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pabloskimg's blog

By pabloskimg, history, 4 weeks ago, In English,
 
 
 
 
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4 weeks ago, # |
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This is the final scoreboard of the contest

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4 weeks ago, # |
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Well, the problem D and M were pretty easy, it was just implementation. A symetrical Pizza was just to solve the equation aXmod 360 = 0, however X was a float integer with two decimals. In the regional contest this was a problem beacause it got you WA even if you multiply X and 360 by 100. So We had to parse the number and remove the zero.

I would like to know how to Solve B,C and L. Thanks in advance

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    4 weeks ago, # ^ |
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    For problem B, you should notice that a rectangle must have two diagonals, and as the rectangle is inscribed in the circle, those diagonals must be diameters of the circle. Therefore you just have to check if there are a least two pairs of opposite points on the circle.

    Problem C can be solved with a DP. DP[i][l][t] is the optimal cost for taking all trips from i, given that we have l minutes left of the discount and there have been taken t trips with the current discount. So the answer is DP[0][0][0].

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      4 weeks ago, # ^ |
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      For problem C you only need i and t for the matrix. there is only one valid l value for every i and t values

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        4 weeks ago, # ^ |
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        Nice observation! I coded the other one and it fit in time so I didn't think more carefully about that, maybe a little modification over limits could have done the problem more interesting :P

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        4 weeks ago, # ^ |
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        In fact, you only need to know the cost for a trip ending at point I, that was my AC in-contest

        If you store optimal cost for trip I, then you can check 3 transitions:

        1-Buy trip I and nothing after

        2-Buy trip I and get next trip with a 50% discount

        3-Previous transition + get any number of trips between i + 2 and i + 6 with a 75% discount

        You also need to check if it's possible to buy by having sum_of_distances(i, i+k) < 120

        EDIT: added solution

        AC Code
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      3 weeks ago, # ^ |
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      Problem C is possible with dijkstra too

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    4 weeks ago, # ^ |
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    Thanks to the peruvian team [UPC] La Señora K-ruskal for the main idea for problem B:

    Two points in the circle are called opposite if are exactly separated by 180° (are at the same distance both clockwise and anti-clockwise). Having said that, if you can find at least two pairs of opposite points, you can asseverate that you can form a rectangle. See the image:

    So, in order to find the pairs of opposite points you can use a set of prefix sums and find the opposite of a point doing: (x + s/2) % s, being x the current distance of the point and s the total distance sum of the circle. The s/2 means "add 180° from this position".

    Code: https://ideone.com/jsIpPM

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I've solved the problem L. Has been basically solved with an offline algorithm for manage the queries by your favorite update/query data structure (BIT, Segment Tree). Code: https://ideone.com/irbeZ3

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Does anyone know what is the general idea to solve problem E? Thanks

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    4 weeks ago, # ^ |
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    Fix a base. Take the edges that project on the left/right of this base. Create a tournament graph where the direction of the edges is (u, v) is directed from u to v if and only if v projects to the left of u. The task is now counting the number of 3-cycles in this graph and that's way easier. Count everything — 3-cycles that don't exist. A 3-cycle that doesn't exists will have exactly 1 vertex that has 2 out edges and exactly 1 vertex that has 2 in edges. This doesn't treat the parallel edges case (the graph isn't exactly a tournament graph) but that's easily counted apart.

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    4 weeks ago, # ^ |
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    Here's another idea, though it may be equivalent. Iterate through the sides of the polygon counter-clockwise. Suppose that we are processing some side (the green side in the picture). Consider the consecutive edges after that one, and continue until the angle with respect to the horizontal has increased by , the relevant portion of the polygon is cut off by the grey line in the picture. Now it's easy to see that any two edges in this section, along with the green edge, form a triple that doesn't work (the red edges in the picture for example).

    With some care you can see that this process actually counts every bad triple exactly once, so just iterate over all n sides, find the number of edges in the relevant section of the polygon and find the number of pairs of these edges. Finally just subtract this from the total number of triples. We didn't solve E during the contest but this did give AC on the online judge.

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      4 weeks ago, # ^ |
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      That's equivalent, the rest of the explanation is for understanding why that counts all the bad triples once.

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    4 weeks ago, # ^ |
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    Let a[i] = Slope of ith segment, now we have to find triples of segments i < j <k such that one of the following conditions holds: a[i] > a[j] > a[k] or a[j] > a[k] > a[i] or a[k] > a[i] > a[j]. To count them easy, one can shift the array such that it only decreases once.

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The problem H just needs dijkstra with a small modification. You store de cost of the arc that arrives to the node an keep the cheapest one. Then add the costs.

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    3 weeks ago, # ^ |
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    Actually I thought the same approach at the contest but I got WA.

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We actually managed to solve problem L online during the contest. The idea is that we can precoumpute the answer for every value of K that is either a prime smaller than 1000 or a multiple of 1000. Then for queries with small values of K we just output the answer we calculated for the largest prime that is less than or equal to K, and for large values of K we take an m such that 1000m ≤ K < 1000m + 1000, take the answer for K = 1000m and then iterate through the primes in the interval (1000m, K]. We can see that for each one of these primes p we only need to add to the answer, because p2 > 105.There are not very many primes in these intervals so each query is fast enough.

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Did anyone solve problem F? Think im missing something in my implementation.

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    4 weeks ago, # ^ |
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    The max answer could be greater than 10^18, that could be your problem.

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    4 weeks ago, # ^ |
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    Didn't solve during the contest, but I got accepted on the online judge.

    The idea is fairly simple, the i-th line describes a function fi from S = [1, 2, ..., Z] to itself. It is not difficult to verify that fiZ(S) = fiZ - 1(S), so fi restricted to Si = fiZ - 1(S) is a permutation. So we can simulate the first Z steps and check whether those work manually. After that, every fi will only move us on some cycle Ci, and we want to check whether they coincide. To do this, we try every element of C1 as a possible common value. If this element is not in Ci for some i, then this is impossible. Otherwise the cycles give us a linear system of congruences such that each solution of it is a valid number of steps, and we can solve this using general CRT.

    The implementation, at least in C++, is not easy, because even though the moduli that we build during CRT do barely fit inside a 64-bit integer, it is possible that even the sum of two numbers of that size won't fit, so you have to be extremely careful with how you do each operation in order to avoid an overflow.

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      4 weeks ago, # ^ |
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      How did you manage the fact that module in CRT could be 100*99*98*97*96*95*94*93*92*91 ~ 6e19, and it doesn't fit even in unsigned long long?. I solved it in Java (post-contest) but it was a pain in my neck.

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        4 weeks ago, # ^ |
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        Well, the modulo in that case is actually lcm[100, 99, ... , 91], which does fit. To be honest I don't know whether there is a case that does not fit in an unsigned long long, if there is then it wasn't included in the official test cases.

        Even if it does work, I would definitely advocate for using big integers rather than trying to make it work with C++ long longs, it was extremely annoying.

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          3 weeks ago, # ^ |
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          The largest modulo I could come up with is with [99, 97, 95, 94, 91, 89, 83, 79, 73, 71] which exceeds 264.

          I don't get the decision of the problem-setter regarding the constraints. Having for example B ≤ 5 would be exactly the same problem, but without that unnecessary annoyance with the overflows...

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        4 weeks ago, # ^ |
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        During the contest, you could use 128 bit integers.

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What is the expected solution for problem C? using C++ and a DP[time][pos] gives me a 0.850 AC with a 1 second time limit I believe it should be optimized further?

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How to solve K?

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    4 weeks ago, # ^ |
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    The greatest positive sum is the sum of all positive numbers. The second greatest positive sum must be the greatest positive sum removing the smallest positive number or adding the smallest (in absolute value) negative number. This can be proven mathematically, but I won't do this right now.

    So, the basic idea is to test both alternatives: removing the smallest positive number or adding the smallest negative number. To test if this candidate number is an option, half the sums must contain this number while the other half must not contain it, so you just need to check if this holds. The easiest way is taking the greatest remaining sum and checking if exists the this sum minus your candidate (if the candidate is positive. If it's negative you must check if exists this sum plus your candidate).

    The sums must reduce by half every step, so you have a binary tree with height N. The complexity to do this is 2^N on every level of the tree, so N*2^N in total.

    I don't think there are multiple solutions to this problem, but if there are you must be able to just store the answers on a set of sets and print them in order, but talking to contestants after the contest they didn't do this and got accepted (I just assumed the answer was unique)

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      4 weeks ago, # ^ |
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      The answer is not unique. See this case for example: -3 -2 -1 0 0 1 2 3

      Here are two possible answers: -3 1 2 and -2 -1 3

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        4 weeks ago, # ^ |
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        Okay, so you must calculate every one of them and print in lexicografical order. I don't think there's a smart way to do this. Using set of sets you end up with O(N^2*2^N) or O(N^3*2^N). Is there any better way?

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          4 weeks ago, # ^ |
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          That's enough. Not every mask will be an answer too so it has good constant. There are at most 2^N answers and inserting one has cost N^2 at most. To make it even faster, you could push them into a vector and sort later, that has way better constant than using a set.

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How to solve problems G, I and J?

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    4 weeks ago, # ^ |
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    For problem I, first we divide a stick man by levels as follows:

    Now, we can use dynamic programming to solve the problem, f(node, level) = maximum number of stick men that we can make if we are in node node and our current stick man is at level level.

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    3 weeks ago, # ^ |
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    Yes, for problem I, dp is good. But, my coach has a pretty greedy solution problem I

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    3 weeks ago, # ^ |
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    This is the original solution of problem G. Ask any questions that you may have. https://1drv.ms/b/s!AqdRUIdEpNOHhnHQazJ_8jU2M-iU

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      2 weeks ago, # ^ |
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      The pdf looks great. Thank you very much. You said it is the original solution. Does that mean there are other original solutions? Do you have explanation pdfs for the other problems as well, by any chance?