Editorial for Canada Day Contest 2021 - Bob and Canada - Olympiads Online Judge

Editorial for Canada Day Contest 2021 - Bob and Canada

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Submitting an official solution before solving the problem yourself is a bannable offence.

Author: Aaeria

Hint

In order to make a string Canadian, you need to change all the Ws in the red sections and the Rs in the white section.

Hint

Try to come up with a formula for the number of edits based on the starting point of the second and third section.

Solution

Let's say the first section (red) goes from index $1$ ~1~ to $a$ ~a~, the second (white) goes from $a+1$ ~a+1~ to $b$ ~b~, and the third (red) from $b+1$ ~b+1~ to $n$ ~n~. You can first try every combination of $a$ ~a~ and $b$ ~b~. In order to count the number of Rs and Ws in a range quickly you can use a prefix sum array $r[]$ ~r[]~ and $w[]$ ~w[]~. The answer for $a$ ~a~ and $b$ ~b~ will be $(w[n]-w[b])+(r[b]-r[a])+(w[a])$ ~(w[n]-w[b])+(r[b]-r[a])+(w[a])~. Looking at this formula you can notice that for any $b$ ~b~ you should always choose the $a$ ~a~ that minimizes $w[a]-r[a]$ ~w[a]-r[a]~. So as you loop through $b$ ~b~ from $2$ ~2~ to $n-1$ ~n-1~, instead of checking every $a$ ~a~, you can just store the minimum value of $w[a]-r[a]$ ~w[a]-r[a]~ so far.

Time complexity: $\mathcal O(n)$ ~\mathcal O(n)~

There are quite a few other accepted solutions so feel free to show your own in the comments.

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