# Minimum cost to make Longest Common Subsequence of length k

Given two string **X**, **Y** and an integer **k**. Now the task is to convert string X with minimum cost such that the Longest Common Subsequence of X and Y after conversion is of length k. The cost of conversion is calculated as XOR of old character value and new character value. Character value of ‘a’ is 0, ‘b’ is 1 and so on.

Examples:

Input : X = "abble", Y = "pie", k = 2 Output : 25 If you changed 'a' to 'z', it will cost 0 XOR 25.

The problem can be solved by slight change in Dynamic Programming problem of Longest Increasing Subsequence. Instead of two states, we maintain three states.

Note, that if k > min(n, m) then it’s impossible to attain LCS of atleast k length, else it’s always possible.

Let dp[i][j][p] stores the minimum cost to achieve LCS of length p in x[0…i] and y[0….j].

With base step as dp[i][j][0] = 0 because we can achieve LCS of 0 legth without any cost and for i < 0 or j 0 in such case).

Else there are 3 cases:

1. Convert x[i] to y[j].

2. Skip i^{th} character from x.

3. Skip j^{th} character from y.

If we convert x[i] to y[j], then cost = f(x[i]) XOR f(y[j]) will be added and LCS will decrease by 1. f(x) will return the character value of x.

Note that the minimum cost to convert a charcater ‘a’ to any character ‘c’ is always f(a) XOR f(c) because f(a) XOR f(c) <= (f(a) XOR f(b) + f(b) XOR f(c)) for all a, b, c.

If you skip i^{th} character from x then i will be decreased by 1, no cost will be added and LCS will remain the same.

If you skip j^{th} character from x then j will be decreased by 1, no cost will be added and LCS will remain the same.

Therefore,

dp[i][j][k] = min(cost + dp[i - 1][j - 1][k - 1], dp[i - 1][j][k], dp[i][j - 1][k]) The minimum cost to make the length of their LCS atleast k is dp[n - 1][m - 1][k]

.

`#include <bits/stdc++.h> ` `using` `namespace` `std; ` `const` `int` `N = 30; ` ` ` `// Return Minimum cost to make LCS of length k ` `int` `solve(` `char` `X[], ` `char` `Y[], ` `int` `l, ` `int` `r, ` ` ` `int` `k, ` `int` `dp[][N][N]) ` `{ ` ` ` `// If k is 0. ` ` ` `if` `(!k) ` ` ` `return` `0; ` ` ` ` ` `// If length become less than 0, return ` ` ` `// big number. ` ` ` `if` `(l < 0 | r < 0) ` ` ` `return` `1e9; ` ` ` ` ` `// If state already calculated. ` ` ` `if` `(dp[l][r][k] != -1) ` ` ` `return` `dp[l][r][k]; ` ` ` ` ` `// Finding the cost ` ` ` `int` `cost = (X[l] - ` `'a'` `) ^ (Y[r] - ` `'a'` `); ` ` ` ` ` `// Finding minimum cost and saving the state value ` ` ` `return` `dp[l][r][k] = min({cost + ` ` ` `solve(X, Y, l - 1, r - 1, k - 1, dp), ` ` ` `solve(X, Y, l - 1, r, k, dp), ` ` ` `solve(X, Y, l, r - 1, k, dp)}); ` `} ` ` ` `// Driven Program ` `int` `main() ` `{ ` ` ` `char` `X[] = ` `"abble"` `; ` ` ` `char` `Y[] = ` `"pie"` `; ` ` ` `int` `n = ` `strlen` `(X); ` ` ` `int` `m = ` `strlen` `(Y); ` ` ` `int` `k = 2; ` ` ` ` ` `int` `dp[N][N][N]; ` ` ` `memset` `(dp, -1, ` `sizeof` `dp); ` ` ` `int` `ans = solve(X, Y, n - 1, m - 1, k, dp); ` ` ` ` ` `cout << (ans == 1e9 ? -1 : ans) << endl; ` ` ` `return` `0; ` `} ` |

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Output:

3

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