Toy Train
Contest: Codeforces Round 542 [Alex Lopashev Thanks-Round] (Div. 1)
Difficulty rating: 1800
Tags: brute force, greedy

Alice received a set of Toy Train™ from Bob. It consists of one train and a connected railway network of $$$n$$$ stations, enumerated from $$$1$$$ through $$$n$$$. The train occupies one station at a time and travels around the network of stations in a circular manner. More precisely, the immediate station that the train will visit after station $$$i$$$ is station $$$i+1$$$ if $$$1 \leq i < n$$$ or station $$$1$$$ if $$$i = n$$$. It takes the train $$$1$$$ second to travel to its next station as described.

Bob gave Alice a fun task before he left: to deliver $$$m$$$ candies that are initially at some stations to their independent destinations using the train. The candies are enumerated from $$$1$$$ through $$$m$$$. Candy $$$i$$$ ($$$1 \leq i \leq m$$$), now at station $$$a_i$$$, should be delivered to station $$$b_i$$$ ($$$a_i \neq b_i$$$).

The blue numbers on the candies correspond to $$$b_i$$$ values. The image corresponds to the $$$1$$$-st example.

The train has infinite capacity, and it is possible to load off any number of candies at a station. However, only at most one candy can be loaded from a station onto the train before it leaves the station. You can choose any candy at this station. The time it takes to move the candies is negligible.

Now, Alice wonders how much time is needed for the train to deliver all candies. Your task is to find, for each station, the minimum time the train would need to deliver all the candies were it to start from there.

Input format:
The first line contains two space-separated integers $$$n$$$ and $$$m$$$ ($$$2 \leq n \leq 5\,000$$$; $$$1 \leq m \leq 20\,000$$$) — the number of stations and the number of candies, respectively.

The $$$i$$$-th of the following $$$m$$$ lines contains two space-separated integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \leq a_i, b_i \leq n$$$; $$$a_i \neq b_i$$$) — the station that initially contains candy $$$i$$$ and the destination station of the candy, respectively.

Output format:
In the first and only line, print $$$n$$$ space-separated integers, the $$$i$$$-th of which is the minimum time, in seconds, the train would need to deliver all the candies were it to start from station $$$i$$$.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
Consider the second sample.

If the train started at station $$$1$$$, the optimal strategy is as follows.

1. Load the first candy onto the train.
2. Proceed to station $$$2$$$. This step takes $$$1$$$ second.
3. Deliver the first candy.
4. Proceed to station $$$1$$$. This step takes $$$1$$$ second.
5. Load the second candy onto the train.
6. Proceed to station $$$2$$$. This step takes $$$1$$$ second.
7. Deliver the second candy.
8. Proceed to station $$$1$$$. This step takes $$$1$$$ second.
9. Load the third candy onto the train.
10. Proceed to station $$$2$$$. This step takes $$$1$$$ second.
11. Deliver the third candy.

Hence, the train needs $$$5$$$ seconds to complete the tasks.

If the train were to start at station $$$2$$$, however, it would need to move to station $$$1$$$ before it could load the first candy, which would take one additional second. Thus, the answer in this scenario is $$$5+1 = 6$$$ seconds.

Example input:
5 7
2 4
5 1
2 3
3 4
4 1
5 3
3 5

Example output:
10 9 10 10 9

Example input:
2 3
1 2
1 2
1 2

Example output:
5 6

Editorial:
We can consider the pickup requests from each station individually. The overall answer for some fixed starting station is simply the maximum time needed to fulfill deliveries among all pickup stations.
Suppose the starting station for the train is fixed at some station $$$s$$$ ($$$1 \leq s \leq n$$$).
Consider some station $$$i$$$ ($$$1 \leq i \leq n$$$). Let $$$out(i)$$$ denote the number of candies that need to be picked up from station $$$i$$$. If $$$out(i) = 0$$$, we do not consider this station. From now on, we assume $$$out(i) \geq 1$$$. Each time that the train visits station $$$i$$$ to pick up a candy, we can choose which candy it should pick up. Therefore, we should find an order that would minimize the time needed to fulfill all deliveries from pickup station $$$i$$$. It turns out, however, that the only thing that matters is the last candy to be delivered. Suppose the last candy is to be delivered to station $$$e$$$, the total time needed to fulfill all pickup requests from station $$$i$$$ would be: $$$dist(s, i) + n*(out(i)-1) + dist(i, e)$$$, where $$$dist(a, b)$$$ represents the time needed to travel from station $$$a$$$ to station $$$b$$$. Take some time to think why this is the case! With this formulated, it is now clear that we have to choose the last candy to deliver that minimizes $$$dist(i, e)$$$.
All of this can be done in $$$\mathcal{O}(n)$$$ (with an $$$\mathcal{O}(n+m)$$$ pre-process only once to find the optimal last candy for each pickup station). To find the answer for every starting station for the train, we can simply run the above algorithm $$$n$$$ times.
The time complexity is $$$\mathcal{O}(n^2+m)$$$.