Hello Everyone,
Given an array of positive integers. We need to make the given array a ‘Palindrome’. The only allowed operation is”merging” (of two adjacent elements). Merging two adjacent elements means replacing them with their sum. The task is to find the minimum number of merge operations required to make the given array a ‘Palindrome’.
To make any array a palindrome, we can simply apply merge operation n-1 times where n is the size of the array (because a single-element array is always palindromic, similar to single-character string). In that case, the size of array will be reduced to 1. But in this problem, we are asked to do it in the minimum number of operations.
Example :
Input : arr[] = {15, 4, 15} Output : 0 Array is already a palindrome. So we do not need any merge operation. Input : arr[] = {1, 4, 5, 1} Output : 1 We can make given array palindrome with minimum one merging (merging 4 and 5 to make 9) Input : arr[] = {11, 14, 15, 99} Output : 3 We need to merge all elements to make a palindrome.
The expected time complexity is O(n).
Let f(i, j) be minimum merging operations to make subarray arr[i…j] a palindrome. If i == j answer is 0. We start i from 0 and j from n-1.
- If arr[i] == arr[j], then there is no need to do any merging operations at index i or index j. Our answer in this case will be f(i+1, j-1).
- Else, we need to do merging operations. Following cases arise.
- If arr[i] > arr[j], then we should do merging operation at index j. We merge index j-1 and j, and update arr[j-1] = arr[j-1] + arr[j]. Our answer in this case will be 1 + f(i, j-1).
- For the case when arr[i] < arr[j], update arr[i+1] = arr[i+1] + arr[i]. Our answer in this case will be 1 + f(i+1, j).
- Our answer will be f(0, n-1), where n is the size of array arr[].
Therefore this problem can be solved iteratively using two pointers (first pointer pointing to start of the array and second pointer pointing to the last element of the array) method and keeping count of total merging operations done till now.
Below is an implementation of the above idea.
// C++ program to find number of operations
// to make an array palindrome
#include <bits/stdc++.h>
using namespace std;
// Returns minimum number of count operations
// required to make arr[] palindrome
int findMinOps( int arr[], int n)
{
int ans = 0; // Initialize result
// Start from two corners
for ( int i=0,j=n-1; i<=j;)
{
// If corner elements are same,
// problem reduces arr[i+1..j-1]
if (arr[i] == arr[j])
{
i++;
j--;
}
// If left element is greater, then
// we merge right two elements
else if (arr[i] > arr[j])
{
// need to merge from tail.
j--;
arr[j] += arr[j+1] ;
ans++;
}
// Else we merge left two elements
else
{
i++;
arr[i] += arr[i-1];
ans++;
}
}
return ans;
}
// Driver program to test above
int main()
{
int arr[] = {1, 4, 5, 9, 1};
int n = sizeof (arr)/ sizeof (arr[0]);
cout << "Count of minimum operations is "
<< findMinOps(arr, n) << endl;
return 0;
}
Output :
Count of minimum operations is 1
The time complexity for the given program is: O(n)