python實現跳錶（Skip List）的查找、插入、刪除操作

跳錶（Skip List）是對鏈表改造後的動態數據結構，利用空間換時間的思想，建立多級索引來提高查找、插入、刪除操作的效率。

一層一層索引有點類似二分查找這種分而治之的思想。

在跳錶中查詢一個數據的時間複雜度是O( log n )，空間複雜度是O( n )。

跳錶通過隨機函數來維護平衡性，即索引大小與原始鏈表大小之間的平衡，不至於性能過度退化，有效平衡執行效率和內存消耗。通過隨機函數，來決定將這個結點插入到哪幾級索引中，比如隨機函數生成了值K，那就將結點添加到第一級到第K級這K級索引中。

以下是用python3的具體實現。

from typing import Optional
import random


class ListNode:
    def __init__(self, data: Optional[int] = None):
        self._data = data
        self._forwards = []


class SkipList:
    _MAX_LEVEL = 16

    def __init__(self):
        self._level_count = 1
        self._head = ListNode()
        self._head._forwards = [None] * type(self)._MAX_LEVEL

    def find(self, value: int) -> Optional[ListNode]:  # 查找
        p = self._head
        for i in range(self._level_count - 1, -1, -1):
            while p._forwards[i] and p._forwards[i]._data < value:
                p = p._forwards[i]
        return p._forwards[0] if p._forwards[0] and p._forwards[0]._data == value else None

    def insert(self, value: int):  # 插入
        level = self._random_level()
        if self._level_count < level:
            self._level_count = level
        new_node = ListNode(value)
        new_node._forwards = [None] * level
        update = [self._head] * level
        p = self._head
        for i in range(level - 1, -1, -1):
            while p._forwards[i] and p._forwards[i]._data < value:
                p = p._forwards[i]
            update[i] = p
        for i in range(level):
            new_node._forwards[i] = update[i]._forwards[i]
            update[i]._forwards[i] = new_node

    def delete(self, value: int):  刪除
        update = [None] * self._level_count
        p = self._head
        for i in range(self._level_count - 1, -1, -1):
            while p._forwards[i] and p._forwards[i]._data < value:
                p = p._forwards[i]
            update[i] = p
        if p._forwards[0] and p._forwards[0]._data == value:
            for i in range(self._level_count - 1, -1, -1):
                if update[i]._forwards[i] and update[i]._forwards[i]._data == value:
                    update[i]._forwards[i] = update[i]._forwards[i]._forwards[i]

    def _random_level(self, p: float = 0.5) -> int:  # 隨機函數
        level = 1
        while random.random() < p and level < type(self)._MAX_LEVEL:
            level += 1
        return level

    def __repr__(self) -> str:
        values = []
        p = self._head
        while p._forwards[0]:
            values.append(str(p._forwards[0]._data))
            p = p._forwards[0]
        return "->".join(values)


if __name__ == "__main__":
    l = SkipList()
    for i in range(10):
        l.insert(i)
    print(l)
    p = l.find(8)
    print(p._data)
    l.delete(4)
    print(l)