Find an element in Bitonic array
Last Updated :
14 Jul, 2022
Given a bitonic sequence of n distinct elements, and an integer x, the task is to write a program to find given element x in the bitonic sequence in O(log n) time.
A Bitonic Sequence is a sequence of numbers that is first strictly increasing then after a point decreasing.
Examples:
Input : arr[] = {-3, 9, 18, 20, 17, 5, 1}, key = 20
Output : Found at index 3
Input : arr[] = {5, 6, 7, 8, 9, 10, 3, 2, 1}, key = 30
Output : Not Found
Naive Approach: A simple solution is to do a linear search. The time complexity of this solution would be O(n).
Efficient Approach: 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 a given sequence.
- If the element to be searched is greater than the maximum element return -1,
- else search the element in both halves.
Below is the step by step algorithm on how to do this.
- 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.
- If the element to be searched is equal to the element at the bitonic point then print the index of the bitonic point.
- If the element to be searched is greater than the element at a bitonic point then the element does not exist in the array.
- If the element to be searched is less than the element at a bitonic point then search for the element in both halves of the array using binary search.
Below is the implementation of the above idea:
C++
// C++ code to search key in bitonic array
#include <iostream>
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;
int bitonicPoint = 0;
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])
{
bitonicPoint = findBitonicPoint(arr, n, mid, r);
}
else if (arr[mid] < arr[mid - 1]
&& arr[mid] > arr[mid + 1])
{
bitonicPoint = findBitonicPoint(arr, n, l, mid);
}
return bitonicPoint;
}
// 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[0]);
l = 0;
r = n - 1;
int index;
// Function call
index = findBitonicPoint(arr, n, l, r);
int x = searchBitonic(arr, n, key, index);
if (x == -1)
cout << "Element Not Found" << endl;
else
cout << "Element Found at index " << x << endl;
return 0;
}
// Java code to search key in bitonic array
public 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;
int bitonicPoint = 0;
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])
{
bitonicPoint = findBitonicPoint(arr, n, mid, r);
}
else {
if (arr[mid] < arr[mid - 1]
&& arr[mid] > arr[mid + 1])
{
bitonicPoint = findBitonicPoint(arr, n, l, mid);
}
}
}
return bitonicPoint;
}
// 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
public static void main(String args[])
{
int arr[] = { -8, 1, 2, 3, 4, 5, -2, -3 };
int key = 5;
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*/
# Python code to search key in bitonic array
# Function for binary search in ascending part
def ascendingBinarySearch(arr, low, high, key):
while low <= high:
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
def descendingBinarySearch(arr, low, high, key):
while low <= high:
mid = low + (high - low) // 2
if arr[mid] == key:
return mid
if arr[mid] < key:
high = mid - 1
else:
low = mid + 1
return -1
# Find bitonic point
def findBitonicPoint(arr, n, l, r):
bitonicPoint = 0
mid = (r + l) // 2
if arr[mid] > arr[mid-1] and arr[mid] > arr[mid+1]:
return mid
elif arr[mid] > arr[mid-1] and arr[mid] < arr[mid+1]:
bitonicPoint = findBitonicPoint(arr, n, mid, r)
else:
bitonicPoint = finsBitonicPoint(arr, n, l, mid)
return bitonicPoint
# Function to search key in bitonic array
def searchBitonic(arr, n, key, index):
if key > arr[index]:
return -1
elif key == arr[index]:
return index
else:
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
def main():
arr = [-8, 1, 2, 3, 4, 5, -2, -3]
key = 1
n = len(arr)
l = 0
r = n - 1
# Function call
index = findBitonicPoint(arr, n, l, r)
x = searchBitonic(arr, n, key, index)
if x == -1:
print("Element Not Found")
else:
print("Element Found at index", x)
main()
# This code is contributed by stutipathak31jan
// 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;
int bitonicPoint = 0;
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]) {
bitonicPoint = findBitonicPoint(arr, n, mid, r);
}
else
{
if (arr[mid] < arr[mid - 1]
&& arr[mid] > arr[mid + 1]) {
bitonicPoint = findBitonicPoint(arr, n, l, mid);
}
}
}
return bitonicPoint;
}
// 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
<script>
// JavaScript code to search key in bitonic array
// Function for binary search in ascending part
function ascendingBinarySearch(arr, low, high, key) {
while (low <= high) {
let mid = Math.floor(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
function descendingBinarySearch(arr, low, high, key) {
while (low <= high) {
let mid = Math.floor(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
function findBitonicPoint(arr, n, l, r) {
let mid;
let bitonicPoint = 0;
mid = Math.floor((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]) {
bitonicPoint = findBitonicPoint(arr, n, mid, r);
}
else if (arr[mid] < arr[mid - 1]
&& arr[mid] > arr[mid + 1]) {
bitonicPoint = findBitonicPoint(arr, n, l, mid);
}
return bitonicPoint;
}
// Function to search key in
// bitonic array
function searchBitonic(arr, n, key, index) {
if (key > arr[index])
return -1;
else if (key == arr[index])
return index;
else {
let 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
let arr = [-8, 1, 2, 3, 4, 5, -2, -3];
let key = 1;
let n, l, r;
n = arr.length;
l = 0;
r = n - 1;
let index;
// Function call
index = findBitonicPoint(arr, n, l, r);
let x = searchBitonic(arr, n, key, index);
if (x == -1)
document.write("Element Not Found" + "<br>");
else
document.write("Element Found at index " + x + "<br>");
</script>
OutputElement Found at index 1
Time complexity: O(log n)
Auxiliary Space: O(1)
Similar Reads
Find element with the maximum set bits in an array Given an array arr[]. The task is to find an element from arr[] which has the maximum count of set bits.Examples: Input: arr[] = {10, 100, 1000, 10000} Output: 1000 Binary(10) = 1010 (2 set bits) Binary(100) = 1100100 (3 set bits) Binary(1000) = 1111101000 (6 set bits) Binary(10000) = 10011100010000
5 min read
Bitwise AND of all the elements of array Given an array, arr[] of N integers, the task is to find out the bitwise AND(&) of all the elements of the array. Examples: Input: arr[] = {1, 3, 5, 9, 11} Output: 1 Input: arr[] = {3, 7, 11, 19, 11} Output: 3 Approach: The idea is to traverse all the array elements and compute the bitwise AND f
4 min read
Search for an element in a Mountain Array Given a mountain array arr[] and an integer X, the task is to find the smallest index of X in the given array. If no such index is found, print -1. Examples: Input: arr = {1, 2, 3, 4, 5, 3, 1}, X = 3Output: 2Explanation: The smallest index of X(= 3) in the array is 2. Therefore, the required output
14 min read
Find Duplicates of array using bit array You have an array of N numbers, where N is at most 32,000. The array may have duplicate entries and you do not know what N is. With only 4 Kilobytes of memory available, how would print all duplicate elements in the array ?. Examples: Input : arr[] = {1, 5, 1, 10, 12, 10} Output : 1 10 1 and 10 appe
8 min read
Convert an array into Bitonic array by right shifting array elements Giver an array arr[] consisting of N integers, the task is to perform right shift operations on array elements to convert the given array into a bitonic array. Examples: Input: arr[] = {7, 3, 4, 5, 3}Output: 56 96 128 80 48Explanation:Perform the operation on the array elements as: 7 ? 00000111 ? 3
11 min read