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### Geothermal's blog

By Geothermal, history, 3 months ago, ,

It's been a while since I've done one of these!

# A — Station and Bus

After parsing the problem statement, we see that we simply need to determine whether the two companies each operate at least one station, since if this is the case, those stations will be connected, and otherwise, if one company operates every station, no connections will be built. This is now a fairly easy task---one way to do it is to check whether every character in the string is the same, printing No if this is the case and Yes otherwise. Alternatively, we can directly compare $S$ to "AAA" and "BBB", rejecting if the input is one of these strings and accepting otherwise. There are a wide variety of approaches, but they're all essentially equivalent.

Runtime: $O(1)$. Click here for my submission.

# B — Count Balls

Given the large limits on $N, A,$ and $B$, a brute force simulation will be vastly too slow---we need a faster approach here. First, we compute $X = \lfloor \frac{N}{A+B} \rfloor$ to determine the number of times the operation is fully completed before we reach $N$ balls. Then, each of these operations adds $A$ blue balls to our answer, so we can initialize our answer as $XA$. Then, we have $Y = N \% (A+B)$ remaining balls to add. If $Y < A$, all of the remaining balls will be blue, and if $Y \geq A$, then we add another full set of blue balls. Thus, we can add $\textbf{min} (Y, A)$ to our answer and print it.

Runtime: $O(1)$. Click here for my submission.

# C — Tax Increase

This initially seems like an annoying but doable math exercise--we could compute the minimum and maximum values at which the taxes are $A$ and $B$, intersect these ranges, and take the minimum value in the intersection. However, implementing this is mildly frustrating, and it turns out that there's an even simpler solution.

Notice that the inputs on $A$ and $B$ are extremely low. Indeed, as soon as we reach a tax of $1010$, we see that the $8\%$ tax will already be greater than $100$, so the answer must be less than $1010$. Thus, we can check the taxes resulting from every potential answer from $1$ to $1009$ and print the first one that works, returning $-1$ if none of them are valid.

Runtime: $O(A+B)$, albeit with a bad constant factor. Click here for my submission. Note that I capped the answer at $10000$ instead of $1009$ in my solution--this doesn't change the output; I just used an unnecessarily large bound in case I did my arithmetic wrong and the correct bound was higher than I thought.

# D — String Formation

We can simulate the process fairly directly. First, we maintain whether or not the string $S$ is currently reversed or in its original configuration. Then, we maintain two strings $B$ and $A$, containing the characters before and after $S$ when $S$ is in its original configuration.

Processing the queries is fairly easy. For queries of type $1$, we just update our variable to indicate that $S$ is in the reverse of its position before the query. For queries of type $2$, we simply append the new character to either $B$ or $A$. If $S$ is in its original configuration, we append to the end specified in the problem. If $S$ is reversed, we append to the other end, because, for example, appending to the end when $S$ is reversed is equivalent to appending to the beginning when $S$ is in its original configuration.

Then, at the end of the process, if $S$ is in its original configuration, we print $B^RSA$, where $B^R$ is $B$ reversed (we define this operator for other strings similarly). We reverse $B$ because we want characters added to the beginning last to show up at the beginning of our string. If $S$ is reversed, we print $A^RS^RB$, which is the reverse of this string.

Runtime: $O(|S|+Q).$ Click here for my submission.

# E — Divisible Substring

We use a common trick in problems asking us to count substrings satisfying a specific criterion. Essentially, we store the residues, modulo $P$, of every prefix of the string, followed by appending zeros until we reach $N$ characters. For example, in the first sample input, the numbers we get are $0000$, $3000$, $3500$, $3540$, and $3543$, giving us residues of $0, 0, 2, 0,$ and $0$ modulo $3$. Then, given that each residue appears $C[i]$ times in the result, we sum $\dbinom{C[i]}{2}$ over all $i$ and print that as our answer. There's one catch: because we're appending extra zeros, we handle the cases where $P = 2$ or $5$ separately. (These are fairly easy because we simply need to count the substrings whose last digits are divisible by $P$.)

Why does this work? Notice that if we subtract the $i$'th value in this array from the $j$'th, we get the substring over the range $i+1 \cdots j$, followed by some zeros. For example, subtracting $3000$ from $3540$ gives us $540$. Moreover, note that as long as $10$ is relatively prime to $P$, appending zeros won't affect whether or not this is divisible by $P$. Thus, we can choose any two of these numbers with the same residue mod $P$ and subtract them to get a substring divisible by $P$, while choosing any two of these numbers with different residues mod $P$ will not give us a substring divisible by $P$. Therefore, this process computes exactly the right answer.

How do we compute these numbers? Note that we don't need to know the numbers exactly (and in fact, computing all of them would require $O(N^2)$ memory): we only need their residues modulo $P$. We can thus compute these values sequentially. First, we compute the residue of each prefix--to compute one prefix's residue from the last, multiply by ten, add the new digit, and take the remainder modulo $P$. Then, to get the residue of a number from its corresponding prefix, multiply by $10^{N-1-i}$, given that we just added the digit in position $i$ in the string. These powers of ten can be computed quickly using any efficient modular exponentiation algorithm.

Once we have the counts of each residue, we can compute the answer easily.

Runtime: $O(N \log P).$ Click here for my submission.

# F — Removing Robots

Start by sorting the robots by position. Realize that each robot $i$, either directly or through a chain reaction, will necessarily activate all robots up to some position $L[i]$, and no robots after this position. As a subproblem, we determine how to compute $L[i]$. We compute the values in reverse. Maintain a segment tree storing the values of $L$. Then, for each robot, use binary search to determine the furthest robot it directly activates. Finally, set $L[i]$ to the maximum of the values of $L[i]$ up to this furthest robot (using the segment tree to query for this value). This works because the furthest robot activated by $i$ is also the furthest robot directly or indirectly activated by any robot $i$ touches.

Now, let's compute the answer. Let $dp[i]$ be the number of sets of robots that could be activated given that we're only activating robots in the range $i \cdots N-1$. Define $dp[N]$ to be $1$. We see that $dp[0]$ is our answer. Then, we compute $dp[i]$ from $N-1$ to $0$. Note that when we consider adding given robot, we have two possible choices. We could avoid activating this robot, giving us $dp[i+1]$ possible sets (since if we're considering robots $i \cdots N-1$ but don't activate robot $i$, we can now pick any valid set from $i+1 \cdots N-1$). Alternatively, we can activate this robot, which adds robots $i \cdots L[i]$ to this set and gives us $dp[L[i]+1]$ possible sets. Notice also that if we're considering robots $i \cdots N-1$ and don't directly activate robot $i$, no higher-numbered robot will activate robot $i$, so the only way we'll ever activate robot $i$ is if we choose it now. Thus, we have $dp[i] = dp[i+1] + dp[L[i] + 1].$

We compute this recursion and print $dp[0]$ to get our answer.

Runtime: $O(N \log N)$. Click here for my submission.

• +81

 » 3 months ago, # |   0 Auto comment: topic has been updated by Geothermal (previous revision, new revision, compare).
 » 3 months ago, # |   0 Very good explanation of E! Thank you :)
 » 3 months ago, # |   0 very clean explanations.
 » 3 months ago, # |   0 these editorial helped a lot for Atcoder...Thanks brother
 » 3 months ago, # |   +5 Can anyone provide more problems that use similar to the prefix substring trick to count conditioned substrings as in problem E?
•  » » 2 months ago, # ^ |   0 adzo261 Plz tell me if you found any. Thanks!
 » 3 months ago, # |   0 Thanks for such a great editoral! btw, the link for your submission on problem F seems doesn't work.
•  » » 3 months ago, # ^ |   0 Fixed, thanks for letting me know!
 » 2 months ago, # |   0 Then, to get the residue of a number from its corresponding prefix, multiply by 10^(N−1−i), given that we just added the digit in position i in the string.Could you please explain what we are actually calculating here? I am not getting this part.