Search an element in a sorted and rotated array

An element in a sorted array can be found in O(log n) time via binary search. But suppose we rotate an ascending order sorted array at some pivot unknown to you beforehand. So for instance, 1 2 3 4 5 might become 3 4 5 1 2. Devise a way to find an element in the rotated array in O(log n) time.

Input  : arr[] = {5, 6, 7, 8, 9, 10, 1, 2, 3};
         key = 3
Output : Found at index 8

Input  : arr[] = {5, 6, 7, 8, 9, 10, 1, 2, 3};
         key = 30
Output : Not found

Input : arr[] = {30, 40, 50, 10, 20}
        key = 10   
Output : Found at index 3

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

All solutions provided here assume that all elements in array are distinct.

The idea is to find the pivot point, divide the array in two sub-arrays and call binary search.
The main idea for finding pivot is – for a sorted (in increasing order) and pivoted array, pivot element is the only element for which next element to it is smaller than it.
Using above criteria and binary search methodology we can get pivot element in O(logn) time

Input arr[] = {3, 4, 5, 1, 2}
Element to Search = 1
  1) Find out pivot point and divide the array in two
      sub-arrays. (pivot = 2) /*Index of 5*/
  2) Now call binary search for one of the two sub-arrays.
      (a) If element is greater than 0th element then
             search in left array
      (b) Else Search in right array
          (1 will go in else as 1 < 0th element(3))
  3) If element is found in selected sub-array then return index
     Else return -1.

Below is the implementation of the above approach :

C++

/* C++ Program to search an element
   in a sorted and pivoted array*/
#include <bits/stdc++.h>
using namespace std;

/* Standard Binary Search function*/
int binarySearch(int arr[], int low, 
                  int high, int key)
{
  if (high < low)
    return -1;
    
  int mid = (low + high)/2; /*low + (high - low)/2;*/
  if (key == arr[mid])
    return mid;
    
  if (key > arr[mid])
    return binarySearch(arr, (mid + 1), high, key);
    
  // else
    return binarySearch(arr, low, (mid -1), key);
}

/* Function to get pivot. For array 3, 4, 5, 6, 1, 2
   it returns 3 (index of 6) */
int findPivot(int arr[], int low, int high)
{
  // base cases
  if (high < low) return -1;
  if (high == low) return low;

   int mid = (low + high)/2; /*low + (high - low)/2;*/
   if (mid < high && arr[mid] > arr[mid + 1])
    return mid;
    
   if (mid > low && arr[mid] < arr[mid - 1])
    return (mid-1);
    
   if (arr[low] >= arr[mid])
    return findPivot(arr, low, mid-1);
    
   return findPivot(arr, mid + 1, high);
}

/* Searches an element key in a pivoted
   sorted array arr[] of size n */
int pivotedBinarySearch(int arr[], int n, int key)
{
  int pivot = findPivot(arr, 0, n-1);

  // If we didn't find a pivot, 
  // then array is not rotated at all
  if (pivot == -1)
    return binarySearch(arr, 0, n-1, key);

  // If we found a pivot, then first compare with pivot
  // and then search in two subarrays around pivot
  if (arr[pivot] == key)
    return pivot;
    
  if (arr[0] <= key)
    return binarySearch(arr, 0, pivot-1, key);
    
    return binarySearch(arr, pivot+1, n-1, key);
}

/* Driver program to check above functions */
int main()
{
  // Let us search 3 in below array
  int arr1[] = {5, 6, 7, 8, 9, 10, 1, 2, 3};
  int n = sizeof(arr1)/sizeof(arr1[0]);
  int key = 3;
  
  // Function calling
  cout << "Index of the element is : " << 
           pivotedBinarySearch(arr1, n, key);
           
  return 0;
}

C

/* Program to search an element in a sorted and pivoted array*/
#include <stdio.h>

int findPivot(int[], int, int);
int binarySearch(int[], int, int, int);

/* Searches an element key in a pivoted sorted array arrp[]
   of size n */
int pivotedBinarySearch(int arr[], int n, int key)
{
   int pivot = findPivot(arr, 0, n-1);

   // If we didn't find a pivot, then array is not rotated at all
   if (pivot == -1)
       return binarySearch(arr, 0, n-1, key);

   // If we found a pivot, then first compare with pivot and then
   // search in two subarrays around pivot
   if (arr[pivot] == key)
       return pivot;
   if (arr[0] <= key)
       return binarySearch(arr, 0, pivot-1, key);
   return binarySearch(arr, pivot+1, n-1, key);
}

/* Function to get pivot. For array 3, 4, 5, 6, 1, 2 it returns
   3 (index of 6) */
int findPivot(int arr[], int low, int high)
{
   // base cases
   if (high < low)  return -1;
   if (high == low) return low;

   int mid = (low + high)/2;   /*low + (high - low)/2;*/
   if (mid < high && arr[mid] > arr[mid + 1])
       return mid;
   if (mid > low && arr[mid] < arr[mid - 1])
       return (mid-1);
   if (arr[low] >= arr[mid])
       return findPivot(arr, low, mid-1);
   return findPivot(arr, mid + 1, high);
}

/* Standard Binary Search function*/
int binarySearch(int arr[], int low, int high, int key)
{
   if (high < low)
       return -1;
   int mid = (low + high)/2;  /*low + (high - low)/2;*/
   if (key == arr[mid])
       return mid;
   if (key > arr[mid])
       return binarySearch(arr, (mid + 1), high, key);
   return binarySearch(arr, low, (mid -1), key);
}

/* Driver program to check above functions */
int main()
{
   // Let us search 3 in below array
   int arr1[] = {5, 6, 7, 8, 9, 10, 1, 2, 3};
   int n = sizeof(arr1)/sizeof(arr1[0]);
   int key = 3;
   printf("Index of the element is : %d",
        pivotedBinarySearch(arr1, n, key));
   return 0;
}

Java

/* Java program to search an element 
   in a sorted and pivoted array*/
 
class Main
{
         
    /* Searches an element key in a 
       pivoted sorted array arrp[]
       of size n */
    static int pivotedBinarySearch(int arr[], int n, int key)
    {
       int pivot = findPivot(arr, 0, n-1);
      
       // If we didn't find a pivot, then 
       // array is not rotated at all
       if (pivot == -1)
           return binarySearch(arr, 0, n-1, key);
      
       // If we found a pivot, then first 
       // compare with pivot and then
       // search in two subarrays around pivot
       if (arr[pivot] == key)
           return pivot;
       if (arr[0] <= key)
           return binarySearch(arr, 0, pivot-1, key);
       return binarySearch(arr, pivot+1, n-1, key);
    }
     
    /* Function to get pivot. For array 
       3, 4, 5, 6, 1, 2 it returns
       3 (index of 6) */
    static int findPivot(int arr[], int low, int high)
    {
       // base cases
       if (high < low)  
            return -1;
       if (high == low) 
            return low;
       
       /* low + (high - low)/2; */
       int mid = (low + high)/2;   
       if (mid < high && arr[mid] > arr[mid + 1])
           return mid;
       if (mid > low && arr[mid] < arr[mid - 1])
           return (mid-1);
       if (arr[low] >= arr[mid])
           return findPivot(arr, low, mid-1);
       return findPivot(arr, mid + 1, high);
    }
     
    /* Standard Binary Search function */
    static int binarySearch(int arr[], int low, int high, int key)
    {
       if (high < low)
           return -1;
       
       /* low + (high - low)/2; */       
       int mid = (low + high)/2;  
       if (key == arr[mid])
           return mid;
       if (key > arr[mid])
           return binarySearch(arr, (mid + 1), high, key);
       return binarySearch(arr, low, (mid -1), key);
    }
  
    // main function
    public static void main(String args[])
    {
       // Let us search 3 in below array
       int arr1[] = {5, 6, 7, 8, 9, 10, 1, 2, 3};
       int n = arr1.length;
       int key = 3;
       System.out.println("Index of the element is : "
                      + pivotedBinarySearch(arr1, n, key));
    }
}

Python3

# Python Program to search an element
# in a sorted and pivoted array

# Searches an element key in a pivoted
# sorted array arrp[] of size n 
def pivotedBinarySearch(arr, n, key):

    pivot = findPivot(arr, 0, n-1);

    # If we didn't find a pivot, 
    # then array is not rotated at all
    if pivot == -1:
        return binarySearch(arr, 0, n-1, key);

    # If we found a pivot, then first
    # compare with pivot and then
    # search in two subarrays around pivot
    if arr[pivot] == key:
        return pivot
    if arr[0] <= key:
        return binarySearch(arr, 0, pivot-1, key);
    return binarySearch(arr, pivot+1, n-1, key);


# Function to get pivot. For array 
# 3, 4, 5, 6, 1, 2 it returns 3 
# (index of 6) 
def findPivot(arr, low, high):
    
    # base cases
    if high < low:
        return -1
    if high == low:
        return low
    
    #low + (high - low)/2;
    mid = int((low + high)/2)
    
    if mid < high and arr[mid] > arr[mid + 1]:
        return mid
    if mid > low and arr[mid] < arr[mid - 1]:
        return (mid-1)
    if arr[low] >= arr[mid]:
        return findPivot(arr, low, mid-1)
    return findPivot(arr, mid + 1, high)

# Standard Binary Search function*/
def binarySearch(arr, low, high, key):

    if high < low:
        return -1
        
    #low + (high - low)/2;    
    mid = int((low + high)/2)
    
    if key == arr[mid]:
        return mid
    if key > arr[mid]:
        return binarySearch(arr, (mid + 1), high,
                                            key);
    return binarySearch(arr, low, (mid -1), key);


# Driver program to check above functions */
# Let us search 3 in below array
arr1 = [5, 6, 7, 8, 9, 10, 1, 2, 3]
n = len(arr1)
key = 3
print("Index of the element is : ", 
      pivotedBinarySearch(arr1, n, key))
      
# This is contributed by Smitha Dinesh Semwal

C#

// C# program to search an element
// in a sorted and pivoted array
using System;

class main {

    // Searches an element key in a
    // pivoted sorted array arrp[]
    // of size n
    static int pivotedBinarySearch(int[] arr,
                                int n, int key)
    {
        int pivot = findPivot(arr, 0, n - 1);

        // If we didn't find a pivot, then
        // array is not rotated at all
        if (pivot == -1)
            return binarySearch(arr, 0, n - 1, key);

        // If we found a pivot, then first
        // compare with pivot and then
        // search in two subarrays around pivot
        if (arr[pivot] == key)
            return pivot;

        if (arr[0] <= key)
            return binarySearch(arr, 0, pivot - 1, key);

        return binarySearch(arr, pivot + 1, n - 1, key);
    }

    /* Function to get pivot. For array 
    3, 4, 5, 6, 1, 2 it returns
    3 (index of 6) */
    static int findPivot(int[] arr, int low, int high)
    {
        // base cases
        if (high < low)
            return -1;
        if (high == low)
            return low;

        /* low + (high - low)/2; */
        int mid = (low + high) / 2;

        if (mid < high && arr[mid] > arr[mid + 1])
            return mid;

        if (mid > low && arr[mid] < arr[mid - 1])
            return (mid - 1);

        if (arr[low] >= arr[mid])
            return findPivot(arr, low, mid - 1);

        return findPivot(arr, mid + 1, high);
    }

    /* Standard Binary Search function */
    static int binarySearch(int[] arr, int low,
                            int high, int key)
    {
        if (high < low)
            return -1;

        /* low + (high - low)/2; */
        int mid = (low + high) / 2;

        if (key == arr[mid])
            return mid;
        if (key > arr[mid])
            return binarySearch(arr, (mid + 1), high, key);

        return binarySearch(arr, low, (mid - 1), key);
    }

    // Driver Code
    public static void Main()
    {
        // Let us search 3 in below array
        int[] arr1 = { 5, 6, 7, 8, 9, 10, 1, 2, 3 };
        int n = arr1.Length;
        int key = 3;
        Console.Write("Index of the element is : "
                    + pivotedBinarySearch(arr1, n, key));
    }
}

// This code is contributed by vt_m.

Output:

Index of the element is : 8

Time Complexity O(logn). Thanks to Ajay Mishra for initial solution.

Improved Solution:
We can search an element in one pass of Binary Search. The idea is to search

1) Find middle point mid = (l + h)/2
2) If key is present at middle point, return mid.
3) Else If arr[l..mid] is sorted
    a) If key to be searched lies in range from arr[l]
       to arr[mid], recur for arr[l..mid].
    b) Else recur for arr[mid+1..r]
4) Else (arr[mid+1..r] must be sorted)
    a) If key to be searched lies in range from arr[mid+1]
       to arr[r], recur for arr[mid+1..r].
    b) Else recur for arr[l..mid]

Below is the implementation of above idea :

C++

// Search an element in sorted and rotated
// array using single pass of Binary Search
#include <bits/stdc++.h>
using namespace std;

// Returns index of key in arr[l..h] if 
// key is present, otherwise returns -1
int search(int arr[], int l, int h, int key)
{
    if (l > h) return -1;

    int mid = (l+h)/2;
    if (arr[mid] == key) return mid;

    /* If arr[l...mid] is sorted */
    if (arr[l] <= arr[mid])
    {
        /* As this subarray is sorted, we can quickly
        check if key lies in half or other half */
        if (key >= arr[l] && key <= arr[mid])
        return search(arr, l, mid-1, key);

        return search(arr, mid+1, h, key);
    }

    /* If arr[l..mid] is not sorted, then arr[mid... r]
    must be sorted*/
    if (key >= arr[mid] && key <= arr[h])
        return search(arr, mid+1, h, key);

    return search(arr, l, mid-1, key);
}

// Driver program
int main()
{
    int arr[] = {4, 5, 6, 7, 8, 9, 1, 2, 3};
    int n = sizeof(arr)/sizeof(arr[0]);
    int key = 6;
    int i = search(arr, 0, n-1, key);

    if (i != -1)
    cout << "Index: " << i << endl;
    else
    cout << "Key not found";
}

Java

/* Java program to search an element in 
   sorted and rotated array using
   single pass of Binary Search*/
 
class Main
{
    // Returns index of key in arr[l..h] 
    // if key is present, otherwise returns -1
    static int search(int arr[], int l, int h, int key)
    {
        if (l > h) 
            return -1;
      
        int mid = (l+h)/2;
        if (arr[mid] == key)
            return mid;
      
        /* If arr[l...mid] is sorted */
        if (arr[l] <= arr[mid])
        {
            /* As this subarray is sorted, we 
               can quickly check if key lies in 
               half or other half */
            if (key >= arr[l] && key <= arr[mid])
               return search(arr, l, mid-1, key);
      
            return search(arr, mid+1, h, key);
        }
      
        /* If arr[l..mid] is not sorted, 
           then arr[mid... r] must be sorted*/
        if (key >= arr[mid] && key <= arr[h])
            return search(arr, mid+1, h, key);
      
        return search(arr, l, mid-1, key);
    }
  
    //main function
    public static void main(String args[])
    {
        int arr[] = {4, 5, 6, 7, 8, 9, 1, 2, 3};
        int n = arr.length;
        int key = 6;
        int i = search(arr, 0, n-1, key);
        if (i != -1) 
            System.out.println("Index: " + i);
        else
            System.out.println("Key not found");
    }
}

Python3

# Search an element in sorted and rotated array using
# single pass of Binary Search

# Returns index of key in arr[l..h] if key is present,
# otherwise returns -1
def search (arr, l, h, key):
    if l > h:
        return -1
    
    mid = (l + h) // 2
    if arr[mid] == key:
        return mid

    # If arr[l...mid] is sorted 
    if arr[l] <= arr[mid]:

        # As this subarray is sorted, we can quickly
        # check if key lies in half or other half 
        if key >= arr[l] and key <= arr[mid]:
            return search(arr, l, mid-1, key)
        return search(arr, mid+1, h, key)

    # If arr[l..mid] is not sorted, then arr[mid... r]
    # must be sorted
    if key >= arr[mid] and key <= arr[h]:
        return search(a, mid+1, h, key)
    return search(arr, l, mid-1, key)

# Driver program
arr = [4, 5, 6, 7, 8, 9, 1, 2, 3]
key = 6
i = search(arr, 0, len(arr)-1, key)
if i != -1:
    print ("Index: %d"%i)
else:
    print ("Key not found")

# This code is contributed by Shreyanshi Arun

PHP


<?php
// Search an element in sorted and rotated
// array using single pass of Binary Search

// Returns index of key in arr[l..h] if 
// key is present, otherwise returns -1
function search($arr, $l, $h, $key)
{
    if ($l > $h) return -1;

    $mid = ($l + $h) / 2;
    if ($arr[$mid] == $key)
        return $mid;

    /* If arr[l...mid] is sorted */
    if ($arr[$l] <= $arr[$mid])
    {
        
        /* As this subarray is 
           sorted, we can quickly
           check if key lies in 
           half or other half */
        if ($key >= $arr[$l] && 
            $key <= $arr[$mid])
                return search($arr, $l, 
                       $mid - 1, $key);

        return search($arr, $mid + 1,
                           $h, $key);
    }

    /* If arr[l..mid] is not 
       sorted, then arr[mid... r]
       must be sorted*/
    if ($key >= $arr[$mid] && 
          $key <= $arr[$h])
        return search($arr, $mid + 1, 
                            $h, $key);

    return search($arr, $l, 
             $mid-1, $key);
}

    // Driver Code
    $arr = array(4, 5, 6, 7, 8, 9, 1, 2, 3);
    $n = sizeof($arr);
    $key = 6;
    $i = search($arr, 0, $n-1, $key);

    if ($i != -1)
        echo "Index: ", floor($i) ," \n";
    else
        echo "Key not found";

// This code is contributed by ajit
?>

Output:

Index: 2

Thanks to Gaurav Ahirwar for suggesting above solution.

How to handle duplicates?
It doesn’t look possible to search in O(Logn) time in all cases when duplicates are allowed. For example consider searching 0 in {2, 2, 2, 2, 2, 2, 2, 2, 0, 2} and {2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}. It doesn’t look possible to decide whether to recur for left half or right half by doing constant number of comparisons at the middle.