Hash Table

Hash Table

A hash table allows you to store and retrieve objects by a "key".

A hash table is used to implement structures, such as a dictionary, a map, and an associative array. These structures can be implemented by a tree or a plain array, but it is efficient to use a hash table.

This should explain why Swift's built-in Dictionary type requires that keys conform to the Hashable protocol: internally it uses a hash table, like the one you will learn about here.

How it works

A hash table is nothing more than an array. Initially, this array is empty. When you put a value into the hash table under a certain key, it uses that key to calculate an index in the array. Here is an example:

hashTable["firstName"] = "Steve"

	The hashTable array:
	+--------------+
	| 0:           |
	+--------------+
	| 1:           |
	+--------------+
	| 2:           |
	+--------------+
	| 3: firstName |---> Steve
	+--------------+
	| 4:           |
	+--------------+

In this example, the key "firstName" maps to array index 3.

Adding a value under a different key puts it at another array index:

hashTable["hobbies"] = "Programming Swift"

	The hashTable array:
	+--------------+
	| 0:           |
	+--------------+
	| 1: hobbies   |---> Programming Swift
	+--------------+
	| 2:           |
	+--------------+
	| 3: firstName |---> Steve
	+--------------+
	| 4:           |
	+--------------+

The trick is how the hash table calculates those array indices. That is where the hashing comes in. When you write the following statement,

hashTable["firstName"] = "Steve"

the hash table takes the key "firstName" and asks it for its hashValue property. Hence, keys must be Hashable.

When you write "firstName".hashValue, it returns a big integer: -4799450059917011053. Likewise, "hobbies".hashValue has the hash value 4799450060928805186. (The values you see may vary.)

These numbers are big to be used as indices into our array, and one of them is even negative! A common way to make these big numbers suitable is to first make the hash positive and then take the modulo with the array size.

Our array has size 5, so the index for the "firstName" key becomes abs(-4799450059917011053) % 5 = 3. You can calculate that the array index for "hobbies" is 1.

Using hashes in this manner is what makes the dictionary efficient: to find an element in the hash table, you must hash the key to get an array index and then look up the element in the underlying array. All these operations take a constant amount of time, so inserting, retrieving, and removing are all O(1).

Note: It is difficult to predict where in the array your objects end up. Hence, dictionaries do not guarantee any particular order of the elements in the hash table.

Avoiding collisions

There is one problem: because we take the modulo of the hash value with the size of the array, it can happen that two or more keys get assigned the same array index. This is called a collision.

One way to avoid collisions is to have a large array which reduces the likelihood of two keys mapping to the same index. Another trick is to use a prime number for the array size. However, collisions are bound to occur, so you need to find a way to handle them.

Because our table is small, it is easy to show a collision. For example, the array index for the key "lastName" is also 3, but we do not want to overwrite the value that is already at this array index.

A common way to handle collisions is to use chaining. The array looks as follows:

	buckets:
	+-----+
	|  0  |
	+-----+     +----------------------------+
	|  1  |---> | hobbies: Programming Swift |
	+-----+     +----------------------------+
	|  2  |
	+-----+     +------------------+     +----------------+
	|  3  |---> | firstName: Steve |---> | lastName: Jobs |
	+-----+     +------------------+     +----------------+
	|  4  |
	+-----+

With chaining, keys and their values are not stored directly in the array. Instead, each array element is a list of zero or more key/value pairs. The array elements are usually called the buckets and the lists are called the chains. Here we have 5 buckets, and two of these buckets have chains. The other three buckets are empty.

If we write the following statement to retrieve an item from the hash table,

let x = hashTable["lastName"]

it first hashes the key "lastName" to calculate the array index, which is 3. Since bucket 3 has a chain, we step through the list to find the value with the key "lastName". This is done by comparing the keys using a string comparison. The hash table checks that the key belongs to the last item in the chain and returns the corresponding value, "Jobs".

Common ways to implement this chaining mechanism are to use a linked list or another array. Since the order of the items in the chain does not matter, you can think of it as a set instead of a list. (Now you can also imagine where the term "bucket" comes from; we just dump all the objects together into the bucket.)

Chains should not become long because looking up items in the hash table would become a slow process. Ideally, we would have no chains at all, but in practice it is impossible to avoid collisions. You can improve the odds by giving the hash table enough buckets using high-quality hash functions.

Note: An alternative to chaining is "open addressing". The idea is this: if an array index is already taken, we put the element in the next unused bucket. This approach has its own upsides and downsides.

The code

Let's look at a basic implementation of a hash table in Swift. We will build it up step-by-step.

public struct HashTable<Key: Hashable, Value> {
  private typealias Element = (key: Key, value: Value)
  private typealias Bucket = [Element]
  private var buckets: [Bucket]

  private(set) public var count = 0
  
  public var isEmpty: Bool { return count == 0 }

  public init(capacity: Int) {
    assert(capacity > 0)
    buckets = Array<Bucket>(repeatElement([], count: capacity))
  }

The HashTable is a generic container, and the two generic types are named Key (which must be Hashable) and Value. We also define two other types: Element is a key/value pair for using in a chain, and Bucket is an array of such Elements.

The main array is named buckets. It has a fixed size, the so-called capacity, provided by the init(capacity) method. We are also keeping track of how many items have been added to the hash table using the count variable.

An example of how to create a new hash table object:

var hashTable = HashTable<String, String>(capacity: 5)

The hash table does not do anything yet, so let's add the remaining functionality. First, add a helper method that calculates the array index for a given key:

  private func index(forKey key: Key) -> Int {
    return abs(key.hashValue % buckets.count)
  }

This performs the calculation you saw earlier: it takes the absolute value of the key's hashValue modulo the size of the buckets array. We have put this in a function of its own because it gets used in a few different places.

There are four common things you will do with a hash table or dictionary:

• insert a new element
• look up an element
• update an existing element
• remove an element

The syntax for these is:

hashTable["firstName"] = "Steve"   // insert
let x = hashTable["firstName"]     // lookup
hashTable["firstName"] = "Tim"     // update
hashTable["firstName"] = nil       // delete

We can do all these things with a subscript function:

  public subscript(key: Key) -> Value? {
    get {
      return value(forKey: key)
    }
    set {
      if let value = newValue {
        updateValue(value, forKey: key)
      } else {
        removeValue(forKey: key)
      }
    }
  }

This calls three helper functions to do the actual work. Let's take a look at value(forKey:)which retrieves an object from the hash table.

  public func value(forKey key: Key) -> Value? {
    let index = self.index(forKey: key)
    for element in buckets[index] {
      if element.key == key {
        return element.value
      }
    }
    return nil  // key not in hash table
  }

First it calls index(forKey:) to convert the key into an array index. That gives us the bucket number, but this bucket may be used by more than one key if there were collisions. The value(forKey:) loops through the chain from that bucket and compares the keys one-by-one. If found, it returns the corresponding value, otherwise it returns nil.

The code to insert a new element or update an existing element lives in updateValue(_:forKey:). This is more complicated:

  public mutating func updateValue(_ value: Value, forKey key: Key) -> Value? {
    let index = self.index(forKey: key)
    
    // Do we already have this key in the bucket?
    for (i, element) in buckets[index].enumerated() {
      if element.key == key {
        let oldValue = element.value
        buckets[index][i].value = value
        return oldValue
      }
    }
    
    // This key isn't in the bucket yet; add it to the chain.
    buckets[index].append((key: key, value: value))
    count += 1
    return nil
  }

Again, the first step is to convert the key into an array index to find the bucket. Then we loop through the chain for that bucket. If we find the key in the chain, we must update it with the new value. If the key is not in the chain, we insert the new key/value pair to the end of the chain.

As you can see, it is important to keep the chains short (by making the hash table large enough). Otherwise, you spend excessive time in these for...in loops and the performance of the hash table will no longer be O(1) but more like O(n).

Removing is similar in that again it loops through the chain:

  public mutating func removeValue(forKey key: Key) -> Value? {
    let index = self.index(forKey: key)

    // Find the element in the bucket's chain and remove it.
    for (i, element) in buckets[index].enumerated() {
      if element.key == key {
        buckets[index].remove(at: i)
        count -= 1
        return element.value
      }
    }
    return nil  // key not in hash table
  }

These are the basic functions of the hash table. They all work the same way: convert the key into an array index using its hash value, find the bucket, then loop through that bucket's chain and perform the desired operation.

Try this stuff out in a playground. It should work just like a standard Swift Dictionary.

Resizing the hash table

This version of HashTable always uses an array of a fixed size or capacity. If you have many items to store in the hash table, for the capacity, choose a prime number greater than the maximum number of items.

The load factor of a hash table is the percentage of the capacity that is currently used. If there are 3 items in a hash table with 5 buckets, then the load factor is 3/5 = 60%.

If the hash table is small, and the chains are long, the load factor can become greater than 1, that is not a good idea.

If the load factor becomes high, greater than 75%, you can resize the hash table. Adding the code for this condition is left as an exercise for the reader. Keep in mind that making the buckets array larger will change the array indices that the keys map to! This requires you to insert all the elements again after resizing the array.

Where to go from here?

HashTable is quite basic. It might be efficient to integrate it with the Swift standard library by making it a SequenceType.

Written for Swift Algorithm Club by Matthijs Hollemans