PikMike's blog

By PikMike, history, 2 years ago, translation, In English,
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2 years ago, # |
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for problem D , after an O(N) preprocessing , check(mid) for binary search can be done in N/mid complexity , the worst case would be (N/(N/2)) + (N/(N/4)) + (N/(N/8)) + ... = N/2 + N/4 + N/8 ... = N , so the actual complexity comes out to be O(N) instead of O(NlogN).

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    2 years ago, # ^ |
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    Can you please elaborate as to why check(mid) will be O(N/mid)? I'm unable to understand the complexity.

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2 years ago, # |
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In problem G... We can use interval tree to maintain the sequence , instead of segmenttree. Each node of the interval tree represents a lazy tag. To modify , we firstly split the sequence and then processes the modification. Note that in one split operation , the numbers of intervals only increases by 2. Thus there is no need to use sparse table... This data structure operates at time Complexity O( n + q( log( k + q ) + log( n ) ) And it costs O( n + k + q ) memory.

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    2 years ago, # ^ |
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    To achieve this complexity you need to keep your interval tree balanced, for example by using a splay tree. I looked at your submission (26724827) and it seems it will time out on a case such as:

    50000 1
    1 2 ... 50000
    100000
    2 1 1
    2 2 2
    ...
    2 50000 50000
    2 1 1
    2 2 2
    ...
    2 50000 50000
    
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      2 years ago, # ^ |
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      Because the test cases are pretty naive... I passed this problem without maintaining the balance of the tree...

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    22 months ago, # ^ |
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    Sorry, I know this is an old post, but can you explain to me the difference between a segment tree and an interval tree? I thought they were the same.

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2 years ago, # |
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Silly question! Realised my mistake!

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2 years ago, # |
Rev. 3   Vote: I like it 0 Vote: I do not like it

Problem A, Python3, TLE #15, 22MBytes. Problem A, the same code, C++ MSVisualStudio2010, AC, 260 ms, 3.6 MBytes.

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    2 years ago, # ^ |
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    Welcome to "Python isn't so fast, as C++" club!

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      2 years ago, # ^ |
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      For some reason, I WANT to use Python. Probably, because Python fashionable...

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        2 years ago, # ^ |
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        Then learn how to optimize your python solutions. If you write it like this, it passes in 400 ms.

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          2 years ago, # ^ |
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            2 years ago, # ^ |
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            You can speed up io by reading and writing directly to streams. Also your answer accumulation works really slow, you create new strings N times and their lengths are big.

            Here is diff
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            2 years ago, # ^ |
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            Oh, join() function, of course. I heard it is faster. I tried to use it first, but catched some error and then write just with concatenation.

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            4 months ago, # ^ |
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            You are using '+' in the answer.

            In Python3 don't use '+' in string because strings are immutable so at every '+' operation it is creating a new string that takes O(length of string) and that is where it throws TLE.

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    2 years ago, # ^ |
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    use pypy

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2 years ago, # |
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For problem D, how do you decide to choose binary search to compute the minimum width? I do understand the code itself, but still have no idea how to pick binary search for this problem.

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    2 years ago, # ^ |
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    Choose m by (l+r)/2, try to do text with m lines, if you can m = l, else r = m. So you get answer in l when became r — l <= 1.

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    2 years ago, # ^ |
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    Define function f(w) — number of lines you will require if its width is limited to w. This function is monotonous, f(w) ≥ f(w + 1) for every w. And binary search is usually applied to monotonous functions.

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2 years ago, # |
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Can anybody please tell that why in Problem C the GCD of resulting sequence is always a divisor of n?

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    2 years ago, # ^ |
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    Let's consider the case when GCD isn't a divisor of n. Let it be some value g. Then resulting sequence looks like x1·g, x2·g, ..., xk·g. Its sum is equal to x1·g + x2·g + ... + xk·g = g·(x1 + x2 + ... + xk). That is divisible by g. The sum is equal to n. Thus n is divisible by g. This lead us to contradiction(? I'm not sure of right word to use).

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      2 years ago, # ^ |
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      I got it. Very thanks for the explanation. :)

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      2 years ago, # ^ |
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      Thank you very much, i always isn't understand this solution until i get you explanation. Maybe my brain short circuit, hhhhh, thanks.

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2 years ago, # |
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In problem C why we need to check that n — s > (k — 1) * d? also we need to check that gcd(n-s,d)=d right ?

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    2 years ago, # ^ |
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    We need to check n — s > (k — 1) * d, because sequence must be ascending. If this inequation is false, then last number is breaking this rule. There is no need of checking for gcd(n — s, d) = d. If n is divisible by d and all other numbers from sequence, which sum is equal to n, are divisible by d, the last number also must be divisible by d.

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2 years ago, # |
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In problem C test:#10, n=24, k=2, answer is "8 16". So, the max divisor is 8 ? But 8 is greater than , isn't it a contradiction of the statement in editorial?

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    2 years ago, # ^ |
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    "Don't forget that you should consider d and n / d, if you check divisors up to square root of n." Maybe, you should get acquainted with fast algorithm to find all divisors.

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2 years ago, # |
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There's like 'didnt want to tnink at all' solution for G.

We can actually build a valid persistent treap over whole range(N * K) — build a treap over given N elements, make K copies of it, merge them. Congratulations, u now have a valid treap with 10^9 elements in it.

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2 years ago, # |
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Can anyone please further explain the solution to F?

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    2 years ago, # ^ |
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    Let's say we have a set S of K elements. The power sets (all possible subsets including null sets is 2^K)

    Now for this problem we are given an array a1, a2, ..., aN and here we have to find subsequences of this array such that all GCD is equal to 1.

    Let's try all g (all possible GCDs) from 100k to 1. ans[g] is how many possible subsets we can make with GCD in this subset is equal to g. Our answer will be in ans[1]

    We now iterate for all possible g, let's denote the number of elements which has GCD equals to g as cnt(g). Then the value of cnt(g) is sum of all numbers of elements which has divisors of g, 2g, 3g, and so on.

    The number of possible subsets with GCD = g is equals to 2^cnt(g) — 1 (since we have to exclude null subset). However for some multiple of g, we have already counted before. That's why by using inclusion-exclusion ans[g] value will be equal to 2^cnt(g) — 1 — ans[2g] — ans[3g] — ...

    To have a better understanding, you can view my solution Submission

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2 years ago, # |
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Can someone post good clean solutions for problems A and B in Java? Everything I saw so far was quite tricky and hard to fathom. Thanks in advance!

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    2 years ago, # ^ |
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    Not sure what you mean by tricky. But here is my Java solution for B. My logic is exactly the same as that in the editorial.

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2 years ago, # |
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How to solve E in linear time?

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    2 years ago, # ^ |
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    Consider the dpi j table from the tutorial. For a given i, all j such that dpi j are true must be in one continuous segment. Therefore, you can do dynamic programming by storing the min and max of this segment for all i instead of storing the boolean for all i by j

    My code (java) might be easier to understand than my explanation: 26787020

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      2 years ago, # ^ |
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      Sorry, but I didn't got your explanation, and I got less the Java code. You are saying that for a given i, all the j such that dpi, j are true are contiguous. Let be i = n ; dpi,  - k and dpi, k would be equal to true, and for all j such that  - k < j < k would be equal to false. Am I making any mistake? Thanks in advance for your clarification

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        2 years ago, # ^ |
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        In the dp table, think about how any column relates the the column before it. On a 'W', 'L', or 'D', the true values in the later columns are either shifted up, down, or not at all from the previous column. Similarly, a '?' results in union of these three. None of these operations ever split up a segment of true values into two segments.

        You also know that the first column is a single segment of length 1. Therefore, induction tells you every column must be a continuous segment of true values (or all false).

        This poor ascii representation of the third example might help

             4 
             3                                       T
             2                                     T T
             1     T                            T  T T
           j 0  T  T  T                      T     T T
            -1     T  T  T                T          T
            -2        T  T  T          T
            -3           T  T  T     T
            -4              T  T  T
                0  1  2  3  4  5  6  7  8  9 10 11 12 13 ...
                 ?  L  L  L  L  L  W  W  W  W  W  ?  ? 
                                    i
        
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          2 years ago, # ^ |
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          Thanks for your cool illustration, and i finished a linear solution in C++. 26798716

          Any suggestions are welcome.

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    2 years ago, # ^ |
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    I was able to do greedy in O(n) with a stack (Submission).

    From the start, keep a running sum of the score. Treat (W) as +1, (L) as -1, and ignore (D)s and (?)s.

    At any point, if the score becomes k or -k, assign the nearest (?) to a (W) if score=-k, or (L) if score=k. Here, you can keep track of the nearest (?) by using something like a stack.

    Now be careful, after changing you'll have to slide the whole window after that (?). If the shift will make a part of the score still hit k or -k, then output NO. Otherwise, keep going.

    Now once you reach the end, we'll need to make the score k or -k so we simulate both options. Making use of the stack of (?), to target score=k, we assign the last k — score unknowns as (W)s. The remaining (?)s can be (D)s. Similarly, do it for (L) with score — (-k) unknowns. Check if it still satisfies everything, then we can output either working answer (otherwise, output NO).

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      14 months ago, # ^ |
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      EDIT: Thanks to @xurz97 for pointing out that a bug in my previous submission. The logic is still correct, but please refer to this submission link instead

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2 years ago, # |
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I appreciate the writers who take their valuable time to write these editorials. This is a educational round, so the editorials should have been more explanatory. These seem to be targeted for the experts I guess!!!!!

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2 years ago, # |
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Can someone please explain the use of the mobius function for problem F?

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2 years ago, # |
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In problem C, I could not apprehend, how we reach described solution. And, why does it give us best answer ? Could someone explain me ?

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    2 years ago, # ^ |
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    Same here. Waiting for someone to explain it

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    2 years ago, # ^ |
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    We're trying to maximize a the GCD of a (positive strictly increasing) sequence of length k that sums to n.

    Consider any sequence that has a GCD greater than 1:

    • 2,4,6,8
    • 3,6,12,15
    • 7,21,49

    Note that if a sequence has a GCD > 1, we can factor that number out:

    • 2{1,2,3,4}
    • 3{1,2,4,5}
    • 7{1,3,7}

    Because we can factor this number out, n (the sum of the sequence) must be divisible by the factor.

    Also note that the smallest possible sequence of length k, {1,2,...,k}, sums to k*(k+1)/2.

    We can combine these two observations in the following way.

    • Iterate from 1 to root(n) to find the factors of n.
    • For every factor x, we can consider if it's possible to make a valid sequence of length k where every element is a multiple of x.
    • We know that for this factor, the smallest possible sum is obtained with the sequence x*{1,2,...,k}
    • Therefore, the sequence is possible if and only if x*[k*(k+1)/2] <= n.
    • That's all there is to this problem. Also note that once you find the largest factor that satisfied the above equality, you can stop searching, since any smaller factor will yield a smaller GCD
    • The edge cases where you should return '-1' are where the equality x*[k*(k+1)/2] <= n doesn't apply for even the smallest factor, x=1.

    Below is an example:

    Consider the input "12 3"

    • The factors of 12 are [1,2,3,4,6,12]
    • k*(k+1)/2 = 6
    • We need to find the largest factor of 12, x, such that 6x <= 12
    • x is obviously 2.
    • The answer will be some sequence 2{a,b,c} that sums to 12
    • The answer is therefore "2 4 6"

    I hope that helps!

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2 years ago, # |
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Here is another practice problem to apply the idea from problem F: http://codeforces.com/problemset/problem/439/E. This problem requires the same idea used on problem F plus some knwoledge about stars and bars formula. Hope it can be useful for those who want to become better :)

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2 years ago, # |
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Could you please explain problem G in details. It is pretty hard to understand. And is there any code example? I didn't get how to build the tree, what whould represent sparse table. Please, help me

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2 years ago, # |
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My AC submission for G

WA with a smaller tree

I made a segment tree that is somewhat "persistent" for the initial period, and I thought that the memory complexity should be around O(Qlog(NK)), yet instead I had to allocate much more memory for my solution to pass. Now I wonder if I had a wrong bound or I just underestimated the constant factors. Would anyone mind prove the memory complexity for me?

Many thanks in advance.

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2 years ago, # |
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In problem C, in test case #21, my output sequence gcd is 3 and it's clearly increasing, the checker says my gcd is 1, perhaps I missed something! 26808542, I also submitted a very close version to the editorial binary searching among the divisors of N and it passed! 26809758

So what's the problem?!

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    20 months ago, # ^ |
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    In submission 26808542 your output is 500000003 1000000006 1500000012. 500000003 is not divisible by 3 so clearly your output's gcd is not 3. Further you can check gcd of your output is 1.

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23 months ago, # |
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Can someone provide the proof of the assertion in Problem C ?

The gcd of k positive integers that sum up to n, is always a divisor of n.

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    21 month(s) ago, # ^ |
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    suppose n is the sum of sequence. if there exist a answer then the sequence can be written as

    n = (gcd*1 + gcd*2 + gcd*3 + .....)

    n = gcd*(1 + 2 + 3 + ....)

    by this we can say that sum n is multiple of gcd. so gcd is a divisor of n.

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4 months ago, # |
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