C++哈希表：开散列与闭散列详解
C++ —— 哈希详解：开散列与闭散列
哈希表是一种高效的数据结构，用于存储键值对，支持快速查找、插入和删除。其核心是哈希函数，将键映射到索引位置。当多个键映射到同一位置时（冲突），需要处理策略。常见方法有闭散列（开放地址法）和开散列（链地址法）。下面我将详细解释这两种方法，包括工作原理、优缺点，并提供C++实现示例。
1. 哈希表基础
哈希表由数组和哈希函数组成。哈希函数 $h(k)$ 将键 $k$ 映射到数组索引，理想情况下均匀分布。冲突发生时，需解决：
- 闭散列：在数组内部通过探测序列寻找空位。
- 开散列：使用外部数据结构（如链表）存储冲突元素。
2. 闭散列（开放地址法）
闭散列在数组内部处理冲突，通过探测序列寻找下一个空槽。常见探测方法：
- 线性探测：冲突后，依次检查下一个位置 $h(k, i) = (h(k) + i) mod m$，其中 $i$ 是探测次数，$m$ 是数组大小。
- 二次探测：避免聚集，使用 $h(k, i) = (h(k) + c_1 cdot i + c_2 cdot i^2) mod m$，$c_1$ 和 $c_2$ 是常数。
- 双重哈希：使用第二个哈希函数 $h_2(k)$，序列为 $h(k, i) = (h(k) + i cdot h_2(k)) mod m$。
优点：
- 内存局部性好，访问速度快。
- 无需额外数据结构，节省空间。
缺点：
- 容易产生聚集现象，降低性能。
- 删除操作复杂，需标记删除位。
- 负载因子高时（如 $> 0.7$），性能下降快。
C++实现示例： 以下是一个使用线性探测的哈希表实现：
#include
#include
using namespace std;
class HashTable {
private:
vector table;
vector isOccupied;
int size;
int capacity;
int hash(int key) {
return key % capacity;
}
int probe(int key, int i) {
return (hash(key) + i) % capacity;
}
public:
HashTable(int cap) : capacity(cap), size(0) {
table.resize(cap, -1); // -1表示空位
isOccupied.resize(cap, false);
}
void insert(int key) {
if (size >= capacity) {
cout << "Table full!" << endl;
return;
}
int index = hash(key);
int i = 0;
while (isOccupied[index] && table[index] != key) {
i++;
index = probe(key, i);
}
if (!isOccupied[index]) {
table[index] = key;
isOccupied[index] = true;
size++;
}
}
bool search(int key) {
int index = hash(key);
int i = 0;
while (isOccupied[index]) {
if (table[index] == key) {
return true;
}
i++;
index = probe(key, i);
}
return false;
}
void remove(int key) {
int index = hash(key);
int i = 0;
while (isOccupied[index]) {
if (table[index] == key) {
isOccupied[index] = false; // 标记为删除
size--;
return;
}
i++;
index = probe(key, i);
}
}
};
int main() {
HashTable ht(10);
ht.insert(5);
ht.insert(15);
cout << "Search 15: " << ht.search(15) << endl; // 输出1 (true)
ht.remove(15);
cout << "Search 15: " << ht.search(15) << endl; // 输出0 (false)
return 0;
}
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3. 开散列（链地址法）
开散列使用链表处理冲突：每个数组槽指向一个链表，冲突元素添加到链表中。哈希函数 $h(k)$ 映射到索引，然后在该索引的链表中操作。
优点：
- 处理冲突简单，性能稳定。
- 删除操作容易，直接移除链表节点。
- 高负载因子下仍高效。
缺点：
- 内存开销大，需额外链表空间。
- 链表遍历可能增加访问时间。
C++实现示例： 以下是一个使用链地址法的哈希表实现：
#include
#include
#include
using namespace std;
class HashTable {
private:
vector> table;
int capacity;
int hash(int key) {
return key % capacity;
}
public:
HashTable(int cap) : capacity(cap) {
table.resize(cap);
}
void insert(int key) {
int index = hash(key);
for (auto it = table[index].begin(); it != table[index].end(); ++it) {
if (*it == key) {
return; // 键已存在
}
}
table[index].push_back(key);
}
bool search(int key) {
int index = hash(key);
for (auto it = table[index].begin(); it != table[index].end(); ++it) {
if (*it == key) {
return true;
}
}
return false;
}
void remove(int key) {
int index = hash(key);
for (auto it = table[index].begin(); it != table[index].end(); ++it) {
if (*it == key) {
table[index].erase(it);
return;
}
}
}
};
int main() {
HashTable ht(10);
ht.insert(5);
ht.insert(15);
cout << "Search 15: " << ht.search(15) << endl; // 输出1 (true)
ht.remove(15);
cout << "Search 15: " << ht.search(15) << endl; // 输出0 (false)
return 0;
}
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4. 闭散列 vs. 开散列比较
- 性能：开散列在高负载因子下更稳定；闭散列在低负载因子时更快。
- 内存：闭散列更节省空间；开散列有额外链表开销。
- 适用场景：闭散列适合内存受限环境；开散列适合高冲突率场景。
- C++标准库：
std::unordered_map通常使用开散列实现。
5. 总结
在C++中，哈希表实现需根据需求选择闭散列或开散列。闭散列简单但易聚集，开散列稳定但占用内存。实践中，C++的 std::unordered_map 基于开散列，提供高效操作。以上代码示例展示了核心逻辑，实际应用中需处理扩容、哈希函数优化等。
