# Find an element in Bitonic array

Given a bitonic sequence of n distinct elements, write a program to find a given element x in the bitonic sequence in O(log n) time. A Bitonic Sequence is a sequence of numbers which is first strictly increasing then after a point strictly decreasing.

Examples:

```Input :  arr[] = {-3, 9, 8, 20, 17, 5, 1};
key = 20
Output : Found at index 3

Input :  arr[] = {5, 6, 7, 8, 9, 10, 1, 2, 3};
key = 30
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A simple solution is to do linear search. Time complexity of this solution would be O(n).

An efficient solution is based on Binary Search. The idea is to find the bitonic point k which is the index of the maximum element of given sequence. If the element to be searched is greater than maximum element return -1, else search the element in both halves. Below is the step by step algorithm on how to do this.

1. Find the bitonic point in the given array, i.e the maximum element in the given bitonic array. This can be done in log(n) time by modifying the binary search algorithm. You can refer to this post on how to do this.
2. If the element to be searched is equal to the element at bitonic point then print the index of bitonic point.
3. If the element to be searched is greater than element at bitonic point then element does not exist in the array.
4. If the element to be searched is less than element at bitonic point then search for element in both half of the array using binary search.

Below is the implementation of above idea:

## C++

 `// CPP code to search key in bitonic array ` `#include ` ` `  `using` `namespace` `std; ` ` `  `// Function for binary search in ascending part ` `int` `ascendingBinarySearch(``int` `arr[], ``int` `low, ` `                           ``int` `high, ``int` `key) ` `{ ` `   ``while` `(low <= high) ` `   ``{ ` `       ``int` `mid = low + (high - low) / 2; ` `       ``if` `(arr[mid] == key) ` `           ``return` `mid; ` `       ``if` `(arr[mid] > key) ` `           ``high = mid - 1; ` `       ``else` `           ``low = mid + 1; ` `   ``} ` `    ``return` `-1; ` `} ` ` `  `// Function for binary search in descending part of array ` `int` `descendingBinarySearch(``int` `arr[], ``int` `low,  ` `                            ``int` `high, ``int` `key) ` `{ ` `   ``while` `(low <= high) ` `   ``{ ` `       ``int` `mid = low + (high - low) / 2; ` `       ``if` `(arr[mid] == key) ` `           ``return` `mid; ` `       ``if` `(arr[mid] < key) ` `           ``high = mid - 1; ` `       ``else` `           ``low = mid + 1; ` `   ``} ` `    ``return` `-1; ` `} ` `  `  `// finding bitonic point ` `int` `findBitonicPoint(``int` `arr[] ,``int` `n, ``int` `l, ``int` `r ) ` `{ ` `    ``int` `mid; ` `     `  `    ``mid = (r + l) / 2; ` `    ``if``(arr[mid] > arr[mid-1] && arr[mid] > arr[mid + 1]) ` `    ``{ ` `        ``return` `mid; ` `    ``} ` `     `  `    ``else`  `    ``if``(arr[mid] > arr[mid - 1] && arr[mid] < arr[mid + 1]) ` `    ``{ ` `        ``findBitonicPoint(arr,n, mid , r); ` `    ``} ` ` `  `    ``else`  `    ``if``(arr[mid] < arr[mid - 1] && arr[mid] > arr[mid + 1]) ` `    ``{ ` `        ``findBitonicPoint(arr, n, l, mid); ` `    ``} ` `} ` ` `  `// Function to search key in bitonic array ` `int` `searchBitonic(``int` `arr[], ``int` `n, ``int` `key, ``int` `index) ` `{ ` `    ``if``(key > arr[index]) ` `        ``return` `-1; ` `     `  `    ``else` `if``(key == arr[index]) ` `        ``return` `index; ` `     `  `    ``else` `    ``{    ``int` `temp = ascendingBinarySearch(arr, 0, index - 1, key); ` `        ``if` `(temp != -1) ` `        ``{ ` `             ``return` `temp; ` `        ``} ` `         `  `        ``// Search in right of k ` `        ``return` `descendingBinarySearch(arr, index + 1, n - 1, key); ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() { ` `    ``int` `arr[] = {-8, 1, 2, 3, 4, 5, -2, -3}; ` `    ``int` `key = 1; ` `    ``int` `n ,l, r; ` `    ``n = ``sizeof``(arr)/``sizeof``(arr); ` `    ``l = 0; ` `    ``r = n - 1; ` `    ``int` `index; ` `    ``index = findBitonicPoint(arr, n, l, r); ` `      `  `    ``int` `x = searchBitonic(arr, n, key, index); ` `     `  `    ``if` `(x == -1) ` `       ``cout << ``"Element Not Found"``<

## Java

 `// Java code to search key in bitonic array ` ` `  `public` `class` `GFG { ` ` `  `// CPP code to search key in bitonic array  ` `// Function for binary search in ascending part  ` `    ``static` `int` `ascendingBinarySearch(``int` `arr[], ``int` `low, ` `            ``int` `high, ``int` `key) { ` `        ``while` `(low <= high) { ` `            ``int` `mid = low + (high - low) / ``2``; ` `            ``if` `(arr[mid] == key) { ` `                ``return` `mid; ` `            ``} ` `            ``if` `(arr[mid] > key) { ` `                ``high = mid - ``1``; ` `            ``} ``else` `{ ` `                ``low = mid + ``1``; ` `            ``} ` `        ``} ` `        ``return` `-``1``; ` `    ``} ` ` `  `// Function for binary search in descending part of array  ` `    ``static` `int` `descendingBinarySearch(``int` `arr[], ``int` `low, ` `            ``int` `high, ``int` `key) { ` `        ``while` `(low <= high) { ` `            ``int` `mid = low + (high - low) / ``2``; ` `            ``if` `(arr[mid] == key) { ` `                ``return` `mid; ` `            ``} ` `            ``if` `(arr[mid] < key) { ` `                ``high = mid - ``1``; ` `            ``} ``else` `{ ` `                ``low = mid + ``1``; ` `            ``} ` `        ``} ` `        ``return` `-``1``; ` `    ``} ` ` `  `// finding bitonic point  ` `    ``static` `int` `findBitonicPoint(``int` `arr[], ``int` `n, ``int` `l, ``int` `r) { ` `        ``int` `mid; ` ` `  `        ``mid = (r + l) / ``2``; ` `        ``if` `(arr[mid] > arr[mid - ``1``] && arr[mid] > arr[mid + ``1``]) { ` `            ``return` `mid; ` `        ``} ``else` `{ ` `            ``if` `(arr[mid] > arr[mid - ``1``] && arr[mid] < arr[mid + ``1``]) { ` `                ``findBitonicPoint(arr, n, mid, r); ` `            ``} ``else` `{ ` `                ``if` `(arr[mid] < arr[mid - ``1``] && arr[mid] > arr[mid + ``1``]) { ` `                    ``findBitonicPoint(arr, n, l, mid); ` `                ``} ` `            ``} ` `        ``} ` `        ``return` `mid; ` `    ``} ` ` `  `// Function to search key in bitonic array  ` `    ``static` `int` `searchBitonic(``int` `arr[], ``int` `n, ``int` `key, ``int` `index) { ` `        ``if` `(key > arr[index]) { ` `            ``return` `-``1``; ` `        ``} ``else` `if` `(key == arr[index]) { ` `            ``return` `index; ` `        ``} ``else` `{ ` `            ``int` `temp = ascendingBinarySearch(arr, ``0``, index - ``1``, key); ` `            ``if` `(temp != -``1``) { ` `                ``return` `temp; ` `            ``} ` ` `  `            ``// Search in right of k  ` `            ``return` `descendingBinarySearch(arr, index + ``1``, n - ``1``, key); ` `        ``} ` `    ``} ` ` `  `// Driver program to test above function  ` `    ``public` `static` `void` `main(String args[]) { ` `        ``int` `arr[] = {-``8``, ``1``, ``2``, ``3``, ``4``, ``5``, -``2``, -``3``}; ` `        ``int` `key = ``1``; ` `        ``int` `n, l, r; ` `        ``n = arr.length; ` `        ``l = ``0``; ` `        ``r = n - ``1``; ` `        ``int` `index; ` `        ``index = findBitonicPoint(arr, n, l, r); ` ` `  `        ``int` `x = searchBitonic(arr, n, key, index); ` ` `  `        ``if` `(x == -``1``) { ` `            ``System.out.println(``"Element Not Found"``); ` `        ``} ``else` `{ ` `            ``System.out.println(``"Element Found at index "` `+ x); ` `        ``} ` ` `  `    ``} ` `} ` ` `  `/*This code is contributed by 29AjayKumar*/`

## C#

 `// C# code to search key in bitonic array ` `using` `System; ` ` `  `class` `GFG ` `{ ` `// Function for binary search in ascending part  ` `static` `int` `ascendingBinarySearch(``int` `[]arr, ``int` `low, ` `                                 ``int` `high, ``int` `key)  ` `{ ` `    ``while` `(low <= high)  ` `    ``{ ` `        ``int` `mid = low + (high - low) / 2; ` `        ``if` `(arr[mid] == key)  ` `        ``{ ` `            ``return` `mid; ` `        ``} ` `        ``if` `(arr[mid] > key)  ` `        ``{ ` `            ``high = mid - 1; ` `        ``} ` `        ``else` `        ``{ ` `            ``low = mid + 1; ` `        ``} ` `    ``} ` `    ``return` `-1; ` `} ` ` `  `// Function for binary search in descending part of array  ` `static` `int` `descendingBinarySearch(``int` `[]arr, ``int` `low, ` `                                  ``int` `high, ``int` `key)  ` `{ ` `    ``while` `(low <= high)  ` `    ``{ ` `        ``int` `mid = low + (high - low) / 2; ` `        ``if` `(arr[mid] == key)  ` `        ``{ ` `            ``return` `mid; ` `        ``} ` `        ``if` `(arr[mid] < key)  ` `        ``{ ` `            ``high = mid - 1; ` `        ``}  ` `        ``else`  `        ``{ ` `            ``low = mid + 1; ` `        ``} ` `    ``} ` `    ``return` `-1; ` `} ` ` `  `// finding bitonic point  ` `static` `int` `findBitonicPoint(``int` `[]arr, ``int` `n,  ` `                            ``int` `l, ``int` `r)  ` `{ ` `    ``int` `mid; ` ` `  `    ``mid = (r + l) / 2; ` `    ``if` `(arr[mid] > arr[mid - 1] &&  ` `        ``arr[mid] > arr[mid + 1]) ` `    ``{ ` `        ``return` `mid; ` `    ``}  ` `    ``else`  `    ``{ ` `        ``if` `(arr[mid] > arr[mid - 1] &&  ` `            ``arr[mid] < arr[mid + 1])  ` `        ``{ ` `            ``findBitonicPoint(arr, n, mid, r); ` `        ``} ` `        ``else`  `        ``{ ` `            ``if` `(arr[mid] < arr[mid - 1] &&  ` `                ``arr[mid] > arr[mid + 1])  ` `            ``{ ` `                ``findBitonicPoint(arr, n, l, mid); ` `            ``} ` `        ``} ` `    ``} ` `    ``return` `mid; ` `} ` ` `  `// Function to search key in bitonic array  ` `static` `int` `searchBitonic(``int` `[]arr, ``int` `n,  ` `                         ``int` `key, ``int` `index)  ` `{ ` `    ``if` `(key > arr[index])  ` `    ``{ ` `        ``return` `-1; ` `    ``}  ` `    ``else` `if` `(key == arr[index])  ` `    ``{ ` `        ``return` `index; ` `    ``} ` `    ``else`  `    ``{ ` `        ``int` `temp = ascendingBinarySearch(arr, 0,  ` `                                         ``index - 1, key); ` `        ``if` `(temp != -1)  ` `        ``{ ` `            ``return` `temp; ` `        ``} ` ` `  `        ``// Search in right of k  ` `        ``return` `descendingBinarySearch(arr, index + 1,  ` `                                      ``n - 1, key); ` `    ``} ` `} ` ` `  `// Driver Code ` `static` `public` `void` `Main () ` `{ ` `    ``int` `[]arr = {-8, 1, 2, 3, 4, 5, -2, -3}; ` `    ``int` `key = 1; ` `    ``int` `n, l, r; ` `    ``n = arr.Length; ` `    ``l = 0; ` `    ``r = n - 1; ` `    ``int` `index; ` `    ``index = findBitonicPoint(arr, n, l, r); ` ` `  `    ``int` `x = searchBitonic(arr, n, key, index); ` ` `  `    ``if` `(x == -1)  ` `    ``{ ` `        ``Console.WriteLine(``"Element Not Found"``); ` `    ``}  ` `    ``else` `    ``{ ` `        ``Console.WriteLine(``"Element Found at index "` `+ x); ` `    ``} ` `} ` `} ` ` `  `// This code is contributed by ajit `

Output:

```Element Found at index 1
```

Time complexity: O(log n)
Auxiliary Space: O(1)
This article is contributed by Vaishali Goyal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Improved By : 29AjayKumar, jit_t

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