搜索与排序

搜索和排序是最基础的算法，在面试中出现频率极高。掌握高效的搜索和排序算法对于优化程序性能至关重要。

二分搜索（Binary Search）

基本二分搜索

def binary_search(arr, target):
    left, right = 0, len(arr) - 1
    
    while left <= right:
        mid = (left + right) // 2
        if arr[mid] == target:
            return mid
        elif arr[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    
    return -1

查找边界

查找第一个等于 target 的位置

def find_first(arr, target):
    left, right = 0, len(arr) - 1
    result = -1
    
    while left <= right:
        mid = (left + right) // 2
        if arr[mid] == target:
            result = mid
            right = mid - 1  # 继续向左查找
        elif arr[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    
    return result

查找最后一个等于 target 的位置

def find_last(arr, target):
    left, right = 0, len(arr) - 1
    result = -1
    
    while left <= right:
        mid = (left + right) // 2
        if arr[mid] == target:
            result = mid
            left = mid + 1  # 继续向右查找
        elif arr[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    
    return result

应用：搜索旋转排序数组

def search_rotated(nums, target):
    left, right = 0, len(nums) - 1
    
    while left <= right:
        mid = (left + right) // 2
        if nums[mid] == target:
            return mid
        
        # 左半部分有序
        if nums[left] <= nums[mid]:
            if nums[left] <= target < nums[mid]:
                right = mid - 1
            else:
                left = mid + 1
        # 右半部分有序
        else:
            if nums[mid] < target <= nums[right]:
                left = mid + 1
            else:
                right = mid - 1
    
    return -1

排序算法

1. 快速排序（Quick Sort）

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)

# 原地排序版本
def quick_sort_inplace(arr, low, high):
    if low < high:
        pi = partition(arr, low, high)
        quick_sort_inplace(arr, low, pi - 1)
        quick_sort_inplace(arr, pi + 1, high)

def partition(arr, low, high):
    pivot = arr[high]
    i = low - 1
    
    for j in range(low, high):
        if arr[j] <= pivot:
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return i + 1

• 时间复杂度：平均 O(n log n)，最坏 O(n²)
• 空间复杂度：O(log n)
• 稳定性：不稳定

2. 归并排序（Merge Sort）

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

• 时间复杂度：O(n log n)
• 空间复杂度：O(n)
• 稳定性：稳定

3. 堆排序（Heap Sort）

import heapq

def heap_sort(arr):
    heap = []
    for num in arr:
        heapq.heappush(heap, num)
    
    result = []
    while heap:
        result.append(heapq.heappop(heap))
    
    return result

# 原地堆排序
def heapify(arr, n, i):
    largest = i
    left = 2 * i + 1
    right = 2 * i + 2
    
    if left < n and arr[left] > arr[largest]:
        largest = left
    if right < n and arr[right] > arr[largest]:
        largest = right
    
    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        heapify(arr, n, largest)

def heap_sort_inplace(arr):
    n = len(arr)
    
    # 构建最大堆
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)
    
    # 逐个提取元素
    for i in range(n - 1, 0, -1):
        arr[0], arr[i] = arr[i], arr[0]
        heapify(arr, i, 0)

• 时间复杂度：O(n log n)
• 空间复杂度：O(1)
• 稳定性：不稳定

4. 计数排序（Counting Sort）

def counting_sort(arr):
    if not arr:
        return arr
    
    min_val, max_val = min(arr), max(arr)
    count = [0] * (max_val - min_val + 1)
    
    for num in arr:
        count[num - min_val] += 1
    
    result = []
    for i, cnt in enumerate(count):
        result.extend([i + min_val] * cnt)
    
    return result

• 时间复杂度：O(n + k)，k 是值域范围
• 空间复杂度：O(k)
• 稳定性：稳定
• 适用场景：整数排序，值域范围较小

5. 基数排序（Radix Sort）

def radix_sort(arr):
    max_val = max(arr)
    exp = 1
    
    while max_val // exp > 0:
        counting_sort_by_digit(arr, exp)
        exp *= 10
    
    return arr

def counting_sort_by_digit(arr, exp):
    n = len(arr)
    output = [0] * n
    count = [0] * 10
    
    for i in range(n):
        index = (arr[i] // exp) % 10
        count[index] += 1
    
    for i in range(1, 10):
        count[i] += count[i - 1]
    
    for i in range(n - 1, -1, -1):
        index = (arr[i] // exp) % 10
        output[count[index] - 1] = arr[i]
        count[index] -= 1
    
    for i in range(n):
        arr[i] = output[i]

• 时间复杂度：O(d × (n + k))，d 是位数
• 空间复杂度：O(n + k)
• 稳定性：稳定

排序算法比较

常见题目

1. 合并两个有序数组

def merge_sorted_arrays(nums1, m, nums2, n):
    i, j, k = m - 1, n - 1, m + n - 1
    
    while i >= 0 and j >= 0:
        if nums1[i] > nums2[j]:
            nums1[k] = nums1[i]
            i -= 1
        else:
            nums1[k] = nums2[j]
            j -= 1
        k -= 1
    
    while j >= 0:
        nums1[k] = nums2[j]
        j -= 1
        k -= 1

2. 寻找峰值

def find_peak_element(nums):
    left, right = 0, len(nums) - 1
    
    while left < right:
        mid = (left + right) // 2
        if nums[mid] > nums[mid + 1]:
            right = mid
        else:
            left = mid + 1
    
    return left

3. 数组中的第 K 个最大元素

import heapq

def find_kth_largest(nums, k):
    return heapq.nlargest(k, nums)[-1]

# 使用快速选择
def find_kth_largest_quickselect(nums, k):
    def quick_select(left, right, k_smallest):
        if left == right:
            return nums[left]
        
        pivot_index = partition(left, right)
        
        if k_smallest == pivot_index:
            return nums[k_smallest]
        elif k_smallest < pivot_index:
            return quick_select(left, pivot_index - 1, k_smallest)
        else:
            return quick_select(pivot_index + 1, right, k_smallest)
    
    def partition(left, right):
        pivot = nums[right]
        i = left
        for j in range(left, right):
            if nums[j] <= pivot:
                nums[i], nums[j] = nums[j], nums[i]
                i += 1
        nums[i], nums[right] = nums[right], nums[i]
        return i
    
    return quick_select(0, len(nums) - 1, len(nums) - k)

总结

搜索和排序是算法的基础：

1. 二分搜索：O(log n) 时间查找，适用于有序数组
2. 快速排序：平均 O(n log n)，适合通用排序
3. 归并排序：稳定排序，O(n log n) 时间
4. 堆排序：原地排序，O(n log n) 时间
5. 特殊排序：计数排序、基数排序适用于特定场景

掌握这些算法能够帮助你高效解决各种问题。

接下来，让我们学习递归与回溯，这是解决复杂问题的重要方法。