M72BTS- Editorial

#1

Author: nmalviya1929
Editorialist: nmalviya1929

EASY-MEDIUM

DP

PROBLEM:

Find the number of ways to divide n days into groups such that each group have at least k days.

QUICK EXPLANATION:

Let dp* represent answer for i days. Then dp[n]=\sum_{i=0}^{n-k} dp*.

EXPLANATION:

Let dp* represent answer for i days.
If we’re solving for n days, in the last group we can keep days from k to n.If we select k days in last group then number of ways is dp[n-k] , if we select k+1 , then number of ways is dp[n-k-1] ans so on. Therefore dp[n]=\sum_{i=0}^{n-k} dp*.

dp[0]=1;
for( i=1 to n) {
dp*=0;
for (j= i-k to 0) {
dp*+=dp[j];
dp*%=MOD;
}
}


However, time complexity for this solution is O(n^2).
We can store cumulative sum of dp to reduce the complexity.
cum*=\sum_{j=0}^{i} dp[j]

dp[0]=1;
cum[0]=1;
for (i=1 to n) {
dp*=0;
if(i-k is greater than equal to 0)
dp*+=(cum[i-k]);
dp*%=MOD;
}
cum*=(cum[i-1]+dp*)%MOD;
}


AUTHOR’S AND TESTER’S SOLUTIONS:

Author’s solution can be found here.