# 归并排序 - 分治经典应用
def merge_sort(arr):
    if len(arr) <= 1:
        return arr
    
    mid = len(arr) // 2
    left = merge_sort(arr[:mid])
    right = merge_sort(arr[mid:])
    
    return merge(left, right)

def merge(left, right):
    result = []
    i = j = 0
    while i < len(left) and j < len(right):
        if left[i] <= right[j]:
            result.append(left[i])
            i += 1
        else:
            result.append(right[j])
            j += 1
    result.extend(left[i:])
    result.extend(right[j:])
    return result

# 示例
arr = [38, 27, 43, 3, 9, 82, 10]
print(merge_sort(arr))
# 输出: [3, 9, 10, 27, 38, 43, 82]

# 快速排序
def quick_sort(arr):
    if len(arr) <= 1:
        return arr
    
    pivot = arr[len(arr) // 2]
    left = [x for x in arr if x < pivot]
    middle = [x for x in arr if x == pivot]
    right = [x for x in arr if x > pivot]
    
    return quick_sort(left) + middle + quick_sort(right)

# 示例
arr = [3, 6, 8, 10, 1, 2, 1]
print(quick_sort(arr))
# 输出: [1, 1, 2, 3, 6, 8, 10]

# 最近点对问题
def closest_pair(points):
    if len(points) <= 3:
        return brute_force_closest(points)
    
    mid = len(points) // 2
    mid_x = points[mid][0]
    
    left = points[:mid]
    right = points[mid:]
    
    dl = closest_pair(left)
    dr = closest_pair(right)
    d = min(dl, dr)
    
    strip = [p for p in points if abs(p[0] - mid_x) < d]
    return min(d, strip_closest(strip, d))