A discrete point minimum circle covering method based on farthest point pairing

Abstract

The application discloses a discrete point minimum circle covering method based on farthest point pairing, and belongs to the technical field of unmanned aerial vehicle cluster covering. The method takes a circle with a fixed radius as a single unmanned aerial vehicle covering unit, and aims at a discrete non-overlapping point set, i.e. a target to be covered. The core logic of the method includes the following steps: reference point selection, Euclidean distance calculation, effective point set screening, farthest effective point pairing, midpoint circle center determination, maximum coverage point selection, circle center position fine adjustment and covered point removal. The method realizes covering of all targets by using the minimum number of unmanned aerial vehicles. The algorithm steps are simple, the execution efficiency is high, and the number of covering circles is close to the theoretical optimal value. The method does not need complex iterative calculation, can directly output the circle center coordinates of all covering circles, is suitable for engineering scenes such as unmanned aerial vehicle cluster deployment, regional monitoring and target covering, and has good engineering practicability and portability.

Description

Technical Field

[0001] This invention relates to a discrete point minimum circle coverage method based on the pairing of farthest points, belonging to the field of UAV swarm coverage technology. Background Technology

[0002] In engineering and academic fields such as drone swarm deployment, area monitoring, target localization and coverage, and computer graphics, it is often necessary to use circles of fixed radius to fully cover a set of discrete points within a specified area, while minimizing the number of circles used for coverage—this is known as the minimum circle coverage problem. The minimum circle coverage problem is a classic combinatorial optimization problem. Its core requirement is to minimize the number of coverage units while ensuring that all discrete points are covered, thereby reducing deployment costs and improving coverage efficiency.

[0003] Existing methods for minimum circle coverage of discrete points mainly include greedy algorithms, exhaustive search, and genetic algorithms. Exhaustive search determines the minimum coverage circle by traversing all combinations of points. While it can obtain the theoretically optimal solution, its algorithm complexity is high, and its execution efficiency is extremely low when the number of points is large, making it difficult to adapt to practical engineering applications. Intelligent optimization algorithms such as genetic algorithms can obtain relatively good solutions, but they require setting a large number of iteration parameters, have cumbersome steps, and suffer from slow convergence speed and a tendency to get trapped in local optima. Traditional greedy algorithms often use random points as the center and update the center by continuously expanding the coverage area. This results in high randomness in center selection and low utilization efficiency of the coverage circles, easily leading to an excessive number of coverage circles and failing to achieve a coverage effect close to the theoretical optimal.

[0004] In drone swarm coverage scenarios, most existing methods, designed for complex environments, integrate redundant logic such as obstacle avoidance and complex area adaptation. When applied to simple, unobstructed designated areas, this redundant logic increases the number of algorithm execution steps, reduces efficiency, and the complex obstacle avoidance judgments can lead to unnecessary fallbacks in center selection, affecting the optimality of the coverage circle. Therefore, for unobstructed designated areas, there is an urgent need for a minimal discrete-point coverage method that is concise, efficient, and has a near-ideal number of coverage circles to reduce the number of drones required for area coverage and lower application costs. Summary of the Invention

[0005] The purpose of this invention is to design a discrete point minimum circle coverage method based on the pairing of farthest points, to solve the problems of high complexity, large number of algorithms, low execution efficiency, and redundant logic in obstacle-free area coverage scenarios of existing UAV swarm area coverage algorithms.

[0006] To solve these problems, the technical solution adopted by this invention is as follows: A discrete point minimum circle cover method based on farthest point pairing includes the following steps: Step 1: Establish an initial point set P for N targets to be covered within the specified area; and set the radius of the UAV coverage circle to be r, where the diameter d = 2r; Step 2: Initialize the set of target points to be covered, Q, let Q=P, and initialize the set of centers of the drone coverage circle, C, where C is an empty set; Step 3: Determine if the set of target points Q to be covered is empty. If it is empty, proceed to step 10; otherwise, proceed to step 4. Step 4: Select the uncovered points in the target point set Q as reference points p0 in sequence; Step 5: Calculate the Euclidean distance from all points in the target point set Q to the reference point p0, and filter out the points whose distance is less than or equal to the diameter d to form the valid point set Q_valid; Step 6: In the valid point set Q_valid, select the point farthest from the reference point p0 as the pairing point p_max; Step 7: Calculate the midpoint coordinates of each reference point p0 and the paired point p_max in sequence, and take the midpoint as the center c of the current covering circle; Step 8: Calculate the Euclidean distance from all points in the target point set Q to the center c of the circle, filter out points whose distance is less than or equal to the radius r, and select the initial center c_max_init that contains the most uncovered points; Step 9: Using c_max_init as the initial center, generate multiple fine-tuning centers in uniform directions around it with a radius of 0.1*r. Select the center c_max with the most uncovered points from the initial center and all fine-tuning centers, add the center c_max to the drone coverage circle center set C, remove the points covered by the center c_max from the target point set Q, and return to step 3. Step 10: The drone covers all the circles with radius r corresponding to the centers in the set C of the covered circles. This is the minimum number of covered circles for the original point set P. Output the set C of the covered circles and the number of covered circles.

[0007] The present invention has the following advantages over the prior art: 1. The core logic of this invention is: reference point selection - Euclidean distance calculation - effective point set screening - farthest effective point pairing - midpoint to determine the center of the circle - selection of the most covered points - fine adjustment of the center position - removal of covered points. The steps are simple, without complex iterations and parameter settings, the algorithm has low complexity and high execution efficiency. 2. This invention maximizes the utilization of the coverage area of ​​a single UAV by screening effective points within the diameter of the reference point and selecting the farthest point for pairing, with the midpoint of the two points as the center. Compared with the traditional greedy algorithm, the number of UAV circles can be effectively reduced, and the coverage effect is close to the theoretical optimal value. 3. This invention performs a secondary screening on the selected coverage circle, selecting the circle containing the most uncovered points as the initial coverage circle, and further fine-tunes the center of the initial coverage circle, selecting the circle containing the most uncovered points again near the initial center, thereby optimizing the coverage algorithm and further reducing the number of drones. Attached Figure Description

[0008] Figure 1 This is a flowchart of the method according to an embodiment of the present invention.

[0009] Figure 2 This is a distribution map of the number of drones obtained from an embodiment of the present invention.

[0010] Figure 3 This is a distribution map of the number of drones obtained using a traditional greedy algorithm.

[0011] Figure 4 This is a distribution map of the number of drones obtained by using the maximum coverage strategy. Detailed Implementation

[0012] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0013] Reference Figure 1 In this embodiment, the method is implemented by simulating a drone swarm area coverage scenario in MATLAB. The coverage area is set as a rectangular area with a length of 20 and a width of 10. 200 discrete non-overlapping random points are generated within the rectangular area as the target to be covered. A circle with a radius of 2 is used as the coverage area of ​​a single drone to achieve coverage with the minimum number of circles.

[0014] Step 1: Parameter initialization. Set the size of the specified rectangular area to 20 in length and 10 in width. Generate 200 discrete, non-overlapping random points within this area as targets to be covered, forming the original point set P. The coverage area of ​​a single UAV is a circle with a radius of r=2 and a diameter of d=2r. Step 2: Initialize the set of target points to be covered, Q, let Q=P, and initialize the set of centers of the UAV coverage circles, C, where C is an empty set; Step 3: Determine if the target point set Q to be covered is empty. If it is empty, proceed to step 10; otherwise, proceed to step 4. Step 4: Select the uncovered points in the target point set Q as reference points p0 in sequence; Step 5: Calculate the Euclidean distance from all points in the target point set Q to the reference point p0, and filter out points whose distance is less than or equal to the diameter d to form the valid point set Q_valid; the calculation method is as follows: for points and points The distance is ; Step 6: In the valid point set Q_valid, select the point farthest from the reference point p0 as the pairing point p_max; Step 7: Calculate the midpoint coordinates of each reference point p0 and the paired point p_max in sequence, and use the midpoint as the center c of the current covering circle, and draw a circle with radius r; the midpoint coordinates are calculated as follows: if the coordinates of the reference point p0 are (x0, y0) and the coordinates of the paired point p_max are (x_max, y_max), then the midpoint coordinates are ((x0+x_max) / 2, (y0+y_max) / 2); Step 8: Calculate the Euclidean distance from all points in the target point set Q to the center c of the circle, filter out the points whose distance is less than or equal to the radius r, and select the initial center c_max_init that contains the most uncovered points; Step 9: Using c_max_init as the initial center, generate multiple fine-tuning centers in uniform directions around it with a radius of 0.1*r. Select the center c_max with the most uncovered points from the initial center and all fine-tuning centers, add the center c_max to the set of covering center C, remove the points covered by the center c_max from the set of points to be covered Q, and return to step 3. Step 10: End the loop, cover all circles with radius r corresponding to the center of the circle in the set of circle centers C, which is the minimum number of drones for the original point set P, and output the set of circle centers C and the number of drones.

[0015] To verify the effectiveness of the algorithm, a comparison was made between the traditional greedy algorithm and the greedy algorithm based on the maximum coverage strategy during the simulation, demonstrating the advantages of the algorithm. Figure 2 The drone quantity distribution diagram obtained in the example shows that there are 21 drones. Figure 3 The image shows the distribution of drone numbers obtained from a simulation using a traditional greedy algorithm, with a total of 31 drones. Figure 4 The image shows the distribution of drones obtained using the maximum coverage strategy, with 26 drones. The core logic of the traditional greedy algorithm is to sequentially select uncovered points as centers to draw circles, removing points covered by those circles until all points are covered. The core logic of the greedy algorithm based on the maximum coverage strategy is to repeatedly find the center of a circle that covers the most uncovered points from all uncovered points, using this as the center of a new circle, and repeating this process until all points are covered. This significantly reduces the number of circles used.

[0016] In summary, this invention optimizes the coverage algorithm by generating random points within a specified area and then optimizing the coverage algorithm through the following steps: reference point selection, Euclidean distance calculation, effective point set filtering, pairing of the farthest effective points, centering the circle at the midpoint, selecting the most covered points, fine-tuning the center position, and removing covered points. This further reduces the number of drones required.

Claims

1. A discrete point minimum circle coverage method based on farthest point pairing, characterized in that, Includes the following steps: Step 1: Establish an initial point set P for N targets to be covered within the specified area; and set the radius of the UAV coverage circle to be r, where the diameter d = 2r; Step 2: Initialize the set of target points to be covered, Q, let Q=P, and initialize the set of centers of the drone coverage circle, C, where C is an empty set; Step 3: Determine if the set of target points Q to be covered is empty. If it is empty, proceed to step 10; otherwise, proceed to step 4. Step 4: Select the uncovered points in the target point set Q as reference points p0 in sequence; Step 5: Calculate the Euclidean distance from all points in the target point set Q to the reference point p0, and filter out the points whose distance is less than or equal to the diameter d to form the valid point set Q_valid; Step 6: In the valid point set Q_valid, select the point farthest from the reference point p0 as the pairing point p_max; Step 7: Calculate the midpoint coordinates of each reference point p0 and the paired point p_max in sequence, and take the midpoint as the center c of the current covering circle; Step 8: Calculate the Euclidean distance from all points in the target point set Q to the center c of the circle, filter out points whose distance is less than or equal to the radius r, and select the initial center c_max_init that contains the most uncovered points; Step 9: Using c_max_init as the initial center, generate multiple fine-tuning centers in uniform directions around it with a radius of 0.1*r. Select the center c_max with the most uncovered points from the initial center and all fine-tuning centers, add the center c_max to the drone coverage circle center set C, remove the points covered by the center c_max from the target point set Q, and return to step 3. Step 10: The drone covers all the circles with radius r corresponding to the centers in the set C of the covered circles. This is the minimum number of covered circles for the original point set P. Output the set C of the covered circles and the number of covered circles.

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