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
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem