Minimum number of palindromes required to express N as a sum | Set 1

Given a number N, we have to find the minimum number of palindromes required to express N as a sum of them.


Input: N = 11
Output: 1
11 is itself a palindrome.

Input: N = 65
Output: 3
65 can be expressed as a sum of three palindromes (55, 9, 1).

We can use Dynamic Programming to solve this problem. The idea is to first generate all the palindromes up to N in a sorted fashion, and then we can treat this problem as a variation of the subset sum problem, where we have to find the size of the smallest subset such that its sum is N.

Below is the implementation of above approach:





// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;
// Declaring the DP table as global variable
vector<vector<long long> > dp;
// A utility for creating palindrome
int createPalindrome(int input, bool isOdd)
    int n = input;
    int palin = input;
    // checks if number of digits is odd or even
    // if odd then neglect the last digit of input in
    // finding reverse as in case of odd number of
    // digits middle element occur once
    if (isOdd)
        n /= 10;
    // Creates palindrome by just appending revers
    // of number to itself
    while (n > 0) {
        palin = palin * 10 + (n % 10);
        n /= 10;
    return palin;
// Function to generate palindromes
vector<int> generatePalindromes(int N)
    vector<int> palindromes;
    int number;
    // Run two times for odd and even length palindromes
    for (int j = 0; j < 2; j++) {
        // Creates palindrome numbers with first half as i.
        // Value of j decides whether we need an odd length
        // or even length palindrome.
        int i = 1;
        while ((number = createPalindrome(i++, j)) <= N)
    return palindromes;
// Function to find the minimum
// number of elements in a sorted
// array A[i..j] such that their sum is N
long long minimumSubsetSize(vector<int>& A, int i, int j, int N)
    if (!N)
        return 0;
    if (i > j || A[i] > N)
        return INT_MAX;
    if (dp[i][N])
        return dp[i][N];
    dp[i][N] = min(1 + minimumSubsetSize(A, i + 1, j,
                                         N - A[i]),
                   minimumSubsetSize(A, i + 1, j, N));
    return dp[i][N];
// Function to find the minimum
// number of palindromes that N
// can be expressed as a sum of
int minimumNoOfPalindromes(int N)
    // Getting the list of all palindromes upto N
    vector<int> palindromes = generatePalindromes(N);
    // Sorting the list of palindromes
    sort(palindromes.begin(), palindromes.end());
    // Initializing the DP table
    dp = vector<vector<long long> >(palindromes.size(),
                                    vector<long long>(N + 1, 0));
    // Returning the required value
    return minimumSubsetSize(palindromes, 0,
                             palindromes.size() - 1, N);
// Driver code
int main()
    int N = 65;
    cout << minimumNoOfPalindromes(N);
    return 0;




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