Count N-length arrays made from first M natural numbers whose subarrays can be made palindromic by replacing less than half of its elements
Given two integer N and M, the task is to find the count of arrays of size N with elements from the range [1, M] in which all subarrays of length greater than 1 can be made palindromic by replacing less than half of its elements i.e., floor(length/2).
Examples:
Input: N = 2, M = 3
Output: 6
Explanation:
There are 9 arrays possible of length 2 using values 1 to 3 i.e. [1, 1], [1, 2], [1, 3], [2, 1][2, 2], [2, 3], [3, 1], [3, 2], [3, 3].
All of these arrays except [1, 1], [2, 2] and [3, 3] have subarrays of length greater than 1 which requires 1 operation to make them palindrome. So the required answer is 9 – 3 = 6.
Input: N = 5, M = 10
Output: 30240
Approach: The problem can be solved based on the following observations:
- It is possible that the maximum permissible number of operations required to make an array a palindrome is floor(size(array)/2).
- It can be observed that by choosing a subarray, starting and ending with the same value, the number of operations needed to make it a palindrome will be less than floor(size of subarray)/2.
- Therefore, the task is reduced to finding the number of arrays of size N using integer values in the range [1, M], which do not contain any duplicate elements, which can be easily done by finding the permutation of M with N i.e. MpN, which is equal to M * (M – 1) * (M – 2) * … * (M – N + 1).
Follow the steps below to solve the problem:
- Initialize an integer variable, say ans = 1.
- Traverse from i = 0 to N – 1 and update ans as ans = ans * (M-i)
- Print ans as the answer.
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
void noOfArraysPossible(ll N, ll M)
{
ll ans = 1;
for (ll i = 0; i < N; ++i) {
ans = ans * (M - i);
}
cout << ans;
}
int main()
{
ll N = 2, M = 3;
noOfArraysPossible(N, M);
return 0;
}
|
Java
class GFG
{
static void noOfArraysPossible( int N, int M)
{
int ans = 1 ;
for ( int i = 0 ; i < N; ++i)
{
ans = ans * (M - i);
}
System.out.print(ans);
}
public static void main(String[] args)
{
int N = 2 , M = 3 ;
noOfArraysPossible(N, M);
}
}
|
Python3
def noOfArraysPossible(N, M):
ans = 1
for i in range (N):
ans = ans * (M - i)
print (ans)
if __name__ = = "__main__" :
N = 2
M = 3
noOfArraysPossible(N, M)
|
C#
using System;
class GFG{
static void noOfArraysPossible( int N, int M)
{
int ans = 1;
for ( int i = 0; i < N; ++i)
{
ans = ans * (M - i);
}
Console.Write(ans);
}
public static void Main()
{
int N = 2, M = 3;
noOfArraysPossible(N, M);
}
}
|
Javascript
<script>
function noOfArraysPossible(N , M)
{
var ans = 1;
for (i = 0; i < N; ++i) {
ans = ans * (M - i);
}
document.write(ans);
}
var N = 2, M = 3;
noOfArraysPossible(N, M);
</script>
|
Time Complexity: O(N)
Auxiliary Space: O(1)
Last Updated :
02 Dec, 2021
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