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Hash Table
A hash table, also known as a hash map, establishes a mapping between keys key and values value, enabling efficient element retrieval. Specifically, when we input a key key into a hash table, we can retrieve the corresponding value value in O(1) time.
As shown in the figure below, given n students, each with two pieces of data: "name" and "student ID". If we want to implement a query function that "inputs a student ID and returns the corresponding name", we can use the hash table shown below.
In addition to hash tables, arrays and linked lists can also implement query functionality. Their efficiency comparison is shown in the following table.
- Adding elements: Simply add elements to the end of the array (linked list), using
O(1)time. - Querying elements: Since the array (linked list) is unordered, all elements need to be traversed, using
O(n)time. - Deleting elements: The element must first be located, then deleted from the array (linked list), using
O(n)time.
Table Comparison of element query efficiency
| Array | Linked List | Hash Table | |
|---|---|---|---|
| Find element | O(n) |
O(n) |
O(1) |
| Add element | O(1) |
O(1) |
O(1) |
| Delete element | O(n) |
O(n) |
O(1) |
As observed, the time complexity for insertion, deletion, search, and modification operations in a hash table is $O(1)$, which is very efficient.
Common Hash Table Operations
Common operations on hash tables include: initialization, query operations, adding key-value pairs, and deleting key-value pairs. Example code is as follows:
=== "Python"
```python title="hash_map.py"
# Initialize hash table
hmap: dict = {}
# Add operation
# Add key-value pair (key, value) to hash table
hmap[12836] = "XiaoHa"
hmap[15937] = "XiaoLuo"
hmap[16750] = "XiaoSuan"
hmap[13276] = "XiaoFa"
hmap[10583] = "XiaoYa"
# Query operation
# Input key into hash table to get value
name: str = hmap[15937]
# Delete operation
# Delete key-value pair (key, value) from hash table
hmap.pop(10583)
```
=== "C++"
```cpp title="hash_map.cpp"
/* Initialize hash table */
unordered_map<int, string> map;
/* Add operation */
// Add key-value pair (key, value) to hash table
map[12836] = "XiaoHa";
map[15937] = "XiaoLuo";
map[16750] = "XiaoSuan";
map[13276] = "XiaoFa";
map[10583] = "XiaoYa";
/* Query operation */
// Input key into hash table to get value
string name = map[15937];
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.erase(10583);
```
=== "Java"
```java title="hash_map.java"
/* Initialize hash table */
Map<Integer, String> map = new HashMap<>();
/* Add operation */
// Add key-value pair (key, value) to hash table
map.put(12836, "XiaoHa");
map.put(15937, "XiaoLuo");
map.put(16750, "XiaoSuan");
map.put(13276, "XiaoFa");
map.put(10583, "XiaoYa");
/* Query operation */
// Input key into hash table to get value
String name = map.get(15937);
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.remove(10583);
```
=== "C#"
```csharp title="hash_map.cs"
/* Initialize hash table */
Dictionary<int, string> map = new() {
/* Add operation */
// Add key-value pair (key, value) to hash table
{ 12836, "XiaoHa" },
{ 15937, "XiaoLuo" },
{ 16750, "XiaoSuan" },
{ 13276, "XiaoFa" },
{ 10583, "XiaoYa" }
};
/* Query operation */
// Input key into hash table to get value
string name = map[15937];
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.Remove(10583);
```
=== "Go"
```go title="hash_map_test.go"
/* Initialize hash table */
hmap := make(map[int]string)
/* Add operation */
// Add key-value pair (key, value) to hash table
hmap[12836] = "XiaoHa"
hmap[15937] = "XiaoLuo"
hmap[16750] = "XiaoSuan"
hmap[13276] = "XiaoFa"
hmap[10583] = "XiaoYa"
/* Query operation */
// Input key into hash table to get value
name := hmap[15937]
/* Delete operation */
// Delete key-value pair (key, value) from hash table
delete(hmap, 10583)
```
=== "Swift"
```swift title="hash_map.swift"
/* Initialize hash table */
var map: [Int: String] = [:]
/* Add operation */
// Add key-value pair (key, value) to hash table
map[12836] = "XiaoHa"
map[15937] = "XiaoLuo"
map[16750] = "XiaoSuan"
map[13276] = "XiaoFa"
map[10583] = "XiaoYa"
/* Query operation */
// Input key into hash table to get value
let name = map[15937]!
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.removeValue(forKey: 10583)
```
=== "JS"
```javascript title="hash_map.js"
/* Initialize hash table */
const map = new Map();
/* Add operation */
// Add key-value pair (key, value) to hash table
map.set(12836, 'XiaoHa');
map.set(15937, 'XiaoLuo');
map.set(16750, 'XiaoSuan');
map.set(13276, 'XiaoFa');
map.set(10583, 'XiaoYa');
/* Query operation */
// Input key into hash table to get value
let name = map.get(15937);
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.delete(10583);
```
=== "TS"
```typescript title="hash_map.ts"
/* Initialize hash table */
const map = new Map<number, string>();
/* Add operation */
// Add key-value pair (key, value) to hash table
map.set(12836, 'XiaoHa');
map.set(15937, 'XiaoLuo');
map.set(16750, 'XiaoSuan');
map.set(13276, 'XiaoFa');
map.set(10583, 'XiaoYa');
console.info('\nAfter adding, hash table is\nKey -> Value');
console.info(map);
/* Query operation */
// Input key into hash table to get value
let name = map.get(15937);
console.info('\nInput student ID 15937, queried name ' + name);
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.delete(10583);
console.info('\nAfter deleting 10583, hash table is\nKey -> Value');
console.info(map);
```
=== "Dart"
```dart title="hash_map.dart"
/* Initialize hash table */
Map<int, String> map = {};
/* Add operation */
// Add key-value pair (key, value) to hash table
map[12836] = "XiaoHa";
map[15937] = "XiaoLuo";
map[16750] = "XiaoSuan";
map[13276] = "XiaoFa";
map[10583] = "XiaoYa";
/* Query operation */
// Input key into hash table to get value
String name = map[15937];
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.remove(10583);
```
=== "Rust"
```rust title="hash_map.rs"
use std::collections::HashMap;
/* Initialize hash table */
let mut map: HashMap<i32, String> = HashMap::new();
/* Add operation */
// Add key-value pair (key, value) to hash table
map.insert(12836, "XiaoHa".to_string());
map.insert(15937, "XiaoLuo".to_string());
map.insert(16750, "XiaoSuan".to_string());
map.insert(13279, "XiaoFa".to_string());
map.insert(10583, "XiaoYa".to_string());
/* Query operation */
// Input key into hash table to get value
let _name: Option<&String> = map.get(&15937);
/* Delete operation */
// Delete key-value pair (key, value) from hash table
let _removed_value: Option<String> = map.remove(&10583);
```
=== "C"
```c title="hash_map.c"
// C does not provide a built-in hash table
```
=== "Kotlin"
```kotlin title="hash_map.kt"
/* Initialize hash table */
val map = HashMap<Int,String>()
/* Add operation */
// Add key-value pair (key, value) to hash table
map[12836] = "XiaoHa"
map[15937] = "XiaoLuo"
map[16750] = "XiaoSuan"
map[13276] = "XiaoFa"
map[10583] = "XiaoYa"
/* Query operation */
// Input key into hash table to get value
val name = map[15937]
/* Delete operation */
// Delete key-value pair (key, value) from hash table
map.remove(10583)
```
=== "Ruby"
```ruby title="hash_map.rb"
# Initialize hash table
hmap = {}
# Add operation
# Add key-value pair (key, value) to hash table
hmap[12836] = "XiaoHa"
hmap[15937] = "XiaoLuo"
hmap[16750] = "XiaoSuan"
hmap[13276] = "XiaoFa"
hmap[10583] = "XiaoYa"
# Query operation
# Input key into hash table to get value
name = hmap[15937]
# Delete operation
# Delete key-value pair (key, value) from hash table
hmap.delete(10583)
```
??? pythontutor "Visualized Execution"
https://pythontutor.com/render.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%9F%A5%E8%AF%A2%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%90%91%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E8%BE%93%E5%85%A5%E9%94%AE%20key%20%EF%BC%8C%E5%BE%97%E5%88%B0%E5%80%BC%20value%0A%20%20%20%20name%20%3D%20hmap%5B15937%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E5%88%A0%E9%99%A4%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap.pop%2810583%29&cumulative=false&curInstr=2&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false
There are three common ways to traverse a hash table: traversing key-value pairs, traversing keys, and traversing values. Example code is as follows:
=== "Python"
```python title="hash_map.py"
# Traverse hash table
# Traverse key-value pairs key->value
for key, value in hmap.items():
print(key, "->", value)
# Traverse keys only
for key in hmap.keys():
print(key)
# Traverse values only
for value in hmap.values():
print(value)
```
=== "C++"
```cpp title="hash_map.cpp"
/* Traverse hash table */
// Traverse key-value pairs key->value
for (auto kv: map) {
cout << kv.first << " -> " << kv.second << endl;
}
// Traverse using iterator key->value
for (auto iter = map.begin(); iter != map.end(); iter++) {
cout << iter->first << "->" << iter->second << endl;
}
```
=== "Java"
```java title="hash_map.java"
/* Traverse hash table */
// Traverse key-value pairs key->value
for (Map.Entry<Integer, String> kv: map.entrySet()) {
System.out.println(kv.getKey() + " -> " + kv.getValue());
}
// Traverse keys only
for (int key: map.keySet()) {
System.out.println(key);
}
// Traverse values only
for (String val: map.values()) {
System.out.println(val);
}
```
=== "C#"
```csharp title="hash_map.cs"
/* Traverse hash table */
// Traverse key-value pairs Key->Value
foreach (var kv in map) {
Console.WriteLine(kv.Key + " -> " + kv.Value);
}
// Traverse keys only
foreach (int key in map.Keys) {
Console.WriteLine(key);
}
// Traverse values only
foreach (string val in map.Values) {
Console.WriteLine(val);
}
```
=== "Go"
```go title="hash_map_test.go"
/* Traverse hash table */
// Traverse key-value pairs key->value
for key, value := range hmap {
fmt.Println(key, "->", value)
}
// Traverse keys only
for key := range hmap {
fmt.Println(key)
}
// Traverse values only
for _, value := range hmap {
fmt.Println(value)
}
```
=== "Swift"
```swift title="hash_map.swift"
/* Traverse hash table */
// Traverse key-value pairs Key->Value
for (key, value) in map {
print("\(key) -> \(value)")
}
// Traverse keys only
for key in map.keys {
print(key)
}
// Traverse values only
for value in map.values {
print(value)
}
```
=== "JS"
```javascript title="hash_map.js"
/* Traverse hash table */
console.info('\nTraverse key-value pairs Key->Value');
for (const [k, v] of map.entries()) {
console.info(k + ' -> ' + v);
}
console.info('\nTraverse keys only Key');
for (const k of map.keys()) {
console.info(k);
}
console.info('\nTraverse values only Value');
for (const v of map.values()) {
console.info(v);
}
```
=== "TS"
```typescript title="hash_map.ts"
/* Traverse hash table */
console.info('\nTraverse key-value pairs Key->Value');
for (const [k, v] of map.entries()) {
console.info(k + ' -> ' + v);
}
console.info('\nTraverse keys only Key');
for (const k of map.keys()) {
console.info(k);
}
console.info('\nTraverse values only Value');
for (const v of map.values()) {
console.info(v);
}
```
=== "Dart"
```dart title="hash_map.dart"
/* Traverse hash table */
// Traverse key-value pairs Key->Value
map.forEach((key, value) {
print('$key -> $value');
});
// Traverse keys only
map.keys.forEach((key) {
print(key);
});
// Traverse values only
map.values.forEach((value) {
print(value);
});
```
=== "Rust"
```rust title="hash_map.rs"
/* Traverse hash table */
// Traverse key-value pairs Key->Value
for (key, value) in &map {
println!("{key} -> {value}");
}
// Traverse keys only
for key in map.keys() {
println!("{key}");
}
// Traverse values only
for value in map.values() {
println!("{value}");
}
```
=== "C"
```c title="hash_map.c"
// C does not provide a built-in hash table
```
=== "Kotlin"
```kotlin title="hash_map.kt"
/* Traverse hash table */
// Traverse key-value pairs key->value
for ((key, value) in map) {
println("$key -> $value")
}
// Traverse keys only
for (key in map.keys) {
println(key)
}
// Traverse values only
for (_val in map.values) {
println(_val)
}
```
=== "Ruby"
```ruby title="hash_map.rb"
# Traverse hash table
# Traverse key-value pairs key->value
hmap.entries.each { |key, value| puts "#{key} -> #{value}" }
# Traverse keys only
hmap.keys.each { |key| puts key }
# Traverse values only
hmap.values.each { |val| puts val }
```
??? pythontutor "Visualized Execution"
https://pythontutor.com/render.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E9%94%AE%E5%80%BC%E5%AF%B9%20key-%3Evalue%0A%20%20%20%20for%20key,%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key,%20%22-%3E%22,%20value%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E9%94%AE%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E5%80%BC%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&cumulative=false&curInstr=8&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false
Simple Hash Table Implementation
Let's first consider the simplest case: implementing a hash table using only an array. In a hash table, each empty position in the array is called a bucket, and each bucket can store a key-value pair. Therefore, the query operation is to find the bucket corresponding to key and retrieve the value from the bucket.
So how do we locate the corresponding bucket based on key? This is achieved through a hash function. The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all keys, and the output space is all buckets (array indices). In other words, given a key, we can use the hash function to obtain the storage location of the key-value pair corresponding to that key in the array.
When inputting a key, the hash function's calculation process consists of the following two steps:
- Calculate the hash value through a hash algorithm
hash(). - Take the modulo of the hash value by the number of buckets (array length)
capacityto obtain the bucket (array index)indexcorresponding to thatkey.
index = hash(key) % capacity
Subsequently, we can use index to access the corresponding bucket in the hash table and retrieve the value.
Assuming the array length is capacity = 100 and the hash algorithm is hash(key) = key, the hash function becomes key % 100. The figure below shows the working principle of the hash function using key as student ID and value as name.
The following code implements a simple hash table. Here, we encapsulate key and value into a class Pair to represent a key-value pair.
[file]{array_hash_map}-[class]{array_hash_map}-[func]{}
Hash Collision and Resizing
Fundamentally, the role of a hash function is to map the input space consisting of all keys to the output space consisting of all array indices, and the input space is often much larger than the output space. Therefore, theoretically there must be cases where "multiple inputs correspond to the same output".
For the hash function in the above example, when the input keys have the same last two digits, the hash function produces the same output. For example, when querying two students with IDs 12836 and 20336, we get:
12836 % 100 = 36
20336 % 100 = 36
As shown in the figure below, two student IDs point to the same name, which is obviously incorrect. We call this situation where multiple inputs correspond to the same output a hash collision.
It's easy to see that the larger the hash table capacity n, the lower the probability that multiple keys will be assigned to the same bucket, and the fewer collisions. Therefore, we can reduce hash collisions by expanding the hash table.
As shown in the figure below, before expansion, the key-value pairs (136, A) and (236, D) collided, but after expansion, the collision disappears.
Similar to array expansion, hash table expansion requires migrating all key-value pairs from the original hash table to the new hash table, which is very time-consuming. Moreover, since the hash table capacity capacity changes, we need to recalculate the storage locations of all key-value pairs through the hash function, further increasing the computational overhead of the expansion process. For this reason, programming languages typically reserve a sufficiently large hash table capacity to prevent frequent expansion.
The load factor is an important concept for hash tables. It is defined as the number of elements in the hash table divided by the number of buckets, and is used to measure the severity of hash collisions. It is also commonly used as a trigger condition for hash table expansion. For example, in Java, when the load factor exceeds 0.75, the system will expand the hash table to 2 times its original size.