跳表（Skip List）是一种概率型数据结构，用于实现有序元素集合，支持高效的插入、删除和查找操作。跳表通过在多个层级上维护链表，使得平均时间复杂度可以达到 $O (lo g n)$ 。下面是一个简单的 Python 实现：

pythonCopy Codeimport randomclass Node:    def __init__(self, value, level):
        self.value = value
        self.forward = [None] * (level + 1)class SkipList:    def __init__(self, max_level, p):
        self.max_level = max_level
        self.p = p  # Probability for random level generation
        self.header = self._create_node(self.max_level, -1)  # Header node, value doesn't matter
        self.level = 0  # Current highest level in the skip list
    def _create_node(self, lvl, value):        return Node(value, lvl)    def _random_level(self):
        lvl = 0
        while random.random() < self.p and lvl < self.max_level:
            lvl += 1
        return lvl    def insert(self, value):
        update = [None] * (self.max_level + 1)
        current = self.header        # Start from the highest level of the skip list
        for i in range(self.level, -1, -1):            while current.forward[i] and current.forward[i].value < value:
                current = current.forward[i]
            update[i] = current
        current = current.forward[0]        # If the value is not already present, insert it
        if current is None or current.value != value:
            rlevel = self._random_level()            if rlevel > self.level:                for i in range(self.level + 1, rlevel + 1):
                    update[i] = self.header
                self.level = rlevel
            node = self._create_node(rlevel, value)            for i in range(rlevel + 1):
                node.forward[i] = update[i].forward[i]
                update[i].forward[i] = node    def search(self, value):
        current = self.header        for i in range(self.level, -1, -1):            while current.forward[i] and current.forward[i].value < value:
                current = current.forward[i]
        current = current.forward[0]        if current and current.value == value:            return True
        return False
    def delete(self, value):
        update = [None] * (self.max_level + 1)
        current = self.header        for i in range(self.level, -1, -1):            while current.forward[i] and current.forward[i].value < value:
                current = current.forward[i]
            update[i] = current
        current = current.forward[0]        if current is not None and current.value == value:            for i in range(self.level + 1):                if update[i].forward[i] != current:                    break
                update[i].forward[i] = current.forward[i]            while self.level > 0 and self.header.forward[self.level] is None:
                self.level -= 1# 示例用法skip_list = SkipList(3, 0.5)
skip_list.insert(3)
skip_list.insert(6)
skip_list.insert(7)
skip_list.insert(9)print(skip_list.search(3))  # 输出 Trueprint(skip_list.search(4))  # 输出 Falseskip_list.delete(3)print(skip_list.search(3))  # 输出 False

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用 Python 实现跳表（Skip List）