Find the longest palindromic DNA sub-sequence that has the most mutations on it

Question

I've been trying to do a Dynamic Programming assignment for university but I had no success so far.

The problem:

Given a DNA string and a list of mutation locations (for exemple, pieces 0 and 2 are mutations), find the longest palindromic sub-sequence that contains the most mutations on it.

Input: a string S with 0 to 2000 chars; an integer N such that 0<=N<=|S| and N positions (numbers from 0 to |S|) of mutations.

Output: an integer representing the size of the longest palindromic sub-sequence containing the maximum number of mutations.

Examples:

Input: CAGACAT 0

Output: 5

Input: GATTACA 1 0

Output: 1

Input: GATTACA 3 0 4 5

Output: 3

Input: TATACTATA 2 4 8

Output: 7

We have to code it in C, but what I really need are ideas, any language or pseudo-code is good to me.

My code to find the LPS (in C)

int find_lps(char *input)
{
    int len = strlen(input), i, cur_len;
    int c[len][len];

    for (i = 0; i < len; i++)
        c[i][i] = 1;

    for (cur_len = 1; cur_len < len; cur_len++) {
        for (i = 0; i < len - cur_len; i++) {
            int j = i + cur_len;

            if (input[i] == input[j]) {
                c[i][j] = c[i + 1][j - 1] + 2;
            } else {
                c[i][j] = max(c[i + 1][j], c[i][j - 1]);
            }
        }
    }

    return c[0][len - 1];
}

What I tried to do for the mutations:

1- Creating an array of places where the LPS is changed. That doesn't work, and really, I have no idea of what to do.

More details about the problem: In a situation where you have n palindromic subsequences, both of them with the same size of mutations inside, I need the longest of them. Given that you have n palindromic subsequences with X mutations, (we have M mutations), I need the longest palindromic subsequence of X mutations, considering you don't have a palindromic subsequence with M mutations. If you do, then you should choose the other subsequence, even if it's shorter. So, first criteria: most mutations in a palindromic subsequence. If we have the same amount, then the longest of the subsequences.

Any help is appreciated, thank you.

Answer 1

Lets define C[i][j] to store 2 values:

1- The length of the longest palindromic sub-sequence in the sub-string S(i,j) that contains the most mutations in it, and lets denote it by C[i][j].len

2- The number of mutations in the longest palindromic sub-sequence in the sub-string S(i,j) that contains the most mutations in it, and lets denote it by C[i][j].ms

Then the result of the problem would be C[0][|S|-1].len

Note: m[i] = 1 means the character s[i] is a mutation, otherwise m[i] = 0

Here is the full code written in c++:

#include <iostream>
#include <string>
using namespace std;


string s;
int m[2001];

struct Node {
    int ms;//number of mutations
    int len;

    Node() {
        ms = len = 0;
    }

    Node(int v1,int v2) {
        ms = v1;
        len = v2;
    }
};

Node C[2001][2001];


Node getBestNode(Node n1, Node n2) {
    if (n1.ms > n2.ms)
        return n1;

    if (n1.ms <  n2.ms)
        return n2;

    if (n1.len > n2.len)
        return n1;

    if (n1.len < n2.len)
        return n2;  

    return n1;
}

void init() {
    for (int i = 0; i < 2001; i++) {
        m[i] = 0;
        for (int j = 0; j < 2001; j++)  C[i][j] = Node(0,0);
    }
}

void solve() {
    int len = s.length();

    // initializing the ranges of length = 1
    for (int i = 0; i < len; i++)
        C[i][i] = Node( m[i],1 );

    // initializing the ranges of length = 2
    for (int i = 0; i < len - 1; i++) 
        if (s[i] == s[i + 1])
            C[i][i + 1] = Node(m[i] + m[i + 1],2);
        else if (m[i] || m[i + 1])
                C[i][i + 1] = Node(1,1) ;

    // for ranges of length >= 3
    for (int cur_len = 3; cur_len <= len; cur_len++)
        for (int i = 0; i <= len - cur_len; i++) {

            int j = i + cur_len - 1;

            C[i][j] = getBestNode(C[i + 1][j], C[i][j-1]);

            if (s[i] == s[j]) {
                Node nn = Node(
                    C[i + 1][j - 1].ms + m[i] + m[j] , 
                    C[i + 1][j - 1].len + 2
                );

                C[i][j] = getBestNode(C[i][j], nn);
            }
        }
}

int main() {

    int n;  
    cin >> s >> n;

    init();//initializing the arrays with zeros

    for (int i = 0; i < n; i++) {
        int x;  cin >> x;
        m[x] = 1;
    }

    solve();

    cout << C[0][s.length()-1].len << endl; 
    return 0;
}

The function getBestNode() is returning the best of 2 solutions by considering the number of mutations then the length of the sub-sequence.

Note: The code can be shorter, but I made it this way for clarity.