最长连续序列：O(n)最优解法详解

最长连续序列问题概述给定一个未排序的整数数组找出数字连续的最长序列不要求序列元素在原数组中连续的长度。要求算法的时间复杂度为 O(n)。示例 输入nums [100,4,200,1,3,2] 输出4 解释最长数字连续序列是 [1, 2, 3, 4]其长度为 4。哈希集合解法使用哈希集合存储所有数字实现快速查找。遍历每个数字时检查是否存在比它小1的数字。如果不存在则以该数字为起点开始向后查找连续序列。def longestConsecutive(nums): num_set set(nums) max_length 0 for num in num_set: if num - 1 not in num_set: current_num num current_length 1 while current_num 1 in num_set: current_num 1 current_length 1 max_length max(max_length, current_length) return max_length时间复杂度分析每个数字最多被访问两次作为序列起点和作为其他序列的中间节点总体时间复杂度为 O(n)。排序解法先对数组进行排序然后遍历排序后的数组统计连续序列的长度。虽然时间复杂度为 O(n log n)但实现简单。def longestConsecutive(nums): if not nums: return 0 nums.sort() max_length 1 current_length 1 for i in range(1, len(nums)): if nums[i] ! nums[i-1]: if nums[i] nums[i-1] 1: current_length 1 max_length max(max_length, current_length) else: current_length 1 return max_length注意处理重复数字的情况避免错误计数。并查集解法将连续的数字视为连通分量使用并查集数据结构维护连通关系。最终统计最大连通分量的大小。class UnionFind: def __init__(self, nums): self.parent {num: num for num in nums} self.size {num: 1 for num in nums} def find(self, x): while self.parent[x] ! x: self.parent[x] self.parent[self.parent[x]] x self.parent[x] return x def union(self, x, y): x_root self.find(x) y_root self.find(y) if x_root y_root: return if self.size[x_root] self.size[y_root]: x_root, y_root y_root, x_root self.parent[y_root] x_root self.size[x_root] self.size[y_root] def longestConsecutive(nums): if not nums: return 0 uf UnionFind(nums) num_set set(nums) for num in num_set: if num 1 in num_set: uf.union(num, num 1) return max(uf.size.values())并查集通过路径压缩和按秩合并优化使得操作接近常数时间复杂度。动态规划解法使用哈希表记录以每个数字为端点的连续序列长度。遍历时检查相邻数字是否存在并更新哈希表。def longestConsecutive(nums): num_dict {} max_length 0 for num in nums: if num not in num_dict: left num_dict.get(num - 1, 0) right num_dict.get(num 1, 0) current_length left right 1 num_dict[num] current_length num_dict[num - left] current_length num_dict[num right] current_length max_length max(max_length, current_length) return max_length这种方法只需一次遍历时间复杂度为 O(n)空间复杂度为 O(n)。边界情况处理空数组应返回0。包含重复数字时不应影响结果。大整数测试时需验证算法效率。# 测试用例 print(longestConsecutive([])) # 0 print(longestConsecutive([1,2,0,1])) # 3 print(longestConsecutive([0,3,7,2,5,8,4,6,0,1])) # 9算法选择建议面试场景推荐哈希集合解法代码简洁且满足时间复杂度要求。实际工程中可根据数据特征选择合适方法如数据规模较小时可用排序解法。复杂度对比哈希集合时间 O(n)空间 O(n) 排序时间 O(n log n)空间 O(1) 或 O(n) 并查集时间接近 O(n)空间 O(n) 动态规划时间 O(n)空间 O(n) 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