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Graham扫描法（Graham Scan）是一种计算平面点集凸包的有效算法。作为一名开发人员，以下是通过Objective-C实现Graham扫描法的详细说明。

Graham扫描法的基本原理

Graham扫描法的核心思想是：通过选择一个凸包中的顶点作为起点，然后依次选择向外扩展的点，构建凸包。具体步骤如下：

Objective-C实现步骤

以下是Objective-C中实现Graham扫描法的具体步骤：

#import 
   
    
#include 
    
     
@interface GrahamScan : NSObject {
    // 存储点的集合
    NSArray *points;
}
// 初始化Graham扫描
- (id)initWithPoints:(NSArray *)points;
- (NSArray *)computeConvexHull;
- (NSArray *)getPointsSortedByPolarAngle:(NSArray *)points;
- (NSArray *)getPointsSortedByAngle;
- (NSArray *)computeConvexHullUsingGrahamScan;
@end
    
   

代码实现

#import 
   
    
#include 
    
     
@interface GrahamScan : NSObject {
    NSArray *points;
    CGPoint *pointArray;
    int n;
}
- (id)initWithPoints:(NSArray *)points;
- (NSArray *)computeConvexHull;
- (NSArray *)getPointsSortedByPolarAngle:(NSArray *)points;
- (NSArray *)getPointsSortedByAngle;
- (NSArray *)computeConvexHullUsingGrahamScan;
@end
@implementation GrahamScan
- (id)initWithPoints:(NSArray *)points {
    self;
    self.points = points;
    self.n = [points count];
    self.pointArray = (CGPoint *)malloc(self.n * sizeof(CGPoint));
    for (int i = 0; i < self.n; i++) {
        self.pointArray[i] = (CGPoint){ [points[i] x, points[i] y ] };
    }
    return self;
}
- (NSArray *)getPointsSortedByPolarAngle:(NSArray *)points {
    if (self.n < 3) return points;
    
    // 计算每个点的极角
    double *theta = malloc(self.n * sizeof(double));
    for (int i = 0; i < self.n; i++) {
        double x = self.pointArray[i].x;
        double y = self.pointArray[i].y;
        if (x == 0 && y == 0) continue;
        double angle = atan2(y, x);
        theta[i] = angle;
    }
    
    // 根据极角排序
    for (int i = 0; i < self.n; i++) {
        for (int j = i + 1; j < self.n; j++) {
            if (theta[i] > theta[j]) {
                CGPoint temp = self.pointArray[i];
                self.pointArray[i] = self.pointArray[j];
                self.pointArray[j] = temp;
                double tempTheta = theta[i];
                theta[i] = theta[j];
                theta[j] = tempTheta;
            }
        }
    }
    
    // 构建结果数组
    NSMutableArray *sortedPoints = [NSMutableArray array];
    for (int i = 0; i < self.n; i++) {
        [sortedPoints addObject: self.pointArray[i]];
    }
    return sortedPoints;
}
- (NSArray *)getPointsSortedByAngle {
    if (self.n < 3) return self.points;
    
    // 计算每个点的极角
    double *theta = malloc(self.n * sizeof(double));
    for (int i = 0; i < self.n; i++) {
        double x = self.pointArray[i].x;
        double y = self.pointArray[i].y;
        if (x == 0 && y == 0) continue;
        double angle = atan2(y, x);
        theta[i] = angle;
    }
    
    // 根据极角排序
    for (int i = 0; i < self.n; i++) {
        for (int j = i + 1; j < self.n; j++) {
            if (theta[i] > theta[j]) {
                CGPoint temp = self.pointArray[i];
                self.pointArray[i] = self.pointArray[j];
                self.pointArray[j] = temp;
                double tempTheta = theta[i];
                theta[i] = theta[j];
                theta[j] = tempTheta;
            }
        }
    }
    
    NSMutableArray *sortedPoints = [NSMutableArray array];
    for (int i = 0; i < self.n; i++) {
        [sortedPoints addObject: self.pointArray[i]];
    }
    return sortedPoints;
}
- (NSArray *)computeConvexHull {
    if (self.n < 3) return self.points;
    
    NSArray *sortedPoints = [self getPointsSortedByPolarAngle:self.points];
    NSMutableArray *hullPoints = [NSMutableArray array];
    
    for (int i = 0; i < sortedPoints.count; i++) {
        bool clockwise = false;
        for (int j = 0; j < hullPoints.count; j++) {
            double crossProduct = [self crossProductBetweenPoint:sortedPoints[i] 
                                                      andPoint:hullPoints[j]
                                                      andPoint:hullPoints[i-1]];
            
            if (crossProduct < 0) {
                clockwise = true;
                break;
            }
        }
        
        if (clockwise) {
            hullPoints[0] = sortedPoints[i];
        } else {
            hullPoints[i] = sortedPoints[i];
        }
    }
    
    return hullPoints;
}
- (double)crossProductBetweenPoint:(CGPoint)point1 
                             andPoint:(CGPoint)point2 
                             andPoint:(CGPoint)point3 {
    return (point1.x - point2.x) * (point3.y - point2.y) - (point1.y - point2.y) * (point3.x - point2.x);
}
- (NSArray *)computeConvexHullUsingGrahamScan {
    if (self.n < 3) return self.points;
    
    NSArray *sortedPoints = [self getPointsSortedByPolarAngle:self.points];
    NSMutableArray *hullPoints = [NSMutableArray array];
    
    for (int i = 0; i < sortedPoints.count; i++) {
        bool lastClockwise = false;
        for (int j = 0; j < hullPoints.count; j++) {
            double crossProduct = [self crossProductBetweenPoint:sortedPoints[i]
                                                      andPoint:hullPoints[j]
                                                      andPoint:hullPoints[j-1]];
            
            if (j == 0 || crossProduct < 0) {
                if (j == 0) {
                    lastClockwise = true;
                }
                continue;
            }
            
            if (lastClockwise != crossProduct < 0) {
                hullPoints[j] = sortedPoints[i];
                lastClockwise = !lastClockwise;
            }
        }
    }
    
    return hullPoints;
}
    
   

代码解释

性能优化

为了提高性能，可以尝试以下优化：

• 减少重复计算：尽量减少不必要的计算步骤，避免重复赋值。
• 使用更高效的数据结构：考虑使用数组代替动态数组，提升访问速度。
• 多线程处理：对于大规模数据集，可以尝试并行处理，减少计算时间。

以上是Graham扫描法在Objective-C中的实现与应用，希望对您有所帮助！