Find an element in Bitonic array (original) (raw)

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.

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 the bitonic point then print the index of the bitonic point.
  3. If the element to be searched is greater than the element at a bitonic point then the element does not exist in the array.
  4. 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

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

// 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/

Python3

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#

// 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

JavaScript

`

Output

Element Found at index 1

Time complexity: O(log n)
Auxiliary Space: O(1)