Codeforces DP problem explanation

I am fairly new to dp and practicing to get better. Can anyone explain this problem from codeforces educational round. There is an editorial but still I am not convinced with the intution. Can anyone help me explain it?

https://codeforces.com/contest/1096/problem/D

Here we can memoize our solution in the DP array, let dp[i][4] be the cost of making the string easy the index till 1<=i<=N where ith character can be any character of β€œhard”.

If the ith character of our string is β€˜h’ then dp[i][0] = dp[i-1][0] + cost[i].

If the ith character of our string is β€˜a’ then dp[i][1] = we have 2 options whether to take dp[i-1][0] i.e. cost of making string easy removing only character β€˜h’ or second option is to add the current cost of β€˜a’ to the previously calculated cost of β€˜a’ i.e. dp[i-1][1] + cost[i].

Similarly, If the ith character of our string is β€˜r’ then dp[i][2] = we have 2 options whether to take dp[i-1][1] i.e. cost of making string easy removing only character β€˜a’ or second option is to add the cost of β€˜r’ to the previously calculated cost of β€˜r’ i.e. dp[i-1][2] + cost[i].

Similarly for β€˜d’ we have dp[i][3] = minimum of dp[i-1][2] // Taking only cost till first i-1 index and having character β€˜r’ or take dp[i-1][3] i.e. adding β€˜d’ cost into in the previous calculated cost of β€˜d’.

for(int i = 1; i <= n; i++){
    for(int j = 0; j < 4; j++){
        dp[i][j] = dp[i-1][j];

        if(arr[i] == 'h'){
            dp[i][0] = dp[i-1][0] + cost[i];
        }
        else if(arr[i] == 'a'){
            dp[i][1] = min(dp[i-1][0], dp[i-1][1] + cost[i]);
        }
        else if(arr[i] == 'r'){
            dp[i][2] = min(dp[i-1][1],dp[i-1][2] + cost[i]);
        }
        else if(arr[i] == 'd'){
            dp[i][3] = min(dp[i-1][2],dp[i-1][3] + cost[i]);
        }
    }
}
 dp[n][3]; //Answer to the problem
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