一、定义
散列表（Hash Table，也叫哈希表），是通过把键值映射成整数来作为数组的索引，并进行访问记录的一种数据结构。
二、基本思想
实现散列表的关键是散列算法，即如何将任意类型的键值转化为数组的索引。通常，使用散列表进行查找分为两步：
- 利用散列函数将被查找的键转化为数组的一个索引。
- 访问索引以得到键对应的值。
三、散列函数
理想情况下，散列函数能将每个不同的键值转换成唯一的索引。但事实上，因为要考虑空间（内存）的使用，会出现碰撞冲突，即两个不同的键值映射到相同的索引，常见的解决碰撞冲突的方法有：拉链法和线性探测法。
优秀的散列函数满足如下条件：
- 一致性——等价的键必然产生相等的散列值
- 高效性——计算简便
- 均匀性——均匀地散列所有的键
对于大小为M的数组，理想的散列函数对任意键处理后，其值分布在0～M-1之间的概率应该相等。针对不同类型的键值，常见的散列函数有如下几种：
1、除留余数法
步骤如下：
①选择大小为M的数组（M应当为素数）；
②对于任意正整数键值k，取k%M作为散列值。
为什么M必须用素数？
因为素数在数学上有很多特殊的性质，使用素数可以使散列后的值分布更均匀。例如，M=10k，N为正整数，N%M后的值为N的后k位。
四、碰撞冲突
4.1 拉链法
基本思想：
将大小为M的数组中的每个元素指向一条链表，链表的每个结点存储了散列值为该元素的索引的键值对。则N个键最终保存在M条链表中，链的平均长度为：N/M。
4-1 拉链法
拉链法实现源码：
public class SeparateChainingHashST<Key, Value> {
private static final int INIT_CAPACITY = 4;
private int n; // number of key-value pairs
private int m; // hash table size
private SequentialSearchST<Key, Value>[] st; // array of linked-list symbol tables
/**
* Initializes an empty symbol table.
*/
public SeparateChainingHashST() {
this(INIT_CAPACITY);
}
/**
* Initializes an empty symbol table with {@code m} chains.
* @param m the initial number of chains
*/
public SeparateChainingHashST(int m) {
this.m = m;
st = (SequentialSearchST<Key, Value>[]) new SequentialSearchST[m];
for (int i = 0; i < m; i++)
st[i] = new SequentialSearchST<Key, Value>();
}
// resize the hash table to have the given number of chains,
// rehashing all of the keys
private void resize(int chains) {
SeparateChainingHashST<Key, Value> temp = new SeparateChainingHashST<Key, Value>(chains);
for (int i = 0; i < m; i++) {
for (Key key : st[i].keys()) {
temp.put(key, st[i].get(key));
}
}
this.m = temp.m;
this.n = temp.n;
this.st = temp.st;
}
// hash value between 0 and m-1
private int hash(Key key) {
return (key.hashCode() & 0x7fffffff) % m;
}
/**
* Returns the number of key-value pairs in this symbol table.
*
* @return the number of key-value pairs in this symbol table
*/
public int size() {
return n;
}
/**
* Returns true if this symbol table is empty.
*
* @return {@code true} if this symbol table is empty;
* {@code false} otherwise
*/
public boolean isEmpty() {
return size() == 0;
}
/**
* Returns true if this symbol table contains the specified key.
*
* @param key the key
* @return {@code true} if this symbol table contains {@code key};
* {@code false} otherwise
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public boolean contains(Key key) {
if (key == null) throw new IllegalArgumentException("argument to contains() is null");
return get(key) != null;
}
/**
* Returns the value associated with the specified key in this symbol table.
*
* @param key the key
* @return the value associated with {@code key} in the symbol table;
* {@code null} if no such value
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public Value get(Key key) {
if (key == null) throw new IllegalArgumentException("argument to get() is null");
int i = hash(key);
return st[i].get(key);
}
/**
* Inserts the specified key-value pair into the symbol table, overwriting the old
* value with the new value if the symbol table already contains the specified key.
* Deletes the specified key (and its associated value) from this symbol table
* if the specified value is {@code null}.
*
* @param key the key
* @param val the value
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public void put(Key key, Value val) {
if (key == null) throw new IllegalArgumentException("first argument to put() is null");
if (val == null) {
delete(key);
return;
}
// double table size if average length of list >= 10
if (n >= 10*m) resize(2*m);
int i = hash(key);
if (!st[i].contains(key)) n++;
st[i].put(key, val);
}
/**
* Removes the specified key and its associated value from this symbol table
* (if the key is in this symbol table).
*
* @param key the key
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public void delete(Key key) {
if (key == null) throw new IllegalArgumentException("argument to delete() is null");
int i = hash(key);
if (st[i].contains(key)) n--;
st[i].delete(key);
// halve table size if average length of list <= 2
if (m > INIT_CAPACITY && n <= 2*m) resize(m/2);
}
// return keys in symbol table as an Iterable
public Iterable<Key> keys() {
Queue<Key> queue = new Queue<Key>();
for (int i = 0; i < m; i++) {
for (Key key : st[i].keys())
queue.enqueue(key);
}
return queue;
}
}
4.2 线性探测法
基本思想：
用大小为M的数组保存N个键值对，其中M>N，即内部索引数组的大小总是大于已经插入的键值对。基于这种策略的所有方法被统称为开放地址散列表。
具体步骤：
- 用散列函数查找键在数组中的索引；
- 如果其中的键和被查找的键相同，则返回键值；如果不同，则继续向后查找（索引+1，遇末尾则折回开头），直到找到该键或遇到空位置。
4-2-1 线性探测法
线性探测法实现源码：
public class LinearProbingHashST<Key, Value> {
private static final int INIT_CAPACITY = 4;
private int n; // number of key-value pairs in the symbol table
private int m; // size of linear probing table
private Key[] keys; // the keys
private Value[] vals; // the values
/**
* Initializes an empty symbol table.
*/
public LinearProbingHashST() {
this(INIT_CAPACITY);
}
/**
* Initializes an empty symbol table with the specified initial capacity.
*
* @param capacity the initial capacity
*/
public LinearProbingHashST(int capacity) {
m = capacity;
n = 0;
keys = (Key[]) new Object[m];
vals = (Value[]) new Object[m];
}
/**
* Returns the number of key-value pairs in this symbol table.
*
* @return the number of key-value pairs in this symbol table
*/
public int size() {
return n;
}
/**
* Returns true if this symbol table is empty.
*
* @return {@code true} if this symbol table is empty;
* {@code false} otherwise
*/
public boolean isEmpty() {
return size() == 0;
}
/**
* Returns true if this symbol table contains the specified key.
*
* @param key the key
* @return {@code true} if this symbol table contains {@code key};
* {@code false} otherwise
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public boolean contains(Key key) {
if (key == null) throw new IllegalArgumentException("argument to contains() is null");
return get(key) != null;
}
// hash function for keys - returns value between 0 and M-1
private int hash(Key key) {
return (key.hashCode() & 0x7fffffff) % m;
}
// resizes the hash table to the given capacity by re-hashing all of the keys
private void resize(int capacity) {
LinearProbingHashST<Key, Value> temp = new LinearProbingHashST<Key, Value>(capacity);
for (int i = 0; i < m; i++) {
if (keys[i] != null) {
temp.put(keys[i], vals[i]);
}
}
keys = temp.keys;
vals = temp.vals;
m = temp.m;
}
/**
* Inserts the specified key-value pair into the symbol table, overwriting the old
* value with the new value if the symbol table already contains the specified key.
* Deletes the specified key (and its associated value) from this symbol table
* if the specified value is {@code null}.
*
* @param key the key
* @param val the value
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public void put(Key key, Value val) {
if (key == null) throw new IllegalArgumentException("first argument to put() is null");
if (val == null) {
delete(key);
return;
}
// double table size if 50% full
if (n >= m/2) resize(2*m);
int i;
for (i = hash(key); keys[i] != null; i = (i + 1) % m) {
if (keys[i].equals(key)) {
vals[i] = val;
return;
}
}
keys[i] = key;
vals[i] = val;
n++;
}
/**
* Returns the value associated with the specified key.
* @param key the key
* @return the value associated with {@code key};
* {@code null} if no such value
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public Value get(Key key) {
if (key == null) throw new IllegalArgumentException("argument to get() is null");
for (int i = hash(key); keys[i] != null; i = (i + 1) % m)
if (keys[i].equals(key))
return vals[i];
return null;
}
/**
* Removes the specified key and its associated value from this symbol table
* (if the key is in this symbol table).
*
* @param key the key
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public void delete(Key key) {
if (key == null) throw new IllegalArgumentException("argument to delete() is null");
if (!contains(key)) return;
// find position i of key
int i = hash(key);
while (!key.equals(keys[i])) {
i = (i + 1) % m;
}
// delete key and associated value
keys[i] = null;
vals[i] = null;
// rehash all keys in same cluster
i = (i + 1) % m;
while (keys[i] != null) {
// delete keys[i] an vals[i] and reinsert
Key keyToRehash = keys[i];
Value valToRehash = vals[i];
keys[i] = null;
vals[i] = null;
n--;
put(keyToRehash, valToRehash);
i = (i + 1) % m;
}
n--;
// halves size of array if it's 12.5% full or less
if (n > 0 && n <= m/8) resize(m/2);
assert check();
}
/**
* Returns all keys in this symbol table as an {@code Iterable}.
* To iterate over all of the keys in the symbol table named {@code st},
* use the foreach notation: {@code for (Key key : st.keys())}.
*
* @return all keys in this symbol table
*/
public Iterable<Key> keys() {
Queue<Key> queue = new Queue<Key>();
for (int i = 0; i < m; i++)
if (keys[i] != null) queue.enqueue(keys[i]);
return queue;
}
// integrity check - don't check after each put() because
// integrity not maintained during a delete()
private boolean check() {
// check that hash table is at most 50% full
if (m < 2*n) {
System.err.println("Hash table size m = " + m + "; array size n = " + n);
return false;
}
// check that each key in table can be found by get()
for (int i = 0; i < m; i++) {
if (keys[i] == null) continue;
else if (get(keys[i]) != vals[i]) {
System.err.println("get[" + keys[i] + "] = " + get(keys[i]) + "; vals[i] = " + vals[i]);
return false;
}
}
return true;
}
}
性能分析：
开放地址类的散列表的性能依赖于α=N/M的值，α称为散列表的使用率（0≤α<1）。
线性探测的平均成本取决于元素在插入符号表后形成的键簇的大小。所谓键簇，就是一条连续的元素组大小，键簇越小，性能越好，如下图：
根据数学分析，在一张大小为M并含有N=αM个键的基于线性探测的散列表中：
命中查找所需的探测次数为：
未命中查找所需的探测次数为：
当α约为0.5时，查找命中所需探测次数约为1.5次，查找未命中所需探测次数约为2.5次。
也就是说当散列表快满的时候，查找所需的探测次数是巨大的（α趋近于1），但当α<0.5时，查找所需的探测次数只在1.5～2.5之间。
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