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 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 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# 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 <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 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 # 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# 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 <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
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