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Is there a simpler way to solve this without calculating the median of the arrangements of the duals? It's O(n^2), but is that the solution the problemsetter was looking for (the implementation is long..)? Thanks for any hints.

I don't know what length will u consider long

My faster solution is 140 Lines
My slower (three times) is 100 lines
Kisman's solution is 40 lines (With STL) which is also quite slow but runs within time limit.

my solution is less than 70 lines, without using any library except stdio.h.

Larry wrote:Is there a simpler way to solve this without calculating the median of the arrangements of the duals? It's O(n^2), but is that the solution the problemsetter was looking for (the implementation is long..)? Thanks for any hints.

I also forgot to reply the actual question. Yes I used median of arrangement as my idea to solve this problem. Kisman's idea was similar I think.

My solution has about 100 lines and uses a tiny bit of STL. I use an iterative approach, improving the selected mines in an alternating manner... And it seems to be quite fast.

I've got an idea of N*log^2(N) solution. If it works, I'll write it here.

P.S: Was that authors' intention of N^2 solution or they just didn't know how to code a better way for the problem they've put, but are aware of its existence?

Well, my method didn't work (tried binary search over convex hull nodes). Though, there still might be another one which consider inner points as well. I was unable yet to find any n*log(n) ordering.

By the way, if the intentional complexity was O(N^2), so you wanted to prevent people from sending O(N^3), why did you set limit to 10'000, but not to 2000? 10'000 is not better than 2000 as O(N^3) blocker, but it is very confusing limit as O(N^2) allower.

Also I won't be surprised if O(N^2) testing of the 1st set via 2nd set will differ from the opposite approach (testing 2nd set via 1st set) 5 or 6 times on time. E.g. TL will turn into AC once you make x=-x. Fair square limit of 2000-3000 won't allow such weird thing to happen.

I don't think the judge solution is O(N^2).

I have two solutions one was O(N^2) and modifying it slightly I made a better solution. But I allowed both the better and the O(N^2) to pass. I did not want a very bad O(N^2) to pass that is why N=10000. And I don't understand what is your problem if N=10000 and not 2000.

Shahriar,

I don't think the judge solution is O(N^2).

If you are the author, they why don't you know? Or you're not?

The problem is that whenever I see N=10'000 or greater, I seek for N*log^k(N) solution. I think that problems are intended to demand some asymptotics, but not optimizing N^2 (or N^3) to make it AC. Because for any such "solution" there is always a test which makes its worst. And all those "optimizatiosn" just increase the contanst, and you get a fair TL. Limits are a way to set an upper limit for demanded asymptotic.

Perhaps I am just too old problem solver, and what before was 1'000 now is 10'000. Well, never mind.

THE QUESTION:
Is there a way to build nested convex hulls of set of points better than at O(N^2)? I.e. once the outer hull is built, delete all its nodes and do it again until no points left (innermost hull may contain 1 or 2 points). If such algoritmh exists, there is possibility of logarithm for this problem.

Yes, here it is. Called "Onion-peeling". http://cgm.cs.mcgill.ca/~orm/ontri.html

To be specific the judge solution is O(cN), where c is a constant of value around 100. "I don't think..." was another way to say "is not".

I never thought of convex hull while setting this problem so cannot comment on your issue.

No, onion-peeling won't work either. Well, it will give correct answer, but we'll have logarithm only for layer length, not on number of layers. So, if each layer has 3 nodes, we'll stay at O(n1*n2) = O(N^2).

What do you mean by C=100? CPU cycles, some fixed time or what? Or is this some sort of function which is 100 for N<=10'000? Would it remain 100 for N<=1'000'000?

I used bisection in this problem. C is the number of iteration required in bisection to find the answer in the worst case. When N=10000000 C will not change.

Uh. I read one of your 1st posts incorrectly. You said that you also seek for the median, but you didn't say that you also perform it at O(N^2)

Anyway, my question about C remains.