Cuckoo Hashing – Worst case O(1) Lookup!

Background : 

There are three basic operations that must be supported by a hash table (or a dictionary):  

• Lookup(key): return true if key is there on the table, else false
• Insert(key): add the item ‘key’ to the table if not already present
• Delete(key): removes ‘key’ from the table

Collisions are very likely even if we have a big table to store keys. Using the results from the birthday paradox: with only 23 persons, the probability that two people share the same birth date is 50%! There are 3 general strategies towards resolving hash collisions: 

• Closed addressing or Chaining: store colliding elements in an auxiliary data structure like a linked list or a binary search tree.
• Open addressing: allow elements to overflow out of their target bucket and into other spaces.

Although above solutions provide expected lookup cost as O(1), the expected worst-case cost of a lookup in Open Addressing (with linear probing) is ?(log n) and ?(log n / log log n) in simple chaining (Source : Standford Lecture Notes). To close the gap of expected time and worst case expected time, two ideas are used: 

• Multiple-choice hashing: Give each element multiple choices for positions where it can reside in the hash table
• Relocation hashing: Allow elements in the hash table to move after being placed

Cuckoo Hashing : 

Cuckoo hashing applies the idea of multiple-choice and relocation together and guarantees O(1) worst case lookup time! 

• Multiple-choice: We give a key two choices the h1(key) and h2(key) for residing.
• Relocation: It may happen that h1(key) and h2(key) are preoccupied. This is resolved by imitating the Cuckoo bird: it pushes the other eggs or young out of the nest when it hatches. Analogously, inserting a new key into a cuckoo hashing table may push an older key to a different location. This leaves us with the problem of re-placing the older key. 
  • If the alternate position of older key is vacant, there is no problem.
  • Otherwise, the older key displaces another key. This continues until the procedure finds a vacant position, or enters a cycle. In the case of a cycle, new hash functions are chosen and the whole data structure is ‘rehashed’. Multiple rehashes might be necessary before Cuckoo succeeds.

Insertion is expected O(1) (amortized) with high probability, even considering the possibility of rehashing, as long as the number of keys is kept below half of the capacity of the hash table, i.e., the load factor is below 50%.

Deletion is O(1) worst-case as it requires inspection of just two locations in the hash table. 

Illustration 

Input:

{20, 50, 53, 75, 100, 67, 105, 3, 36, 39}

Hash Functions:

h1(key) = key%11
h2(key) = (key/11)%11

Let’s start by inserting 20 at its possible position in the first table determined by h1(20):

Next: 50

Next: 53. h1(53) = 9. But 20 is already there at 9. We place 53 in table 1 & 20 in table 2 at h2(20) 

Next: 75. h1(75) = 9. But 53 is already there at 9. We place 75 in table 1 & 53 in table 2 at h2(53) 

Next: 100. h1(100) = 1. 

Next: 67. h1(67) = 1. But 100 is already there at 1. We place 67 in table 1 & 100 in table 2 

Next: 105. h1(105) = 6. But 50 is already there at 6. We place 105 in table 1 & 50 in table 2 at h2(50) = 4. Now 53 has been displaced. h1(53) = 9. 75 displaced: h2(75) = 6.

Next: 3. h1(3) = 3.

Next: 36. h1(36) = 3. h2(3) = 0.

Next: 39. h1(39) = 6. h2(105) = 9. h1(100) = 1. h2(67) = 6. h1(75) = 9. h2(53) = 4. h1(50) = 6. h2(39) = 3.
Here, the new key 39 is displaced later in the recursive calls to place 105, which it displaced.

Implementation: 

Below is the implementation of Cuckoo hashing

C++

// C++ program to demonstrate working of Cuckoo
// hashing.
#include<bits/stdc++.h>

// upper bound on number of elements in our set
#define MAXN 11

// choices for position
#define ver 2

// Auxiliary space bounded by a small multiple
// of MAXN, minimizing wastage
int hashtable[ver][MAXN];

// Array to store possible positions for a key
int pos[ver];

/* function to fill hash table with dummy value
 * dummy value: INT_MIN
 * number of hashtables: ver */
void initTable()
{
    for (int j=0; j<MAXN; j++)
        for (int i=0; i<ver; i++)
            hashtable[i][j] = INT_MIN;
}

/* return hashed value for a key
 * function: ID of hash function according to which
    key has to hashed
 * key: item to be hashed */
int hash(int function, int key)
{
    switch (function)
    {
        case 1: return key%MAXN;
        case 2: return (key/MAXN)%MAXN;
    }
}

/* function to place a key in one of its possible positions
 * tableID: table in which key has to be placed, also equal
   to function according to which key must be hashed
 * cnt: number of times function has already been called
   in order to place the first input key
 * n: maximum number of times function can be recursively
   called before stopping and declaring presence of cycle */
void place(int key, int tableID, int cnt, int n)
{
    /* if function has been recursively called max number
       of times, stop and declare cycle. Rehash. */
    if (cnt==n)
    {
        printf("%d unpositioned\n", key);
        printf("Cycle present. REHASH.\n");
        return;
    }

    /* calculate and store possible positions for the key.
     * check if key already present at any of the positions.
      If YES, return. */
    for (int i=0; i<ver; i++)
    {
        pos[i] = hash(i+1, key);
        if (hashtable[i][pos[i]] == key)
           return;
    }

    /* check if another key is already present at the
       position for the new key in the table
     * If YES: place the new key in its position
     * and place the older key in an alternate position
       for it in the next table */
    if (hashtable[tableID][pos[tableID]]!=INT_MIN)
    {
        int dis = hashtable[tableID][pos[tableID]];
        hashtable[tableID][pos[tableID]] = key;
        place(dis, (tableID+1)%ver, cnt+1, n);
    }
    else //else: place the new key in its position
       hashtable[tableID][pos[tableID]] = key;
}

/* function to print hash table contents */
void printTable()
{
    printf("Final hash tables:\n");

    for (int i=0; i<ver; i++, printf("\n"))
        for (int j=0; j<MAXN; j++)
            (hashtable[i][j]==INT_MIN)? printf("- "):
                     printf("%d ", hashtable[i][j]);

    printf("\n");
}

/* function for Cuckoo-hashing keys
 * keys[]: input array of keys
 * n: size of input array */
void cuckoo(int keys[], int n)
{
    // initialize hash tables to a dummy value (INT-MIN)
    // indicating empty position
    initTable();

    // start with placing every key at its position in
    // the first hash table according to first hash
    // function
    for (int i=0, cnt=0; i<n; i++, cnt=0)
        place(keys[i], 0, cnt, n);

    //print the final hash tables
    printTable();
}

/* driver function */
int main()
{
    /* following array doesn't have any cycles and
       hence  all keys will be inserted without any
       rehashing */
    int keys_1[] = {20, 50, 53, 75, 100, 67, 105,
                    3, 36, 39};

    int n = sizeof(keys_1)/sizeof(int);

    cuckoo(keys_1, n);

    /* following array has a cycle and hence we will
       have to rehash to position every key */
    int keys_2[] = {20, 50, 53, 75, 100, 67, 105,
                    3, 36, 39, 6};

    int m = sizeof(keys_2)/sizeof(int);

    cuckoo(keys_2, m);

    return 0;
}

Java

// Java program to demonstrate working of
// Cuckoo hashing.
import java.util.*;

class GFG
{

// upper bound on number of elements in our set
static int MAXN = 11;

// choices for position
static int ver = 2;

// Auxiliary space bounded by a small multiple
// of MAXN, minimizing wastage
static int [][]hashtable = new int[ver][MAXN];

// Array to store possible positions for a key
static int []pos = new int[ver];

/* function to fill hash table with dummy value
* dummy value: INT_MIN
* number of hashtables: ver */
static void initTable()
{
    for (int j = 0; j < MAXN; j++)
        for (int i = 0; i < ver; i++)
            hashtable[i][j] = Integer.MIN_VALUE;
}

/* return hashed value for a key
* function: ID of hash function according to which
    key has to hashed
* key: item to be hashed */
static int hash(int function, int key)
{
    switch (function)
    {
        case 1: return key % MAXN;
        case 2: return (key / MAXN) % MAXN;
    }
    return Integer.MIN_VALUE;
}

/* function to place a key in one of its possible positions
* tableID: table in which key has to be placed, also equal
  to function according to which key must be hashed
* cnt: number of times function has already been called
  in order to place the first input key
* n: maximum number of times function can be recursively
  called before stopping and declaring presence of cycle */
static void place(int key, int tableID, int cnt, int n)
{
    /* if function has been recursively called max number
    of times, stop and declare cycle. Rehash. */
    if (cnt == n)
    {
        System.out.printf("%d unpositioned\n", key);
        System.out.printf("Cycle present. REHASH.\n");
        return;
    }

    /* calculate and store possible positions for the key.
    * check if key already present at any of the positions.
    If YES, return. */
    for (int i = 0; i < ver; i++)
    {
        pos[i] = hash(i + 1, key);
        if (hashtable[i][pos[i]] == key)
        return;
    }

    /* check if another key is already present at the
       position for the new key in the table
    * If YES: place the new key in its position
    * and place the older key in an alternate position
    for it in the next table */
    if (hashtable[tableID][pos[tableID]] != Integer.MIN_VALUE)
    {
        int dis = hashtable[tableID][pos[tableID]];
        hashtable[tableID][pos[tableID]] = key;
        place(dis, (tableID + 1) % ver, cnt + 1, n);
    }
    else // else: place the new key in its position
    hashtable[tableID][pos[tableID]] = key;
}

/* function to print hash table contents */
static void printTable()
{
    System.out.printf("Final hash tables:\n");

    for (int i = 0; i < ver; i++, System.out.printf("\n"))
        for (int j = 0; j < MAXN; j++)
            if(hashtable[i][j] == Integer.MIN_VALUE)
                System.out.printf("- ");
            else
                System.out.printf("%d ", hashtable[i][j]);

    System.out.printf("\n");
}

/* function for Cuckoo-hashing keys
* keys[]: input array of keys
* n: size of input array */
static void cuckoo(int keys[], int n)
{
    // initialize hash tables to a dummy value
    // (INT-MIN) indicating empty position
    initTable();

    // start with placing every key at its position in
    // the first hash table according to first hash
    // function
    for (int i = 0, cnt = 0; i < n; i++, cnt = 0)
        place(keys[i], 0, cnt, n);

    // print the final hash tables
    printTable();
}

// Driver Code
public static void main(String[] args)
{
    /* following array doesn't have any cycles and
    hence all keys will be inserted without any
    rehashing */
    int keys_1[] = {20, 50, 53, 75, 100,
                    67, 105, 3, 36, 39};

    int n = keys_1.length;

    cuckoo(keys_1, n);

    /* following array has a cycle and hence we will
    have to rehash to position every key */
    int keys_2[] = {20, 50, 53, 75, 100,
                    67, 105, 3, 36, 39, 6};

    int m = keys_2.length;

    cuckoo(keys_2, m);
}
}

// This code is contributed by Princi Singh

Python3

# upper bound on number of elements in our set
MAXN = 11

# choices for position
ver = 2

# Auxiliary space bounded by a small multiple
# of MAXN, minimizing wastage
hashtable = [[float('inf')] * MAXN for _ in range(ver)]

# Array to store possible positions for a key
pos = [0] * ver

def init_table():
    """function to fill hash table with dummy value
    dummy value: float('inf')
    number of hashtables: ver"""
    for i in range(ver):
        for j in range(MAXN):
            hashtable[i][j] = float('inf')

def hash(function, key):
    """return hashed value for a key
    function: ID of hash function according to which key has to hashed
    key: item to be hashed"""
    if function == 1:
        return key % MAXN
    elif function == 2:
        return (key // MAXN) % MAXN

def place(key, table_id, cnt, n):
    """function to place a key in one of its possible positions
    table_id: table in which key has to be placed, also equal to function
    according to which key must be hashed
    cnt: number of times function has already been called in order to place
    the first input key
    n: maximum number of times function can be recursively called before
    stopping and declaring presence of cycle"""
    # if function has been recursively called max number of times, stop
    # and declare cycle. Rehash.
    if cnt == n:
        print(f"{key} unpositioned")
        print("Cycle present. REHASH.")
        return

    # calculate and store possible positions for the key. check if key
    # already present at any of the positions. If YES, return.
    for i in range(ver):
        pos[i] = hash(i + 1, key)
        if hashtable[i][pos[i]] == key:
            return

    # check if another key is already present at the position for the
    # new key in the table
    # If YES: place the new key in its position and place the older key
    # in an alternate position for it in the next table
    if hashtable[table_id][pos[table_id]] != float('inf'):
        dis = hashtable[table_id][pos[table_id]]
        hashtable[table_id][pos[table_id]] = key
        place(dis, (table_id + 1) % ver, cnt + 1, n)
    else: # else: place the new key in its position
        hashtable[table_id][pos[table_id]] = key

def print_table():
    """function to print hash table contents"""
    print("Final hash tables:")
    for i in range(ver):
        print()
        for j in range(MAXN):
            if hashtable[i][j] == float('inf'):
                print("- ", end="")
            else:
                print(f"{hashtable[i][j]} ", end="")
    print()

def cuckoo(keys, n):
    # initialize hash tables to a dummy value (float('inf'))
    # indicating empty position
    init_table()

    # start with placing every key at its position in the first
    # hash table according to first hash function
    for i in range(n):
        cnt = 0
        place(keys[i], 0, cnt, n)

    # print the final hash tables
    print_table()

# driver function
def main():
    # following array doesn't have any cycles and
    # hence  all keys will be inserted without any
    # rehashing
    keys_1 = [20, 50, 53, 75, 100, 67, 105, 3, 36, 39]

    cuckoo(keys_1, len(keys_1))

    # following array has a cycle and hence we will
    # have to rehash to position every key
    keys_2 = [20, 50, 53, 75, 100, 67, 105, 3, 36, 39, 6]

    cuckoo(keys_2, len(keys_2))

if __name__ == "__main__":
    main()

# This code is contributed by vikramshirsath177

C#

// C# program to demonstrate working of
// Cuckoo hashing.
using System;

class GFG
{

// upper bound on number of
// elements in our set
static int MAXN = 11;

// choices for position
static int ver = 2;

// Auxiliary space bounded by a small
// multiple of MAXN, minimizing wastage
static int [,]hashtable = new int[ver, MAXN];

// Array to store
// possible positions for a key
static int []pos = new int[ver];

/* function to fill hash table
with dummy value
* dummy value: INT_MIN
* number of hashtables: ver */
static void initTable()
{
    for (int j = 0; j < MAXN; j++)
        for (int i = 0; i < ver; i++)
            hashtable[i, j] = int.MinValue;
}

/* return hashed value for a key
* function: ID of hash function
according to which key has to hashed
* key: item to be hashed */
static int hash(int function, int key)
{
    switch (function)
    {
        case 1: return key % MAXN;
        case 2: return (key / MAXN) % MAXN;
    }
    return int.MinValue;
}

/* function to place a key in one of
its possible positions
* tableID: table in which key
has to be placed, also equal to function
according to which key must be hashed
* cnt: number of times function has already
been called in order to place the first input key
* n: maximum number of times function
can be recursively called before stopping and
declaring presence of cycle */
static void place(int key, int tableID,
                  int cnt, int n)
{
    /* if function has been recursively
    called max number of times,
    stop and declare cycle. Rehash. */
    if (cnt == n)
    {
        Console.Write("{0} unpositioned\n", key);
        Console.Write("Cycle present. REHASH.\n");
        return;
    }

    /* calculate and store possible positions
    * for the key. Check if key already present
    at any of the positions. If YES, return. */
    for (int i = 0; i < ver; i++)
    {
        pos[i] = hash(i + 1, key);
        if (hashtable[i, pos[i]] == key)
        return;
    }

    /* check if another key is already present
    at the position for the new key in the table
    * If YES: place the new key in its position
    * and place the older key in an alternate position
    for it in the next table */
    if (hashtable[tableID,
              pos[tableID]] != int.MinValue)
    {
        int dis = hashtable[tableID, pos[tableID]];
        hashtable[tableID, pos[tableID]] = key;
        place(dis, (tableID + 1) % ver, cnt + 1, n);
    }
    else // else: place the new key in its position
    hashtable[tableID, pos[tableID]] = key;
}

/* function to print hash table contents */
static void printTable()
{
    Console.Write("Final hash tables:\n");

    for (int i = 0; i < ver;
             i++, Console.Write("\n"))
        for (int j = 0; j < MAXN; j++)
            if(hashtable[i, j] == int.MinValue)
                Console.Write("- ");
            else
                Console.Write("{0} ",
                        hashtable[i, j]);

    Console.Write("\n");
}

/* function for Cuckoo-hashing keys
* keys[]: input array of keys
* n: size of input array */
static void cuckoo(int []keys, int n)
{
    // initialize hash tables to a
    // dummy value (INT-MIN)
    // indicating empty position
    initTable();

    // start with placing every key
    // at its position in the first
    // hash table according to first
    // hash function
    for (int i = 0, cnt = 0;
             i < n; i++, cnt = 0)
        place(keys[i], 0, cnt, n);

    // print the final hash tables
    printTable();
}

// Driver Code
public static void Main(String[] args)
{
    /* following array doesn't have
    any cycles and hence all keys
    will be inserted without any rehashing */
    int []keys_1 = {20, 50, 53, 75, 100,
                    67, 105, 3, 36, 39};

    int n = keys_1.Length;

    cuckoo(keys_1, n);

    /* following array has a cycle and
    hence we will have to rehash to
    position every key */
    int []keys_2 = {20, 50, 53, 75, 100,
                    67, 105, 3, 36, 39, 6};

    int m = keys_2.Length;

    cuckoo(keys_2, m);
}
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

// Javascript program to demonstrate working of
// Cuckoo hashing.

// upper bound on number of elements in our set
let MAXN = 11;

// choices for position
let ver = 2;

// Auxiliary space bounded by a small multiple
// of MAXN, minimizing wastage
let hashtable = new Array(ver);
for (var i = 0; i < hashtable.length; i++) {
    hashtable[i] = new Array(2);
}

// Array to store possible positions for a key
let pos = Array(ver).fill(0);

/* function to fill hash table with dummy value
* dummy value: let_MIN
* number of hashtables: ver */
function initTable()
{
    for (let j = 0; j < MAXN; j++)
        for (let i = 0; i < ver; i++)
            hashtable[i][j] = Number.MIN_VALUE;
}

/* return hashed value for a key
* function: ID of hash function according to which
    key has to hashed
* key: item to be hashed */
function hash(function, key)
{
    switch (function)
    {
        case 1: return key % MAXN;
        case 2: return (Math.floor(key / MAXN)) % MAXN;
    }
    return Number.MIN_VALUE;
}

/* function to place a key in one of its possible positions
* tableID: table in which key has to be placed, also equal
  to function according to which key must be hashed
* cnt: number of times function has already been called
  in order to place the first input key
* n: maximum number of times function can be recursively
  called before stopping and declaring presence of cycle */
function place(key, tableID, cnt, n)
{
    /* if function has been recursively called max number
    of times, stop and declare cycle. Rehash. */
    if (cnt == n)
    {
        document.write(key + " unpositioned" + "<br/>");
        document.write("Cycle present. REHASH." + "<br/>");
        return;
    }

    /* calculate and store possible positions for the key.
    * check if key already present at any of the positions.
    If YES, return. */
    for (let i = 0; i < ver; i++)
    {
        pos[i] = hash(i + 1, key);
        if (hashtable[i][pos[i]] == key)
            return;
    }

    /* check if another key is already present at the
       position for the new key in the table
    * If YES: place the new key in its position
    * and place the older key in an alternate position
    for it in the next table */
    if (hashtable[tableID][pos[tableID]] != Number.MIN_VALUE)
    {
        let dis = hashtable[tableID][pos[tableID]];
        hashtable[tableID][pos[tableID]] = key;
        place(dis, (tableID + 1) % ver, cnt + 1, n);
    }
    else // else: place the new key in its position
        hashtable[tableID][pos[tableID]] = key;
}

/* function to print hash table contents */
function printTable()
{
    document.write("Final hash tables:" + "<br/>");

    for (let i = 0; i < ver; i++, document.write("<br/>"))
        for (let j = 0; j < MAXN; j++)
            if(hashtable[i][j] == Number.MIN_VALUE)
                document.write("- ");
            else
                document.write(hashtable[i][j] + " ");

    document.write("<br/>");
}

/* function for Cuckoo-hashing keys
* keys[]: input array of keys
* n: size of input array */
function cuckoo(keys, n)
{
    // initialize hash tables to a dummy value
    // (let-MIN) indicating empty position
    initTable();

    // start with placing every key at its position in
    // the first hash table according to first hash
    // function
    for (let i = 0, cnt = 0; i < n; i++, cnt = 0)
        place(keys[i], 0, cnt, n);

    // print the final hash tables
    printTable();
}

// Driver program

      /* following array doesn't have any cycles and
    hence all keys will be inserted without any
    rehashing */
    let keys_1 = [20, 50, 53, 75, 100,
                    67, 105, 3, 36, 39];

    let n = keys_1.length;

    cuckoo(keys_1, n);

    /* following array has a cycle and hence we will
    have to rehash to position every key */
    let keys_2 = [20, 50, 53, 75, 100,
                    67, 105, 3, 36, 39, 6];

    let m = keys_2.length;

    cuckoo(keys_2, m);

</script>

Output

Final hash tables:
- 100 - 36 - - 50 - - 75 - 
3 20 - 39 53 - 67 - - 105 - 

105 unpositioned
Cycle present. REHASH.
Final hash tables:
- 67 - 3 - - 39 - - 53 - 
6 20 - 36 50 - 75 - - 100 - 

Time Complexity: O(N), the time complexity of the Cuckoo Hashing algorithm is O(N), where N is the number of keys to be stored in the hash table. This is because the algorithm requires only one pass over the list of keys to place them in the hash table.

Auxiliary Space: O(N), the space complexity of the Cuckoo Hashing algorithm is O(N), where N is the number of keys stored in the hash table. This is because the algorithm requires an auxiliary space of size equal to the hash table, where all the keys are stored.

Generalizations of cuckoo hashing that use more than 2 alternative hash functions can be expected to utilize a larger part of the capacity of the hash table efficiently while sacrificing some lookup and insertion speed. Example: if we use 3 hash functions, it’s safe to load 91% and still be operating within expected bounds.