the demand point p is served from a stationary facility p i  ∈ S if the distance of p from p i is less than or equal to the distance of p from all other members in S and also from the boundary of ψ. The objective is to place the stationary facilities in S, inside ψ, such that the total area served by them is maximized. We consider a restricted version of this problem where the members in S are placed equidistantly from the center o of ψ. It is shown that the maximum area is obtained when the members in S lie on the vertices of a regular n-gon, with its circumcenter at o. The distance of the members in S from o and the optimum area increases with n, and at the limit approaches the radius and the area of the circle ψ, respectively. We also consider another variation of this problem where a set of n points is placed inside ψ, and the task is to locate a new point q inside ψ such that the area of the Voronoi region of q is maximized. We give an exact solution of this problem when n = 1 and a (1 − ε)-approximation algorithm for the general case

Locating waste pipelines to minimize their impact on marine environment

A waste pipeline, considered as an undesirable facility, is to be located in a coastal region. Two criteria are taken into account, the Euclidean distance from a given set of protected areas (coral reefs and sandbanks) and a utility function related to the pipe length, both to be maximized. The paper describes a methodology to obtain an efficient set of points where the extreme of a marine pipeline should be located. Since the formulation of the model is based on the zone Voronoi diagram, the computational complexity of the solving procedure is low

The Discrete Voronoi game in ℝ\u3csup\u3e2\u3c/sup\u3e

In this paper we study the last round of the discrete Voronoi game in ℝ2, a problem which is also of independent interest in competitive facility location. The game consists of two players P1 and P2, and a finite set U of users in the plane. The players have already placed two disjoint sets of facilities F and S, respectively, in the plane. The game begins with P1 placing a new facility followed by P2 placing another facility, and the objective of both the players is to maximize their own total payoffs. In this paper we propose polynomial time algorithms for determining the optimal strategies of both the players for arbitrarily located existing facilities F and S. We show that in the L1 and the L∞ metrics, the optimal strategy of P2, given any placement of P1, can be found in O(n log n) time, and the optimal strategy of P1 can be found in O(n5 log n) time. In the L2 metric, the optimal strategies of P2 and P1 can be obtained in O(n2) and O(n2) and O(n8) times, respectively

New Variations of the Maximum Coverage Facility Location Problem

Consider a competitive facility location scenario where, given a set U of n users and a set F of m facilities in the plane, the objective is to place a new facility in an appropriate place such that the number of users served by the new facility is maximized. Here users and facilities are considered as points in the plane, and each user takes service from its nearest facility, where the distance between a pair of points is measured in either L1 or L2 or L∞ metric. This problem is also known as the maximum coverage (MaxCov) problem. In this paper, we will consider the k-MaxCov problem, where the objective is to place k (⩾1) new facilities such that the total number of users served by these k new facilities is maximized. We begin by proposing an O(nlogn) time algorithm for the k-MaxCov problem, when the existing facilities are all located on a single straight line and the new facilities are also restricted to lie on the same line. We then study the 2-MaxCov problem in the plane, and propose an O(n2) time and space algorithm in the L1 and L∞ metrics. In the L2 metric, we solve the 2-MaxCov problem in the plane in O(n3logn) time and O(n2logn) space. Finally, we consider the 2-Farthest-MaxCov problem, where a user is served by its farthest facility, and propose an algorithm that runs in O(nlogn) time, in all the three metrics

Topology Network Optimization of Facility Planning and Design Problems

The research attempts to provide a graphical theory-based approach to solve the facility layout problem. Which has generally been approached using Quadratic Assignment Problem (QAP) in the past, an algebraic method. It is a very complex problem and is part of the NP-Hard optimization class, because of the nonlinear quadratic objective function and (0,1) binary variables. The research is divided into three phases which together provide an optimal facility layout, block plan solution consisting of MHS (material handling solution) projected onto the block plan. In phase one, we solve for the position of departments in a facility based on flow and utility factor (weight for location). The position of all the departments is identified on the vertices of MPG (maximal planar graph), which maximizes the possibility of flow. We use named MPG produced in literature, throughout the research. The grouping of the department is achieved through GMAFLAD, a QSP (quadratic set packing) based optimizer. In Phase 2, the dual for the MPG’s is solved consisting of department location as per phase 1, to generate Voronoi graphs. These graphs are then, expanded by an ingenious parameter optimization formulation to achieve area fitting for individual cases. Optimization modeling software, Lingo17.0 is used for solving the parameter optimization for generating coordinates of the block plan. The plotting of coordinates for the block plan graphics is done via Autodesk inventor 2019. In phase 3, the solution for MHS is achieved using an RSMT (Rectilinear Steiner minimal tree) graph approach. The Voronoi seed coordinates produced through phase 2 results are computed by GeoSteiner package to generated the RSMT graph for projection onto the block plan (Also, done by Inventor 2019). The graphical method employed in this research, itself has complex and NP-hard problem segments in it, which have been relaxed to polynomial time complexity by fragmenting into groups and solving them in sections. Solving for MPG & RSMT are a class of NP-Hard problem, which have been restricted to N=32 here. Finally, to validate the research and its methodology a real-life case study of a shipyard building for the data set of PDVSA, Venezuela is performed and verified

Reverse Thinking in Spatial Queries

In recent years, an increasing number of researches are conducted on spatial queries regarding the influence of query objects. Among these queries, reverse k nearest neighbors (RkNN) query is the one studied the most extensively. Reverse k furthest neighbors (RkFN) queries is the natural complement of RkNN queries. RkNN query is introduced to reflect the influence of the query object. Since this representation is intuitive, RkNN query has attracted significant attention among the database community. Later, reverse top-k queries was introduced, and also used extensively to represent influence. In many scenarios, when we consider the influence of an spatial object, reverse thinking is involved. That is, whether an object is influential to another object is depending on how the other object assess this object, other than how this object considers the other object. In this thesis, we study three problems involves reverse thinking. We first study the problem of efficiently computing RkFN queries. We are the first to propose a solution for arbitrary value of k. Based on several interesting observations, we present an efficient algorithm to process the RkFN queries. We also present a rigorous theoretical analysis to study various important aspects of the problem and our algorithm. An extensive experimental study demonstrates that our algorithm outperforms the state-of-the-art algorithm even for k=1. The accuracy of our theoretical analysis is also verified. We then study the problem of selecting set of representative products considering both diversity and coverage based on reverse top-k queries. Since this problem is NP-hard, we employ a greedy algorithm. We adopt MinHash and KMV Synopses to assist set operations. Our experimental study demonstrates the performance of the proposed algorithm. We also study the problem of maximizing spatial influence of facility bundle based on RkNN queries. We are the first to study this problem. We prove its NP-hardness, and propose a branch-and-bound best first search algorithm that greedily select the currently best facility until we get the required number of facilities. We introduce the concept of kNN region. It allows us to avoid redundant calculation with dynamic programming technique. Experiments show that our algorithm is orders of magnitudes better than our baseline algorithm

Football player dominant region determined by a novel model based on instantaneous kinematics variables

Dominant regions are defined as regions of the pitch where a player can reach before any other and are commonly determined without considering the free-spaces in the pitch. We presented an approach to football players’ dominant regions analysis, based on movement models created from players’ positions, displacement, velocity, and acceleration vectors. 109 Brazilian male professional football players were analysed during official matches, computing over 15 million positional data obtained by video-based tracking system. Movement models were created based on players’ instantaneous vectorial kinematics variables, then probabilities models and dominant regions were determined. Accuracy in determining dominant regions by the proposed model was tested for different time-lag windows. We calculated the areas of dominant, free-spaces, and Voronoi regions. Mean correct predictions of dominant region were 96.56%, 88.64%, and 72.31% for one, two, and three seconds, respectively. Dominant regions areas were lower than the ones computed by Voronoi, with median values of 73 and 171 m2, respectively. A median value of 5537 m2 was presented for free-space regions, representing a large part of the pitch. The proposed movement model proved to be more realistic, representing the match dynamics and can be a useful method to evaluate the players’ tactical behaviours during matches

Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on Computational Geometry 200

Coordinated control of mixed robot and sensor networks in distributed area exploration

Recent advancements in wireless communication and electronics has enabled the development of multifunctional sensor nodes that are small in size and communicate untethered in short distances. In the last decade, significant advantages have been made in the field of robotics, and robots have become more feasible in systems design. Therefore, we trust that a number of open problems with wireless sensor networks can be solved or diminished by including mobility capabilities in agents

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered