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
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; } |
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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; } |
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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)); } } |
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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 = 3print("Index of the element is : ", pivotedBinarySearch(arr1, n, key)) # This is contributed by Smitha Dinesh Semwal |
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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. |
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PHP
<?php // PHP Program to search an element // in a sorted and pivoted array // Standard Binary Search function function binarySearch($arr, $low, $high, $key) { if ($high < $low) return -1; /*low + (high - low)/2;*/ $mid = floor($low + $high) / 2; if ($key == $arr[$mid]) return $mid; if ($key > $arr[$mid]) return binarySearch($arr, ($mid + 1), $high, $key); else return binarySearch($arr, $low, ($mid -1), $key); } // Function to get pivot. // For array 3, 4, 5, 6, 1, 2 // it returns 3 (index of 6) function findPivot($arr, $low, $high) { // base cases if ($high < $low) return -1; if ($high == $low) return $low; /*low + (high - low)/2;*/ $mid = ($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, $high); } // Searches an element key // in a pivoted sorted array // arr[] of size n */ function pivotedBinarySearch($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); } // Driver Code // Let us search 3 // in below array $arr1 = array(5, 6, 7, 8, 9, 10, 1, 2, 3); $n = count($arr1); $key = 3; // Function calling echo "Index of the element is : " , pivotedBinarySearch($arr1, $n, $key); // This code is contributed by anuj_67. ?> |
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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"; } |
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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"); } } |
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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 = 6i = 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 |
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C#
/* C# program to search an element in sorted and rotated array using single pass of Binary Search*/using System; class GFG { // 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() { 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) Console.WriteLine("Index: " + i); else Console.WriteLine("Key not found"); } } // This code is contributed by anuj_67. |
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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 ?> |
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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.
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