Randomized Binary Search Algorithm

Last Updated : 25 Jan, 2022

We are given a sorted array A[] of n elements. We need to find if x is present in A or not.In binary search we always used middle element, here we will randomly pick one element in given range.
In Binary Search we had

middle = (start + end)/2

In Randomized binary search we do following

Generate a random number t
Since range of number in which we want a random
number is [start, end]
Hence we do, t = t % (end-start+1)
Then, t = start + t;
Hence t is a random number between start and end

It is a Las Vegas randomized algorithm as it always finds the correct result.

Expected Time complexity of Randomized Binary Search Algorithm
For n elements let say expected time required be T(n), After we choose one random pivot, array size reduces to say k. Since pivot is chosen with equal probability for all possible pivots, hence p = 1/n.
T(n) is sum of time of all possible sizes after choosing pivot multiplied by probability of choosing that pivot plus time take to generate random pivot index.Hence

T(n) = p*T(1) + p*T(2) + ..... + p*T(n) + 1
putting p = 1/n
T(n) = ( T(1) + T(2) + ..... + T(n) ) / n + 1
n*T(n) = T(1) + T(2) + .... + T(n) + n      .... eq(1)
Similarly for n-1
(n-1)*T(n-1) = T(1) + T(2) + ..... + T(n-1) + n-1    .... eq(2)
Subtract eq(1) - eq(2)
n*T(n) - (n-1)*T(n-1) = T(n) + 1
(n-1)*T(n) - (n-1)*T(n-1) =  1
(n-1)*T(n) = (n-1)*T(n-1) + 1
T(n) = 1/(n-1) + T(n-1)
T(n) = 1/(n-1) + 1/(n-2) + T(n-2)
T(n) = 1/(n-1) + 1/(n-2) + 1/(n-3) + T(n-3)
Similarly,
T(n) = 1 + 1/2 + 1/3 + ... + 1/(n-1)
Hence T(n) is equal to (n-1)th Harmonic number, 
n-th harmonic number is O(log n)
Hence T(n) is O(log n)

Recursive implementation of Randomized Binary Search

C++

// C++ program to implement recursive
// randomized algorithm.
#include <iostream>
#include <ctime>
using namespace std;
 
// To generate random number
// between x and y ie.. [x, y]
int getRandom(int x, int y)
{
    srand(time(NULL));
    return (x + rand() % (y-x+1));
}
 
// A recursive randomized binary search function.
// It returns location of x in
// given array arr[l..r] is present, otherwise -1
int randomizedBinarySearch(int arr[], int l,
                            int r, int x)
{
    if (r >= l)
    {
        // Here we have defined middle as
        // random index between l and r ie.. [l, r]
        int mid = getRandom(l, r);
 
        // If the element is present at the
        // middle itself
        if (arr[mid] == x)
            return mid;
 
        // If element is smaller than mid, then
        // it can only be present in left subarray
        if (arr[mid] > x)
          return randomizedBinarySearch(arr, l,
                                    mid-1, x);
 
        // Else the element can only be present
        // in right subarray
        return randomizedBinarySearch(arr, mid+1,
                                         r, x);
    }
 
    // We reach here when element is not present
    // in array
    return -1;
}
 
// Driver code
int main(void)
{
    int arr[] = {2, 3, 4, 10, 40};
    int n = sizeof(arr)/ sizeof(arr[0]);
    int x = 10;
    int result = randomizedBinarySearch(arr, 0, n-1, x);
    (result == -1)? printf("Element is not present in array")
    : printf("Element is present at index %d", result);
    return 0;
}

Java

// Java program to implement recursive
// randomized algorithm.
public class RandomizedBinarySearch
{
 
    // To generate random number
    // between x and y ie.. [x, y]
    public static int getRandom(int x, int y) 
    {
        return (x + (int)(Math.random() % (y-x+1)));
    }
 
    // A recursive randomized binary search function.
    // It returns location of x in
    // given array arr[l..r] is present, otherwise -1
    public static int randomizedBinarySearch(int arr[], 
                            int low, int high, int key)
    {
        if (high >= low) 
        {
            // Here we have defined middle as
            // random index between l and r ie.. [l, r]
            int mid = getRandom(low, high);
 
            // If the element is present at the
            // middle itself
            if (arr[mid] == key)
                return mid;
 
            // If element is smaller than mid, then
            // it can only be present in left subarray
            if (arr[mid] > key)
                return randomizedBinarySearch(arr, low, mid-1, key);
 
            // Else the element can only be present
            // in right subarray
            return randomizedBinarySearch(arr, mid+1, high, key);
        }
 
        // We reach here when element is not present
        // in array
        return -1;
    }
 
    // Driver code
    public static void main(String[] args) 
    {
        int arr[] = {2, 3, 4, 10, 40};
        int n = arr.length;
        int key = 10;
        int result = randomizedBinarySearch(arr, 0, n-1, key);
        System.out.println((result == -1)?"Element is not present in array":
                "Element is present at index " + result);
    }
}
 
// This code is contributed by JEREM

Python3

# Python3 program to implement recursive 
# randomized algorithm. 
# To generate random number 
# between x and y ie.. [x, y] 
 
import random
def getRandom(x,y):
    tmp=(x + random.randint(0,100000) % (y-x+1))
    return tmp
     
# A recursive randomized binary search function. 
# It returns location of x in 
# given array arr[l..r] is present, otherwise -1 
 
def randomizedBinarySearch(arr,l,r,x) :
    if r>=l:
         
        # Here we have defined middle as 
        # random index between l and r ie.. [l, r] 
        mid=getRandom(l,r)
         
        # If the element is present at the 
        # middle itself
        if arr[mid] == x:
            return mid
             
        # If element is smaller than mid, then 
        # it can only be present in left subarray
        if arr[mid]>x:
            return randomizedBinarySearch(arr, l, mid-1, x)
             
        # Else the element can only be present 
        # in right subarray 
        return randomizedBinarySearch(arr, mid+1,r, x)
         
    # We reach here when element is not present 
    # in array
    return -1
     
# Driver code 
if __name__=='__main__':
    arr = [2, 3, 4, 10, 40]
    n=len(arr)
    x=10
    result = randomizedBinarySearch(arr, 0, n-1, x)
    if result==-1:
        print('Element is not present in array')
    else:
        print('Element is present at index ', result)
         
# This code is contributes by sahilshelangia

C#

// C# program to implement recursive
// randomized algorithm.
using System;
 
class RandomizedBinarySearch
{
 
    // To generate random number
    // between x and y ie.. [x, y]
    public static int getRandom(int x, int y) 
    {
        Random r = new Random();
        return (x + (int)(r.Next() % (y - x + 1)));
    }
 
    // A recursive randomized binary search function.
    // It returns location of x in
    // given array arr[l..r] is present, otherwise -1
    public static int randomizedBinarySearch(int []arr, 
                            int low, int high, int key)
    {
        if (high >= low) 
        {
            // Here we have defined middle as
            // random index between l and r ie.. [l, r]
            int mid = getRandom(low, high);
 
            // If the element is present at the
            // middle itself
            if (arr[mid] == key)
                return mid;
 
            // If element is smaller than mid, then
            // it can only be present in left subarray
            if (arr[mid] > key)
                return randomizedBinarySearch(arr, low, mid - 1, key);
 
            // Else the element can only be present
            // in right subarray
            return randomizedBinarySearch(arr, mid + 1, high, key);
        }
 
        // We reach here when element is not present
        // in array
        return -1;
    }
 
    // Driver code
    public static void Main(String[] args) 
    {
        int []arr = {2, 3, 4, 10, 40};
        int n = arr.Length;
        int key = 10;
        int result = randomizedBinarySearch(arr, 0, n - 1, key);
        Console.WriteLine((result == -1)?"Element is not present in array":
                "Element is present at index " + result);
    }
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
 
// Javascript program to implement recursive
 
// To generate random number
// between x and y ie.. [x, y]
function getRandom(x, y) {
    return (x + Math.floor(Math.random() % (y - x + 1)));
}
 
// A recursive randomized binary search function.
// It returns location of x in
// given array arr[l..r] is present, otherwise -1
function randomizedBinarySearch(arr, l, r, x) 
{
    if (r >= l)
    {
     
        // Here we have defined middle as
        // random index between l and r ie.. [l, r]
        let mid = getRandom(l, r);
 
        // If the element is present at the
        // middle itself
        if (arr[mid] == x)
            return mid;
 
        // If element is smaller than mid, then
        // it can only be present in left subarray
        if (arr[mid] > x)
            return randomizedBinarySearch(arr, l,
                mid - 1, x);
 
        // Else the element can only be present
        // in right subarray
        return randomizedBinarySearch(arr, mid + 1,
            r, x);
    }
 
    // We reach here when element is not present
    // in array
    return -1;
}
 
// Driver code
let arr = [2, 3, 4, 10, 40];
let n = arr.length;
let x = 10;
let result = randomizedBinarySearch(arr, 0, n - 1, x);
(result == -1) ? document.write("Element is not present in array")
    : document.write("Element is present at index " + result);
     
    // This code is contributed by saurabh_jaiswal.
</script>

Output:

Element is present at index 3

Iterative implementation of Randomized Binary Search

C++

// C++ program to implement iterative
// randomized algorithm.
#include <iostream>
#include <ctime>
using namespace std;
 
// To generate random number
// between x and y ie.. [x, y]
int getRandom(int x, int y)
{
    srand(time(NULL));
    return (x + rand()%(y-x+1));
}
 
// A iterative randomized binary search function.
// It returns location of x in
// given array arr[l..r] if present, otherwise -1
int randomizedBinarySearch(int arr[], int l,
                               int r, int x)
{
    while (l <= r)
    {
        // Here we have defined middle as
        // random index between l and r ie.. [l, r]
        int m = getRandom(l, r);
 
        // Check if x is present at mid
        if (arr[m] == x)
            return m;
 
        // If x greater, ignore left half
        if (arr[m] < x)
            l = m + 1;
 
        // If x is smaller, ignore right half
        else
            r = m - 1;
    }
 
    // if we reach here, then element was
    // not present
    return -1;
}
 
// Driver code
int main(void)
{
    int arr[] = {2, 3, 4, 10, 40};
    int n = sizeof(arr)/ sizeof(arr[0]);
    int x = 10;
    int result = randomizedBinarySearch(arr, 0, n-1, x);
    (result == -1)? printf("Element is not present in array")
        : printf("Element is present at index %d", result);
    return 0;
}

Java

// Java program to implement iterative
// randomized algorithm.
class GFG
{
 
// To generate random number
// between x and y ie.. [x, y]
static int getRandom(int x, int y)
{
     
    return (int) (x + Math.random() * 10 % (y - x + 1));
}
 
// A iterative randomized binary search function.
// It returns location of x in
// given array arr[l..r] if present, otherwise -1
static int randomizedBinarySearch(int arr[], int l,
                                    int r, int x)
{
    while (l <= r)
    {
        // Here we have defined middle as
        // random index between l and r ie.. [l, r]
        int m = getRandom(l, r);
 
        // Check if x is present at mid
        if (arr[m] == x)
            return m;
 
        // If x greater, ignore left half
        if (arr[m] < x)
            l = m + 1;
 
        // If x is smaller, ignore right half
        else
            r = m - 1;
    }
 
    // if we reach here, then element was
    // not present
    return -1;
}
 
// Driver code
public static void main(String []args)
{
    int arr[] = {2, 3, 4, 10, 40};
    int n = arr.length;
    int x = 10;
    int result = randomizedBinarySearch(arr, 0, n - 1, x);
    if(result == -1)
        System.out.printf("Element is not present in array");
    else
        System.out.printf("Element is present at index %d", result);
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python program to implement iterative 
# randomized algorithm. 
 
# To generate random number 
# between x and y ie.. [x, y] 
 
from random import randint
 
def getRandom(x, y): 
    return randint(x,y)
 
# A iterative randomized binary search function. 
# It returns location of x in 
# given array arr[l..r] if present, otherwise -1 
def randomizedBinarySearch(arr, l, r, x):
    while (l <= r):
        # Here we have defined middle as 
        # random index between l and r ie.. [l, r] 
        m = getRandom(l, r) 
 
        # Check if x is present at mid 
        if (arr[m] == x):
            return m
 
        # If x greater, ignore left half 
        if (arr[m] < x):
            l = m + 1
 
        # If x is smaller, ignore right half 
        else:
            r = m - 1
    # if we reach here, then element was 
    # not present 
    return -1
 
# Driver code 
arr = [2, 3, 4, 10, 40] 
n = len(arr)
x = 10
result = randomizedBinarySearch(arr, 0, n-1, x)
if result == 1:
    print("Element is not present in array")
else: 
    print("Element is present at index", result)
 
# This code is contributed by ankush_953

C#

// C# program to implement iterative
// randomized algorithm.
using System;
using System.Collections.Generic;
 
class GFG
{
 
// To generate random number
// between x and y ie.. [x, y]
static int getRandom(int x, int y)
{
     
    return (int) (x + new Random(10).Next(1) * 10 % (y - x + 1));
}
 
// A iterative randomized binary search function.
// It returns location of x in
// given array arr[l..r] if present, otherwise -1
static int randomizedBinarySearch(int []arr, int l,
                                    int r, int x)
{
    while (l <= r)
    {
        // Here we have defined middle as
        // random index between l and r ie.. [l, r]
        int m = getRandom(l, r);
 
        // Check if x is present at mid
        if (arr[m] == x)
            return m;
 
        // If x greater, ignore left half
        if (arr[m] < x)
            l = m + 1;
 
        // If x is smaller, ignore right half
        else
            r = m - 1;
    }
 
    // if we reach here, then element was
    // not present
    return -1;
}
 
// Driver code
public static void Main(String []args)
{
    int []arr = {2, 3, 4, 10, 40};
    int n = arr.Length;
    int x = 10;
    int result = randomizedBinarySearch(arr, 0, n - 1, x);
    if(result == -1)
        Console.Write("Element is not present in array");
    else
        Console.Write("Element is present at index {0}", result);
}
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
 
// Javascript program to implement iterative
// randomized algorithm.
 
// To generate random number
// between x and y ie.. [x, y]
function getRandom(x,y)
{
     return Math.floor(x + Math.floor(Math.random() * 10) %
                      (y - x + 1));
}
 
// A iterative randomized binary search function.
// It returns location of x in
// given array arr[l..r] if present, otherwise -1
function randomizedBinarySearch(arr,l,r,x)
{
    while (l <= r)
    {
         
        // Here we have defined middle as
        // random index between l and r ie.. [l, r]
        let m = getRandom(l, r);
   
        // Check if x is present at mid
        if (arr[m] == x)
            return m;
   
        // If x greater, ignore left half
        if (arr[m] < x)
            l = m + 1;
   
        // If x is smaller, ignore right half
        else
            r = m - 1;
    }
   
    // If we reach here, then element was
    // not present
    return -1;
}
 
// Driver code
let arr = [ 2, 3, 4, 10, 40 ];
let n = arr.length;
let x = 10;
let result = randomizedBinarySearch(arr, 0, n - 1, x);
 
if (result == -1)
    document.write("Element is not present in array");
else
    document.write("Element is present at index ", 
                   result);
 
// This code is contributed by rag2127
 
</script>

Output:

Element is present at index 3

Abstraction of Binary Search

Pratik Chhajer

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Randomized Binary Search Algorithm

C++

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