Required Substrings problem on CSES

Revision en1, by harsh-apcr, 2023-11-03 12:55:37

Can someone explain how to solve Required Substring problem in CSES : https://cses.fi/problemset/task/1112

Your task is to calculate the number of strings of length n having a given pattern of length m as their substring. All strings consist of characters A–Z.

My approach :

I was thinking of some dp approach and instead of counting the required quantity directly, I'd count the complement and subtract from $26^n$

But, I am stuck with this approach, I thought of the following recursion

$dp(i, j) = # of strings of length i not containing s, whose suffix of length i is a prefix of s (i.e. s_1\ldots s_j)$

$dp(i, j) = \sum_{j'\in S} dp(i-1, j') + dp(i-1, j-1)$, where $S = {j' \mid s_1\ldots s_{j'} has a prefix of length j-1 which is also its suffix}$ and then finally compute : $dp(i) = 26*dp(i-1)-dp(i-1,m-1)$ (supposed to count strings of length i not containing s)

but I cannot figure out how to instantiate the base case $dp(i, 0)$ for $i > m$ or $dp(i, 1)$

Any help is appreciated, thank you

#### History

Revisions

Rev. Lang. By When Δ Comment
en3 harsh-apcr 2023-11-03 13:00:33 47 Tiny change: 're $S = \{j' \mid s_' -> 're$S = \{ j' \mid s_'
en2 harsh-apcr 2023-11-03 12:56:53 10
en1 harsh-apcr 2023-11-03 12:55:37 1064 Initial revision (published)