Minimum removals required to make a given array Bitonic

Given an array arr[] of size N, the task is to find the minimum number of array elements required to be removed from the array such that the given array is converted to a bitonic array.

Examples:

Input: arr[] = { 2, 1, 1, 5, 6, 2, 3, 1 } 
Output: 3 
Explanation:
Removing arr[0], arr[1] and arr[5] modifies arr[] to { 1, 5, 6, 3, 1 } 
Since the array elements follow an increasing order followed by a decreasing order, the required output is 3.

Input: arr[] = { 1, 3, 1 } 
Output: 0 
Explanation:
The given array is already a bitonic array. Therefore, the required output is 3.

Approach: The problem can be solved based on the concept of the longest increasing subsequence problem. Follow the steps below to solve the problem:

• Initialize a variable, say left[], such that left[i] stores the length of the longest increasing subsequence up to the i^th index.
• Initialize a variable, say right[], such that right[i] stores the length of the longest decreasing subsequence over the range [i, N].
• Traverse left[] and right[] array using variable i and find the maximum value of (left[i] + right[i] – 1) and store it in a variable, say maxLen.
• Finally, print the value of N – maxLen.

Below is the implementation of the above approach:

C++14

// C++ program to implement
// the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to count minimum array elements
// required to be removed to make an array bitonic
void min_element_removal(int arr[], int N)
{
    // left[i]: Stores the length
    // of LIS up to i-th index
    vector<int> left(N, 1);

    // right[i]: Stores the length
    // of decreasing subsequence
    // over the range [i, N]
    vector<int> right(N, 1);

    // Calculate the length of LIS
    // up to i-th index
    for (int i = 1; i < N; i++) {

        // Traverse the array
        // upto i-th index
        for (int j = 0; j < i; j++) {

            // If arr[j] is less than arr[i]
            if (arr[j] < arr[i]) {

                // Update left[i]
                left[i] = max(left[i],
                              left[j] + 1);
            }
        }
    }

    // Calculate the length of decreasing
    // subsequence over the range [i, N]
    for (int i = N - 2; i >= 0;
         i--) {

        // Traverse right[] array
        for (int j = N - 1; j > i;
             j--) {

            // If arr[i] is greater
            // than arr[j]
            if (arr[i] > arr[j]) {

                // Update right[i]
                right[i] = max(right[i],
                               right[j] + 1);
            }
        }
    }

    // Stores length of the
    // longest bitonic array
    int maxLen = 0;

    // Traverse left[] and right[] array
    for (int i = 1; i < N - 1; i++) {

        // Update maxLen
        maxLen = max(maxLen, left[i] + right[i] - 1);
    }

    cout << (N - maxLen) << "\n";
}

// Function to print minimum removals
// required to make given array bitonic
void makeBitonic(int arr[], int N)
{
    if (N == 1) {
        cout << "0" << endl;
        return;
    }

    if (N == 2) {
        if (arr[0] != arr[1])
            cout << "0" << endl;
        else
            cout << "1" << endl;

        return;
    }

    min_element_removal(arr, N);
}

// Driver Code
int main()
{
    int arr[] = { 2, 1, 1, 5, 6, 2, 3, 1 };
    int N = sizeof(arr) / sizeof(arr[0]);

    makeBitonic(arr, N);

    return 0;
}

C

// C program to implement
// the above approach
#include <stdio.h>
#define max(a,b) ((a) > (b) ? (a) : (b)) //defining max

// Function to count minimum array elements
// required to be removed to make an array bitonic
void min_element_removal(int arr[], int N)
{

  // left[i]: Stores the length
  // of LIS up to i-th index
  int left[N];
  for (int i = 0; i < N; i++)
    left[i] = 1;

  // right[i]: Stores the length
  // of decreasing subsequence
  // over the range [i, N]
  int right[N];
  for (int i = 0; i < N; i++)
    right[i] = 1;

  // Calculate the length of LIS
  // up to i-th index
  for (int i = 1; i < N; i++) {

    // Traverse the array
    // upto i-th index
    for (int j = 0; j < i; j++) {

      // If arr[j] is less than arr[i]
      if (arr[j] < arr[i]) {

        // Update left[i]
        left[i] = max(left[i], left[j] + 1);
      }
    }
  }

  // Calculate the length of decreasing
  // subsequence over the range [i, N]
  for (int i = N - 2; i >= 0;
       i--) {

    // Traverse right[] array
    for (int j = N - 1; j > i;
         j--) {

      // If arr[i] is greater
      // than arr[j]
      if (arr[i] > arr[j]) {

        // Update right[i]
        right[i] = max(right[i],  right[j] + 1);
      }
    }
  }

  // Stores length of the
  // longest bitonic array
  int maxLen = 0;

  // Traverse left[] and right[] array
  for (int i = 1; i < N - 1; i++) {

    // Update maxLen
    maxLen = max(maxLen, left[i] + right[i] - 1);
  }

  printf("%d\n", (N - maxLen));
}

// Function to print minimum removals
// required to make given array bitonic
void makeBitonic(int arr[], int N)
{
  if (N == 1) {
    printf("0\n");
    return;
  }

  if (N == 2) {
    if (arr[0] != arr[1])
      printf("0\n");
    else
      printf("1\n");

    return;
  }

  min_element_removal(arr, N);
}

// Driver Code
int main()
{
  int arr[] = { 2, 1, 1, 5, 6, 2, 3, 1 };
  int N = sizeof(arr) / sizeof(arr[0]);

  makeBitonic(arr, N);

  return 0;
}

// This code is contributed by phalashi.

Java

// Java program to implement
// the above approach
class GFG {

    // Function to count minimum array elements
    // required to be removed to make an array bitonic
    static void min_element_removal(int arr[], int N)
    {
        // left[i]: Stores the length
        // of LIS up to i-th index
        int left[] = new int[N];

        for(int i = 0; i < N; i++)
            left[i] = 1;

        // right[i]: Stores the length
        // of decreasing subsequence
        // over the range [i, N]
        int right[] = new int[N];

        for(int i = 0; i < N; i++)
            right[i] = 1;

        // Calculate the length of LIS
        // up to i-th index
        for (int i = 1; i < N; i++) {

            // Traverse the array
            // upto i-th index
            for (int j = 0; j < i; j++) {

                // If arr[j] is less than arr[i]
                if (arr[j] < arr[i]) {

                    // Update left[i]
                    left[i] = Math.max(left[i],
                                  left[j] + 1);
                }
            }
        }

        // Calculate the length of decreasing
        // subsequence over the range [i, N]
        for (int i = N - 2; i >= 0;
             i--) {

            // Traverse right[] array
            for (int j = N - 1; j > i;
                 j--) {

                // If arr[i] is greater
                // than arr[j]
                if (arr[i] > arr[j]) {

                    // Update right[i]
                    right[i] = Math.max(right[i],
                                   right[j] + 1);
                }
            }
        }

        // Stores length of the
        // longest bitonic array
        int maxLen = 0;

        // Traverse left[] and right[] array
        for (int i = 1; i < N - 1; i++) {

            // Update maxLen
            maxLen = Math.max(maxLen, left[i] + right[i] - 1);
        }

        System.out.println(N - maxLen);
    }

    // Function to print minimum removals
    // required to make given array bitonic
    static void makeBitonic(int arr[], int N)
    {
        if (N == 1) {
            System.out.println("0");
            return;
        }

        if (N == 2) {
            if (arr[0] != arr[1])
                System.out.println("0");
            else
                System.out.println("1");

            return;
        }

        min_element_removal(arr, N);
    }

    // Driver Code
    public static void main (String[] args) {

        int arr[] = { 2, 1, 1, 5, 6, 2, 3, 1 };

        int N = arr.length;

        makeBitonic(arr, N);
    }
}

// This code is contributed by AnkitRai01

Python3

# Python3 program to implement
# the above approach

# Function to count minimum array elements
# required to be removed to make an array bitonic
def min_element_removal(arr, N):

    # left[i]: Stores the length
    # of LIS up to i-th index
    left = [1] * N

    # right[i]: Stores the length
    # of decreasing subsequence
    # over the range [i, N]
    right = [1] * (N)

    #Calculate the length of LIS
    #up to i-th index
    for i in range(1, N):

        #Traverse the array
        #upto i-th index
        for j in range(i):

            #If arr[j] is less than arr[i]
            if (arr[j] < arr[i]):

                #Update left[i]
                left[i] = max(left[i], left[j] + 1)

    # Calculate the length of decreasing
    # subsequence over the range [i, N]
    for i in range(N - 2, -1, -1):

        # Traverse right[] array
        for j in range(N - 1, i, -1):

            # If arr[i] is greater
            # than arr[j]
            if (arr[i] > arr[j]):

                # Update right[i]
                right[i] = max(right[i], right[j] + 1)

    # Stores length of the
    # longest bitonic array
    maxLen = 0

    # Traverse left[] and right[] array
    for i in range(1, N - 1):

        # Update maxLen
        maxLen = max(maxLen, left[i] + right[i] - 1)

    print((N - maxLen))

# Function to print minimum removals
# required to make given array bitonic
def makeBitonic(arr, N):
    if (N == 1):
        print("0")
        return

    if (N == 2):
        if (arr[0] != arr[1]):
            print("0")
        else:
            print("1")

        return

    min_element_removal(arr, N)

# Driver Code
if __name__ == '__main__':
    arr=[2, 1, 1, 5, 6, 2, 3, 1]
    N = len(arr)

    makeBitonic(arr, N)

    # This code is contributed by mohit kumar 29

C#

// C# program to implement
// the above approach
using System;

class GFG{

// Function to count minimum array elements
// required to be removed to make an array bitonic
static void min_element_removal(int []arr, int N)
{

    // left[i]: Stores the length
    // of LIS up to i-th index
    int []left = new int[N];

    for(int i = 0; i < N; i++)
        left[i] = 1;

    // right[i]: Stores the length
    // of decreasing subsequence
    // over the range [i, N]
    int []right = new int[N];

    for(int i = 0; i < N; i++)
        right[i] = 1;

    // Calculate the length of LIS
    // up to i-th index
    for(int i = 1; i < N; i++)
    {

        // Traverse the array
        // upto i-th index
        for(int j = 0; j < i; j++)
        {

            // If arr[j] is less than arr[i]
            if (arr[j] < arr[i])
            {

                // Update left[i]
                left[i] = Math.Max(left[i],
                                   left[j] + 1);
            }
        }
    }

    // Calculate the length of decreasing
    // subsequence over the range [i, N]
    for(int i = N - 2; i >= 0; i--)
    {

        // Traverse right[] array
        for(int j = N - 1; j > i; j--)
        {

            // If arr[i] is greater
            // than arr[j]
            if (arr[i] > arr[j])
            {

                // Update right[i]
                right[i] = Math.Max(right[i],
                                    right[j] + 1);
            }
        }
    }

    // Stores length of the
    // longest bitonic array
    int maxLen = 0;

    // Traverse left[] and right[] array
    for(int i = 1; i < N - 1; i++)
    {

        // Update maxLen
        maxLen = Math.Max(maxLen, left[i] +
                                 right[i] - 1);
    }
    Console.WriteLine(N - maxLen);
}

// Function to print minimum removals
// required to make given array bitonic
static void makeBitonic(int []arr, int N)
{
    if (N == 1)
    {
        Console.WriteLine("0");
        return;
    }

    if (N == 2)
    {
        if (arr[0] != arr[1])
            Console.WriteLine("0");
        else
            Console.WriteLine("1");

        return;
    }
    min_element_removal(arr, N);
}

// Driver Code
public static void Main(String[] args)
{
    int []arr = { 2, 1, 1, 5, 6, 2, 3, 1 };
    int N = arr.Length;

    makeBitonic(arr, N);
}
}

// This code is contributed by AnkThon

Javascript

<script>

// Javascript program to implement
// the above approach

// Function to count minimum array elements
// required to be removed to make an array bitonic
function min_element_removal(arr, N)
{
    // left[i]: Stores the length
    // of LIS up to i-th index
    var left = Array(N).fill(1);

    // right[i]: Stores the length
    // of decreasing subsequence
    // over the range [i, N]
    var right = Array(N).fill(1);

    // Calculate the length of LIS
    // up to i-th index
    for (var i = 1; i < N; i++) {

        // Traverse the array
        // upto i-th index
        for (var j = 0; j < i; j++) {

            // If arr[j] is less than arr[i]
            if (arr[j] < arr[i]) {

                // Update left[i]
                left[i] = Math.max(left[i],
                              left[j] + 1);
            }
        }
    }

    // Calculate the length of decreasing
    // subsequence over the range [i, N]
    for (var i = N - 2; i >= 0;
         i--) {

        // Traverse right[] array
        for (var j = N - 1; j > i;
             j--) {

            // If arr[i] is greater
            // than arr[j]
            if (arr[i] > arr[j]) {

                // Update right[i]
                right[i] = Math.max(right[i],
                               right[j] + 1);
            }
        }
    }

    // Stores length of the
    // longest bitonic array
    var maxLen = 0;

    // Traverse left[] and right[] array
    for (var i = 1; i < N - 1; i++) {

        // Update maxLen
        maxLen = Math.max(maxLen, left[i] + right[i] - 1);
    }

    document.write((N - maxLen) + "<br>");
}

// Function to print minimum removals
// required to make given array bitonic
function makeBitonic(arr, N)
{
    if (N == 1) {
        document.write( "0" + "<br>");
        return;
    }

    if (N == 2) {
        if (arr[0] != arr[1])
            document.write( "0" + "<br>");
        else
            document.write( "1" + "<br>");

        return;
    }

    min_element_removal(arr, N);
}

// Driver Code
var arr = [2, 1, 1, 5, 6, 2, 3, 1];
var N = arr.length;
makeBitonic(arr, N);

// This code is contributed by rutvik_56.
</script>

Output:

3

Time Complexity: O(N²)
Auxiliary Space: O(N)