第三阶段：核心算法

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

def two_pointers(arr):
    left, right = 0, len(arr) - 1
    while left < right:
        if condition:
            left += 1
        else:
            right -= 1

def sliding_window(s):
    window = {}
    left = 0
    result = 0
    
    for right in range(len(s)):
        # 1. 扩大窗口
        c = s[right]
        window[c] = window.get(c, 0) + 1
        
        # 2. 收缩窗口
        while need_shrink:
            d = s[left]
            window[d] -= 1
            left += 1
        
        # 3. 更新结果
        result = max(result, right - left + 1)
    
    return result

def backtrack(path, choices):
    if 满足结束条件:
        result.append(path[:])
        return
    
    for choice in choices:
        path.append(choice)      # 做选择
        backtrack(path, new_choices)  # 递归
        path.pop()               # 撤销选择

def dfs(node, visited):
    if node in visited:
        return
    visited.add(node)
    for neighbor in graph[node]:
        dfs(neighbor, visited)

from collections import deque

def bfs(start):
    queue = deque([start])
    visited = {start}
    while queue:
        node = queue.popleft()
        for neighbor in graph[node]:
            if neighbor not in visited:
                visited.add(neighbor)
                queue.append(neighbor)