Python实战开发及案例分析（13）—— 散列表

贾贾20232024-05-10 15:44

散列表（Hash Table）是一种高效的键值对数据结构，具有快速的查找、插入和删除操作。它通过哈希函数将键映射到散列表的索引，实现对数据的快速访问。

散列表的实现

  • 哈希函数：将键转换为整数索引。
  • 冲突处理 ：解决哈希冲突（不同的键映射到同一索引）。
    • 链地址法（拉链法）：每个索引位置用链表存储多个键值对。
    • 开放寻址法：当冲突发生时，使用线性探测、二次探测或双重散列等策略寻找新的空闲位置。

Python 实现：链地址法（拉链法）

我们将使用链地址法来实现一个简单的散列表。

散列表类实现：
python 复制代码 
class HashTable:
    def __init__(self, size=10):
        """初始化散列表"""
        self.size = size
        self.table = [[] for _ in range(size)]

    def _hash_function(self, key):
        """哈希函数：将键转换为表索引"""
        return hash(key) % self.size

    def insert(self, key, value):
        """插入键值对"""
        index = self._hash_function(key)
        # 检查键是否已经存在
        for pair in self.table[index]:
            if pair[0] == key:
                pair[1] = value  # 更新现有键的值
                return
        # 插入新键值对
        self.table[index].append([key, value])

    def search(self, key):
        """根据键查找值"""
        index = self._hash_function(key)
        for pair in self.table[index]:
            if pair[0] == key:
                return pair[1]
        return None  # 找不到对应键

    def delete(self, key):
        """删除键值对"""
        index = self._hash_function(key)
        for i, pair in enumerate(self.table[index]):
            if pair[0] == key:
                del self.table[index][i]
                return True
        return False  # 找不到对应键

    def display(self):
        """显示散列表"""
        for i, pairs in enumerate(self.table):
            print(f"Index {i}: {pairs}")

# 示例使用
hash_table = HashTable(size=7)

# 插入键值对
hash_table.insert("Alice", 25)
hash_table.insert("Bob", 30)
hash_table.insert("Charlie", 35)
hash_table.insert("Diana", 40)
hash_table.insert("Eve", 45)
hash_table.insert("Frank", 50)
hash_table.insert("Grace", 55)
hash_table.insert("Heidi", 60)

# 显示散列表
print("Initial Hash Table:")
hash_table.display()

# 查找键
print("\nSearching for 'Diana':", hash_table.search("Diana"))
print("Searching for 'Eve':", hash_table.search("Eve"))
print("Searching for 'Zara':", hash_table.search("Zara"))

# 删除键值对
hash_table.delete("Alice")
hash_table.delete("Charlie")
hash_table.delete("Heidi")

# 显示删除后的散列表
print("\nHash Table after deletions:")
hash_table.display()

Python 实现：开放寻址法

我们还可以使用开放寻址法来实现散列表。这里使用线性探测作为冲突处理策略。

散列表类实现：
python 复制代码 
class OpenAddressingHashTable:
    def __init__(self, size=10):
        """初始化散列表"""
        self.size = size
        self.table = [None] * size
        self.deleted = object()  # 标记删除位置

    def _hash_function(self, key):
        """哈希函数：将键转换为表索引"""
        return hash(key) % self.size

    def _probe(self, key, index):
        """线性探测"""
        return (index + 1) % self.size

    def insert(self, key, value):
        """插入键值对"""
        index = self._hash_function(key)
        while self.table[index] is not None and self.table[index] != self.deleted:
            if self.table[index][0] == key:
                self.table[index] = (key, value)  # 更新现有键的值
                return
            index = self._probe(key, index)
        self.table[index] = (key, value)

    def search(self, key):
        """根据键查找值"""
        index = self._hash_function(key)
        start_index = index
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                return self.table[index][1]
            index = self._probe(key, index)
            if index == start_index:
                break
        return None  # 找不到对应键

    def delete(self, key):
        """删除键值对"""
        index = self._hash_function(key)
        start_index = index
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                self.table[index] = self.deleted
                return True
            index = self._probe(key, index)
            if index == start_index:
                break
        return False  # 找不到对应键

    def display(self):
        """显示散列表"""
        for i, pair in enumerate(self.table):
            print(f"Index {i}: {pair}")

# 示例使用
open_addressing_hash_table = OpenAddressingHashTable(size=7)

# 插入键值对
open_addressing_hash_table.insert("Alice", 25)
open_addressing_hash_table.insert("Bob", 30)
open_addressing_hash_table.insert("Charlie", 35)
open_addressing_hash_table.insert("Diana", 40)
open_addressing_hash_table.insert("Eve", 45)
open_addressing_hash_table.insert("Frank", 50)
open_addressing_hash_table.insert("Grace", 55)
open_addressing_hash_table.insert("Heidi", 60)

# 显示散列表
print("Initial Open Addressing Hash Table:")
open_addressing_hash_table.display()

# 查找键
print("\nSearching for 'Diana':", open_addressing_hash_table.search("Diana"))
print("Searching for 'Eve':", open_addressing_hash_table.search("Eve"))
print("Searching for 'Zara':", open_addressing_hash_table.search("Zara"))

# 删除键值对
open_addressing_hash_table.delete("Alice")
open_addressing_hash_table.delete("Charlie")
open_addressing_hash_table.delete("Heidi")

# 显示删除后的散列表
print("\nOpen Addressing Hash Table after deletions:")
open_addressing_hash_table.display()

总结

  1. 链地址法（拉链法）：使用链表解决哈希冲突，简单易用。
  2. 开放寻址法：通过线性探测或其他策略处理哈希冲突，但需保持装载因子不过高。
  3. Python 实现 ：Python 中可以使用 dict 类实现高效的键值对存储，但手动实现散列表有助于更好理解其原理和应用。

我们继续扩展和深入探讨散列表的实现与应用，可以考虑：

  1. 改进的开放寻址法：实现更复杂的探测策略，如二次探测和双重散列。
  2. 扩展的链地址法：使用红黑树或 AVL 树作为链表中的数据结构，提高查询效率。
  3. 实际应用案例：例如，统计英文文章中每个单词的出现次数。

改进的开放寻址法：二次探测和双重散列

二次探测实现

在二次探测中，探测位置依次为：

​​​​​​​

其中， ℎ(𝑘)为哈希函数， 𝑖是探测次数， 𝑚为表的大小。

Python 实现：

python 复制代码 
class QuadraticProbingHashTable:
    def __init__(self, size=10):
        """初始化散列表"""
        self.size = size
        self.table = [None] * size
        self.deleted = object()

    def _hash_function(self, key):
        """哈希函数：将键转换为表索引"""
        return hash(key) % self.size

    def _probe(self, key, index, attempt):
        """二次探测"""
        return (index + attempt**2) % self.size

    def insert(self, key, value):
        """插入键值对"""
        index = self._hash_function(key)
        attempt = 0
        while self.table[index] is not None and self.table[index] != self.deleted:
            if self.table[index][0] == key:
                self.table[index] = (key, value)  # 更新现有键的值
                return
            attempt += 1
            index = self._probe(key, index, attempt)
        self.table[index] = (key, value)

    def search(self, key):
        """根据键查找值"""
        index = self._hash_function(key)
        start_index = index
        attempt = 0
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                return self.table[index][1]
            attempt += 1
            index = self._probe(key, index, attempt)
            if index == start_index:
                break
        return None  # 找不到对应键

    def delete(self, key):
        """删除键值对"""
        index = self._hash_function(key)
        start_index = index
        attempt = 0
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                self.table[index] = self.deleted
                return True
            attempt += 1
            index = self._probe(key, index, attempt)
            if index == start_index:
                break
        return False  # 找不到对应键

    def display(self):
        """显示散列表"""
        for i, pair in enumerate(self.table):
            print(f"Index {i}: {pair}")

# 示例使用
quadratic_probing_hash_table = QuadraticProbingHashTable(size=7)

# 插入键值对
quadratic_probing_hash_table.insert("Alice", 25)
quadratic_probing_hash_table.insert("Bob", 30)
quadratic_probing_hash_table.insert("Charlie", 35)
quadratic_probing_hash_table.insert("Diana", 40)
quadratic_probing_hash_table.insert("Eve", 45)
quadratic_probing_hash_table.insert("Frank", 50)
quadratic_probing_hash_table.insert("Grace", 55)
quadratic_probing_hash_table.insert("Heidi", 60)

# 显示散列表
print("Initial Quadratic Probing Hash Table:")
quadratic_probing_hash_table.display()

# 查找键
print("\nSearching for 'Diana':", quadratic_probing_hash_table.search("Diana"))
print("Searching for 'Eve':", quadratic_probing_hash_table.search("Eve"))
print("Searching for 'Zara':", quadratic_probing_hash_table.search("Zara"))

# 删除键值对
quadratic_probing_hash_table.delete("Alice")
quadratic_probing_hash_table.delete("Charlie")
quadratic_probing_hash_table.delete("Heidi")

# 显示删除后的散列表
print("\nQuadratic Probing Hash Table after deletions:")
quadratic_probing_hash_table.display()
双重散列实现

在双重散列中，我们使用两个不同的哈希函数来确定探测位置：

​​​​​​​ ​​​​​​​

其中， ℎ1(𝑘)和 ℎ2(𝑘)分别是两个不同的哈希函数。

Python 实现：

python 复制代码 
class DoubleHashingHashTable:
    def __init__(self, size=10):
        """初始化散列表"""
        self.size = size
        self.table = [None] * size
        self.deleted = object()

    def _hash_function1(self, key):
        """主哈希函数：将键转换为表索引"""
        return hash(key) % self.size

    def _hash_function2(self, key):
        """第二哈希函数：确保非零返回值"""
        return 1 + (hash(key) % (self.size - 1))

    def _probe(self, key, index, attempt):
        """双重散列探测"""
        return (index + attempt * self._hash_function2(key)) % self.size

    def insert(self, key, value):
        """插入键值对"""
        index = self._hash_function1(key)
        attempt = 0
        while self.table[index] is not None and self.table[index] != self.deleted:
            if self.table[index][0] == key:
                self.table[index] = (key, value)  # 更新现有键的值
                return
            attempt += 1
            index = self._probe(key, index, attempt)
        self.table[index] = (key, value)

    def search(self, key):
        """根据键查找值"""
        index = self._hash_function1(key)
        start_index = index
        attempt = 0
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                return self.table[index][1]
            attempt += 1
            index = self._probe(key, index, attempt)
            if index == start_index:
                break
        return None  # 找不到对应键

    def delete(self, key):
        """删除键值对"""
        index = self._hash_function1(key)
        start_index = index
        attempt = 0
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                self.table[index] = self.deleted
                return True
            attempt += 1
            index = self._probe(key, index, attempt)
            if index == start_index:
                break
        return False  # 找不到对应键

    def display(self):
        """显示散列表"""
        for i, pair in enumerate(self.table):
            print(f"Index {i}: {pair}")

# 示例使用
double_hashing_hash_table = DoubleHashingHashTable(size=7)

# 插入键值对
double_hashing_hash_table.insert("Alice", 25)
double_hashing_hash_table.insert("Bob", 30)
double_hashing_hash_table.insert("Charlie", 35)
double_hashing_hash_table.insert("Diana", 40)
double_hashing_hash_table.insert("Eve", 45)
double_hashing_hash_table.insert("Frank", 50)
double_hashing_hash_table.insert("Grace", 55)
double_hashing_hash_table.insert("Heidi", 60)

# 显示散列表
print("Initial Double Hashing Hash Table:")
double_hashing_hash_table.display()

# 查找键
print("\nSearching for 'Diana':", double_hashing_hash_table.search("Diana"))
print("Searching for 'Eve':", double_hashing_hash_table.search("Eve"))
print("Searching for 'Zara':", double_hashing_hash_table.search("Zara"))

# 删除键值对
double_hashing_hash_table.delete("Alice")
double_hashing_hash_table.delete("Charlie")
double_hashing_hash_table.delete("Heidi")

# 显示删除后的散列表
print("\nDouble Hashing Hash Table after deletions:")
double_hashing_hash_table.display()

实际应用案例：单词计数

利用散列表来统计英文文章中每个单词的出现次数。

Python 实现：

python 复制代码 
import string

class WordCounter:
    def __init__(self, size=100):
        """初始化散列表"""
        self.size = size
        self.table = [None] * size
        self.deleted = object()

    def _hash_function(self, key):
        """哈希函数：将键转换为表索引"""
        return hash(key) % self.size

    def insert(self, key, value):
        """插入键值对"""
        index = self._hash_function(key)
        while self.table[index] is not None and self.table[index] != self.deleted:
            if self.table[index][0] == key:
                self.table[index] = (key, value)  # 更新现有键的值
                return
            index = (index + 1) % self.size
        self.table[index] = (key, value)

    def search(self, key):
        """根据键查找值"""
        index = self._hash_function(key)
        start_index = index
        while self.table[index] is not None:
            if self.table[index] != self.deleted and self.table[index][0] == key:
                return self.table[index][1]
            index = (index + 1) % self.size
            if index == start_index:
                break
        return None  # 找不到对应键

    def increment(self, key):
        """增加单词的计数"""
        count = self.search(key)
        if count is None:
            self.insert(key, 1)
        else:
            self.insert(key, count + 1)

    def display(self):
        """显示单词计数"""
        for pair in self.table:
            if pair is not None and pair != self.deleted:
                print(f"{pair[0]}: {pair[1]}")

# 示例使用
text = """
This is a sample text. It is designed to test the hash table implementation. 
The hash table is supposed to count the occurrence of each word.
"""

# 清理文本，转换为小写
translator = str.maketrans('', '', string.punctuation)
clean_text = text.translate(translator).lower()
words = clean_text.split()

# 使用散列表计数单词
word_counter = WordCounter(size=50)
for word in words:
    word_counter.increment(word)

# 显示单词计数结果
print("Word Count:")
word_counter.display()

总结

  1. 开放寻址法改进：二次探测和双重散列是两种有效的开放寻址方法。
  2. 链地址法扩展：可以用红黑树、AVL 树等数据结构代替链表，提高查找效率。
  3. 实际应用案例：散列表在单词计数、字典存储等应用中具有广泛的实际价值。

让我们继续深入探讨散列表的其他改进和实际应用案例，包括：

  1. 基于红黑树或 AVL 树的链地址法实现：提高冲突处理效率。
  2. Bloom Filter：利用散列表和布隆过滤器进行快速集合成员测试。
  3. LRU Cache：结合散列表与双向链表实现的缓存系统。

基于红黑树的链地址法实现

链地址法通常使用链表来处理冲突。我们可以使用更高效的红黑树或 AVL 树来替代链表，提高冲突处理效率。

Python 实现：红黑树节点类
python 复制代码 
class RedBlackTreeNode:
    def __init__(self, key, value, color='RED', left=None, right=None, parent=None):
        """初始化红黑树节点"""
        self.key = key
        self.value = value
        self.color = color  # 'RED' or 'BLACK'
        self.left = left
        self.right = right
        self.parent = parent

class RedBlackTree:
    def __init__(self):
        """初始化红黑树"""
        self.NIL = RedBlackTreeNode(key=None, value=None, color='BLACK')
        self.root = self.NIL

    def _rotate_left(self, node):
        """左旋转"""
        right = node.right
        node.right = right.left
        if right.left != self.NIL:
            right.left.parent = node
        right.parent = node.parent
        if node.parent is None:
            self.root = right
        elif node == node.parent.left:
            node.parent.left = right
        else:
            node.parent.right = right
        right.left = node
        node.parent = right

    def _rotate_right(self, node):
        """右旋转"""
        left = node.left
        node.left = left.right
        if left.right != self.NIL:
            left.right.parent = node
        left.parent = node.parent
        if node.parent is None:
            self.root = left
        elif node == node.parent.right:
            node.parent.right = left
        else:
            node.parent.left = left
        left.right = node
        node.parent = left

    def _fix_insert(self, node):
        """修复插入操作"""
        while node != self.root and node.parent.color == 'RED':
            if node.parent == node.parent.parent.left:
                uncle = node.parent.parent.right
                if uncle.color == 'RED':
                    node.parent.color = 'BLACK'
                    uncle.color = 'BLACK'
                    node.parent.parent.color = 'RED'
                    node = node.parent.parent
                else:
                    if node == node.parent.right:
                        node = node.parent
                        self._rotate_left(node)
                    node.parent.color = 'BLACK'
                    node.parent.parent.color = 'RED'
                    self._rotate_right(node.parent.parent)
            else:
                uncle = node.parent.parent.left
                if uncle.color == 'RED':
                    node.parent.color = 'BLACK'
                    uncle.color = 'BLACK'
                    node.parent.parent.color = 'RED'
                    node = node.parent.parent
                else:
                    if node == node.parent.left:
                        node = node.parent
                        self._rotate_right(node)
                    node.parent.color = 'BLACK'
                    node.parent.parent.color = 'RED'
                    self._rotate_left(node.parent.parent)
        self.root.color = 'BLACK'

    def insert(self, key, value):
        """插入键值对"""
        new_node = RedBlackTreeNode(key, value, color='RED', left=self.NIL, right=self.NIL, parent=None)
        parent = None
        current = self.root
        while current != self.NIL:
            parent = current
            if key < current.key:
                current = current.left
            else:
                current = current.right
        new_node.parent = parent
        if parent is None:
            self.root = new_node
        elif key < parent.key:
            parent.left = new_node
        else:
            parent.right = new_node
        self._fix_insert(new_node)

    def search(self, key):
        """根据键查找值"""
        current = self.root
        while current != self.NIL and key != current.key:
            if key < current.key:
                current = current.left
            else:
                current = current.right
        if current == self.NIL:
            return None
        return current.value

    def display(self, node=None, depth=0):
        """显示红黑树"""
        if node is None:
            node = self.root
        if node != self.NIL:
            print(" " * 4 * depth + f"Key: {node.key}, Value: {node.value}, Color: {node.color}")
            self.display(node.left, depth + 1)
            self.display(node.right, depth + 1)

# 示例使用
rbt = RedBlackTree()
rbt.insert("Alice", 25)
rbt.insert("Bob", 30)
rbt.insert("Charlie", 35)
rbt.insert("Diana", 40)
rbt.insert("Eve", 45)

print("Initial Red-Black Tree:")
rbt.display()
Python 实现：基于红黑树的链地址法
python 复制代码 
class RedBlackTreeHashTable:
    def __init__(self, size=10):
        """初始化散列表"""
        self.size = size
        self.table = [RedBlackTree() for _ in range(size)]

    def _hash_function(self, key):
        """哈希函数：将键转换为表索引"""
        return hash(key) % self.size

    def insert(self, key, value):
        """插入键值对"""
        index = self._hash_function(key)
        self.table[index].insert(key, value)

    def search(self, key):
        """根据键查找值"""
        index = self._hash_function(key)
        return self.table[index].search(key)

    def display(self):
        """显示散列表"""
        for i, tree in enumerate(self.table):
            print(f"Index {i}:")
            tree.display()

# 示例使用
rbt_hash_table = RedBlackTreeHashTable(size=7)
rbt_hash_table.insert("Alice", 25)
rbt_hash_table.insert("Bob", 30)
rbt_hash_table.insert("Charlie", 35)
rbt_hash_table.insert("Diana", 40)
rbt_hash_table.insert("Eve", 45)

print("Initial Red-Black Tree Hash Table:")
rbt_hash_table.display()

布隆过滤器

布隆过滤器是一种基于散列表的概率性数据结构，用于测试集合成员资格。它具有空间效率高的特点，但可能会产生误报。

Python 实现：
python 复制代码 
from bitarray import bitarray
import hashlib

class BloomFilter:
    def __init__(self, size, hash_count):
        """初始化布隆过滤器"""
        self.size = size
        self.hash_count = hash_count
        self.bit_array = bitarray(size)
        self.bit_array.setall(0)

    def _hashes(self, item):
        """计算哈希函数值"""
        result = []
        for i in range(self.hash_count):
            hash_result = int(hashlib.md5((str(i) + item).encode()).hexdigest(), 16)
            result.append(hash_result % self.size)
        return result

    def add(self, item):
        """添加元素到布隆过滤器"""
        for index in self._hashes(item):
            self.bit_array[index] = 1

    def __contains__(self, item):
        """检查元素是否在布隆过滤器中"""
        return all(self.bit_array[index] for index in self._hashes(item))

# 示例使用
bloom_filter = BloomFilter(size=1000, hash_count=7)
bloom_filter.add("Alice")
bloom_filter.add("Bob")
bloom_filter.add("Charlie")

print("Checking membership in Bloom Filter:")
print("Alice in filter:", "Alice" in bloom_filter)
print("Bob in filter:", "Bob" in bloom_filter)
print("Charlie in filter:", "Charlie" in bloom_filter)
print("Diana in filter:", "Diana" in bloom_filter)

LRU Cache

LRU Cache（Least Recently Used Cache）是一种常见的缓存策略，它通过淘汰最少使用的数据保持缓存的最新状态。结合散列表和双向链表，可以有效实现该策略。

Python 实现：
python 复制代码 
class ListNode:
    """双向链表节点类"""
    def __init__(self, key=None, value=None):
        self.key = key
        self.value = value
        self.prev = None
        self.next = None

class LRUCache:
    """LRU Cache 实现类"""
    def __init__(self, capacity=10):
        self.capacity = capacity
        self.cache = {}
        self.head = ListNode()
        self.tail = ListNode()
        self.head.next = self.tail
        self.tail.prev = self.head

    def _remove(self, node):
        """移除节点"""
        prev = node.prev
        next = node.next
        prev.next = next
        next.prev = prev

    def _add_to_head(self, node):
        """添加节点到头部"""
        node.next = self.head.next
        node.prev = self.head
        self.head.next.prev = node
        self.head.next = node

    def _move_to_head(self, node):
        """移动节点到头部"""
        self._remove(node)
        self._add_to_head(node)

    def _pop_tail(self):
        """弹出尾部节点"""
        node = self.tail.prev
        self._remove(node)
        return node

    def get(self, key):
        """获取缓存中的值"""
        node = self.cache.get(key)
        if not node:
            return -1
        self._move_to_head(node)
        return node.value

    def put(self, key, value):
        """存储键值对"""
        node = self.cache.get(key)
        if not node:
            new_node = ListNode(key, value)
            self.cache[key] = new_node
            self._add_to_head(new_node)
            if len(self.cache) > self.capacity:
                tail = self._pop_tail()
                del self.cache[tail.key]
        else:
            node.value = value
            self._move_to_head(node)

    def display(self):
        """显示缓存状态"""
        current = self.head.next
        while current != self.tail:
            print(f"({current.key}, {current.value})", end=" ")
            current = current.next
        print()

# 示例使用
lru_cache = LRUCache(capacity=3)
lru_cache.put("Alice", 25)
lru_cache.put("Bob", 30)
lru_cache.put("Charlie", 35)
print("Initial Cache:")
lru_cache.display()

# 更新并添加新的键值对
lru_cache.put("Diana", 40)
lru_cache.get("Alice")
lru_cache.put("Eve", 45)

print("Cache after updates:")
lru_cache.display()

总结

  1. 红黑树链地址法：使用红黑树替代链表，提高冲突处理效率。
  2. 布隆过滤器：高效的集合成员测试结构。
  3. LRU Cache：结合散列表与双向链表，实现高效缓存策略。

通过这些改进与应用案例，我们可以更深入地了解散列表的实际使用场景与优化策略。

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