**Update 2019/01/02:** The problemset is finally uploaded to CF Gym. If you didn't participated before, this is a great chance to practice on one of Petr's best problem candidates. I hope you enjoy the contest. Happy New Year!

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OpenCup GP of Korea (third edition) is scheduled at 2018/10/14 Sunday, 11:00 MSK.

Both Div1 / Div2 problemset will feature Korean problems.

Problemsetters: ainta alex9801 Cauchy_Function HYEA jh05013 ko_osaga OnionPringles Togekiss .o.

Enjoy!

Can you please give a link to the contest?

opencup.ru

My browser freezes when I click "link" button (OSX / Safari). Is it only for me?

now it works

How to solve A,E,M?

M: Trick from IOI 2016 Aliens

how to prove that this trick works here?

You can think of getting answers for k=1, 2, 3, ..., as computing some partial solutions in maxflow-mincost in bipartite graph formed by the tree. Even in general MFMC every following augmenting path gives you profit which is not larger than previous profit what translates to convexity of f(k) function what is what we need.

A: Build heavy-light decomposition and solve for each path independently. Now we need to recolor some prefix of path and maintain for each color number of edges with this color. It can be done with stack.

E: Delete all multiple edges. Take some vertex of degree 2, delete it, and connect its neighbours. Do this while there is at least one such vertex. In the end check that the remaining graph is 2 vertices connected by edge.

A: Use heavy-light decomposition. Now every update introduces at most one new point where the color changes in each of the O(log(n)) paths. Therefore we can afford to keep track of all segments with the same color.

E: Keep removing nodes of degree 2 until only a single edge remains.

How to solve G?

You can see the solution in the attached PDF.

I can't believe that this was solved by 58 teams. This is intended to be the medium-hard level in this GP.

I've seen this problem a couple of times

I remember that there was a similar problem in Open Cup (swap

Ktimes and do something for an array, solve it inO(Npoly(K))), but I can't find the link now.I see. There was some notorious coincidence :(

We just wrote

O(nk^{3}) solution with array rotation hoping that caching would help. Work like a charm.I've just read this problem and I think I haven't seen anything similar but I knew how to solve it immediately after reading the statement (which was not the case for any other problem I've read from this contest (ABCEIJKM)) and I think it was similar for Marcin who solved it in my team. I think it's rather you overestimating difficulty rather than half of people knowing this problem.

Thank you so much for participation!

Problem author:

You can see the solution for large subset of Div1 / Div2 problems, here.

If I don't make a mistake, C is equivalent to some generalization of Catalan numbers.

You are given points (0,

h_{0}), (1,h_{1}), ..., (X,h_{X}). How many ways are there to go from (0, 0) to (X,Y) by passing exactlyKof these points? You are not allowed to go above the points.How to solve it (in quadratic time)?

All surviving hands will be to the left of all surviving flowers. Also, the rightest surviving hand will be adjacent to the leftest surviving flower. If we fix this position, then the problems for prefix and suffix become independent and also on prefix we only care about

Aand on suffix we care aboutB. Letl_{i, j}be equal probability that if we start with prefixiandjhands coming from the right, in the end there will be exactlyAhands and no flowers. We can definer_{i, j}similarly for suffixes and flowers. Then the answer is .Will it be possible to share test cases: M15, M22, M35? Slightly different implementation of Alien trick gets WA in these cases.

We have WA35 because of small INF in binary search.

You are right. Taking MaxW gives WA35. Taking some big number, may be N*MaxW gives AC. Why? I thought MaxW is enough.

Line graph with cost ∞, - ∞, ∞, - ∞, ..., ∞

We had taken 3MaxW. So: W, -W, W, ... -> 4W, 3W, 4W .. should not be problem right? It will rightfully pick all 4W's.

I don't understand what you are talking about, but in those cases,

f(X) = -X×W,f(X+ 1) = (X+ 1) ×W. There is no wayf(X+ 1) could be found, when the range of binary search isO(W).I see, my sign was opposite, so for me the anti case was -inf, inf, -inf...

I also use MaxW+10 and got WA35 and don't know why it is not enough. :(

We had WA9 because of small INF in ternary search (I thought optimum might be non-integer even if the answer obviously is). I felt that there must be binary search solution too but couldn't figure it out. I just used standard approach — try to reduce problem to integer linear problem; check if integer requirement can be thrown out; if it is, construct a dual problem and solve it. In this case dual problem can be easily solved if one parameter (dual to the number of edges equality) is fixed. Since we have linear programming, the function must be convex and so I used ternary search to optimize this parameter.

This looks similar to the binary search solution but I don't understand how the connection works in general case. Is it some kind of partial-dual trick?

Can you please more elaborately explain how you throw out integer constraint? Or Any link to a problem using same technique?

It is called a relaxation.

I try to prove that optimum solution has integer values. In this case equations are organized in a tree and all their coefficients are +-1 and so it can be easily seen that any non-integer optimum can be converted to an integer optimum in this problem.

Another example of this idea is the assignment problem. If you have non-integer solution, you can construct a cycle of non-integer edges. The move half of them one direction while moving the other half in the opposite direction until one of edges hits 0 or 1 and number of non-integer edges decreases. This proves that any linear programming solution of the assignment problem either has integer coordinates or is equivalent to a solution with integer coordinates. This point of view explains where potentials in assignment algorithms come from — they are dual variables for linear programming formulation.

Btw, you can now download tests here. (600MB)

How to upsolve problems from div-2? Link on opencup.ru is broken and contains empty contest_id.

How solve K?

UPD: Same but faster solution is described by ko_osaga below. My one has an extra factor and may get TL.

Forget about the starting position for a moment.

Assume we've just taken the drug at position

iand it wasC_{i}-th in order. That means that either the whole segment [i-C_{i}+ 1,i] is taken, or the whole segment [i,i+C_{i}- 1]. Let us do the backward DP over such segments:d(l,r) equals to the maximum amount of damage we can get if we have eaten all drugs fromltor. Transition:d(l,r) =max_{l', r'}d(l',r') +last_drug, wherel' ≤l<r≤r', andlast_drugis the score of eating the drug that gave us the segment [l,r].Now we consider all segments in decreasing order by their size and compute the DP using some 2-dimensional data structure. Finally, the answer for the initial problem when we start at the position

iisd(i,i), that is, the maximum DP value over all segments such thatl≤i≤r.You must be careful with off-by-one errors: when considering [

i-C_{i}+ 1,i], you should take maximum over segments covering [i-C_{i}+ 1,i- 1] and store this value intodp(i-C_{i}+ 1,i). This stopped me from accepting the problem at the last minute.P.S. Curious fact: I don't know how to solve the problem for a single starting point faster than for all of them at once.

Spoileri,i] "expanding" to interval [1,N]. This gives a simpleO(N^{2}) DP, which is not enough.i, there exists only two cases of drug being absorbed: Either interval [i-C_{i}+ 1,i- 1] is expanded to [i-C_{i}+ 1,i], or [i+ 1,i+C_{i}- 1] is expanded to [i,i+C_{i}- 1]. Thus, there is onlyO(N) "interesting" intervals.O(N^{2}) time complexity.DP_{i}=max_{Sj ≤ Si, Ei < Ej}(DP_{j}) +A_{i}, so one can sweep through increasing order ofS_{i}, and maintain a max segment tree onE_{i}.i. This is easy.How to solve B?

(by ilyakor)

Let's find strongly connected components of cells. Now each star can be attributed to at most two SCCs: one corresponding to the cells going the most to bottom and top from it, and one corresponding to cells the most to left and right from it.

Now the problem is: we have a new acyclic graph (of SCCs), and need to find a path that passes through at least one vertex in each star pair. This can be reduced to 2-SAT if we say that a set of vertices belongs to a path if and only if it doesn't have two "incomparable" vertices.

I guess the same can be done without SCCs directly via 2-SAT: each star gets a boolean variable depending on whether we pass it horizontally or vertically, and we get constraints that prohibit using two segments such that neither can be reached from the other one, and also prohibiting segments not reachable from the start.

What is the solution of game on plane?

Here's a document with my solution to this problem, and several other problems from this contest.

What’s the idea behind Div2 L (Timsort)?

If you preprocess the next position of each

i, you can answer one query inO(N / MINRUN)and save the result for that MINRUN. Doing this, the worst case is when the queries are 2, 3, 4, 5 ... N, but this is(N / 2) + (N / 3) + ... + (N / N)which is of orderO(N log N). If you don't save the result after each new query and the queries be 2, 2, 2, 2..., the solution isO(N^2).Is it possible to upload the contest to Codeforces?

Let me set a deadline for myself — I'll process this till the end of the year.

Promise made (although I was 2 days late). Enjoy!

Nice job

Thank you very much!

Auto comment: topic has been updated by ko_osaga (previous revision, new revision, compare).Edit: It is on vjudge already. Thanks!

Hi ko_osaga, do you mind to invite vjudge accounts (vjudge1, vjudge2, vjudge3, vjudge4, vjudge5) to this gym contest, so vjudge users can create contests with these awesome problems. Thanks!