awoo's blog

Educational Codeforces Round 142 Editorial
By awoo, history, 22 months ago, translation, In English

1792A - GamingForces

Idea: BledDest

Tutorial
Solution (Neon)

1792B - Stand-up Comedian

Idea: BledDest

Tutorial
Solution (awoo)

1792C - Min Max Sort

Idea: BledDest

Tutorial
Solution (Neon)

1792D - Fixed Prefix Permutations

Idea: BledDest

Tutorial
Solution (awoo)

1792E - Divisors and Table

Idea: adedalic

Tutorial
Solution (adedalic)

1792F1 - Graph Coloring (easy version)

Idea: BledDest

Tutorial
Solution (BledDest)

1792F2 - Graph Coloring (hard version)

Idea: BledDest

Tutorial
Solution (BledDest)
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22 months ago, # |
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Isn't it possible to have an O(n) solution for problem C (Min Max Sort)?

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    22 months ago, # ^ |
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    yeah, there's an easy $$$O(n)$$$ solution for problem C, involving using the array $$$pos[i]$$$, to get the position of index $$$i$$$ in the permutation, then start at the middle value and finding the $$$LIS$$$ to both side. My submission: 190340250

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    22 months ago, # ^ |
      Vote: I like it +13 Vote: I do not like it

    maybe you read only the last line.

    read the third last line.

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22 months ago, # |
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can somebody explain how to come up with the equation in problem B.

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    22 months ago, # ^ |
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    Here, the Optimal strategy to solve the problem is by first saying all the jokes that both Alice and bob likes. (type 1 jokes) then by saying jokes alternately that (Alice likes,bob doesn't) and vice versa.

    Equation: a1 + min(a2,a3) x 2

    Final strategy is to say remaining jokes : first by saying remaining type 2, type 3 jokes and then by saying the jokes that both doesn't like (Type 4). this should be compared with the points that he already acquired hence a1 is taken. We take a1 + 1 as the judges goes when points become negative. Hence we add 1 to make it -1. Finally, We take the minimum of both these values to form the final equation.

    Equation: a1 + min(a2,a3).2 + min(a1 + 1,abs(a2 - a3) + a4)
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22 months ago, # |
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Does anyone have an idea on proving the efficiency of bisearch-on-divisors approach for prob. E? Never expected it will be this fast = =

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22 months ago, # |
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Can somebody explain what is the meaning of run binary search on k in problem C tutorial?

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22 months ago, # |
← Rev. 2   Vote: I like it +62 Vote: I do not like it

For whatever it's worth, here's my $$$O(n \log n)$$$ solution for problem F.

First, we notice that the property for the set $$$S$$$ is the same as having a one-colored cut in $$$S$$$ (in other words, the set of vertices can be split into $$$S_1$$$ and $$$S_2$$$ so that all edges between $$$S_1$$$ and $$$S_2$$$ have the same color). The only issue is that a given set $$$S$$$ can possibly be split into $$$S_1$$$ and $$$S_2$$$ in several ways. So I introduce $$$f(n)$$$ to be equal to the number of graphs on $$$n$$$ vertices where the global cut is blue. Then the answer is $$$2 f(n) - 2$$$, since the global cut may be red, but $$$2$$$ extra cases arise when all edges have the same color.

To find $$$f(n)$$$ we need to partition the set of $$$n$$$ vertices into $$$k \geq 2$$$ subsets so that vertices from different subsets are connected by blue edges, but the subsets themselves obey the rules recursively (they have global cuts of red edges). Since $$$f_{\text{red}} = f_{\text{blue}}$$$, we have

$$$ f(n) = \sum\limits_{k \geq 2}~ \sum\limits_{a_1 + \ldots + a_k = n} C(a_1, \ldots, a_k) \cdot f(a_1) \cdot \ldots \cdot f(a_k),$$$

where $$$C(a_1, \ldots, a_k)$$$ roughly means the number of ways to choose subsets of sizes $$$a_1, \ldots, a_k$$$ from a set of size $$$n = a_1 + \ldots + a_k$$$. If $$$b_1$$$ occurs $$$c_1$$$ times, $$$\ldots$$$, $$$b_m$$$ occurs $$$c_m$$$ times within the multiset $$${ a_1, \ldots, a_k}$$$, then

$$$C(a_1, \ldots, a_k) = \frac{n!}{a_1! \cdot \ldots \cdot a_k! \cdot c_1! \cdot \ldots \cdot c_m!}.$$$

Define $h(n) = \frac{f(n)}{n!}.$ The base values are $$$h(0) = 0, h(1) = 1$$$. Via $$$H(x)$$$ we denote the generating function of $$$h$$$: $$$H(x) = h(0) + h(1) \cdot x + h(2) \cdot x^2 + \ldots$$$. After a careful examination (I don't know how to prove rigorously it though) we obtain

$$$ e^{H(x)} = 2H(x) + 1 - x.$$$

Everything else is the standard approach of how to solve these recurrences: if we know $$$H(x) \bmod{x^{m}}$$$, then from the equality above we can obtain $$$H(x) \bmod{x^{2m}}$$$. Underneath we need FFT (NTT) and an exponential generating function. Each step from $$$m$$$ to $$$2m$$$ takes $$$O(m \log m)$$$ time, so the overall complexity if $$$O(n \log n)$$$.

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    22 months ago, # ^ |
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    I also found this solution, thanks for sharing it!

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22 months ago, # |
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I don't get why we have to make sure that no set of vertices is connected by both colors in F1. Doesn't the lemma proved it impossible to connect a set of vertices with both colors?

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    22 months ago, # ^ |
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    The lemma proves the other thing: it is impossible to have a set of vertices which is neither blue-connected nor red-connected.

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      22 months ago, # ^ |
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      Thanks. I made an oversight while reading the lemma.

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22 months ago, # |
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A-F1 video editorial for Chinese:

BiliBili

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22 months ago, # |
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In fact, we can calculate the convolution-like sequences (such as those in problem F) in $$$O(n \log^2 n)$$$ or $$$O(n \log^2 n / \log \log n)$$$ or even faster. One can find the approach from Elegia's report. The implementations usually has lower constant factor in time than those of the Newton iteration (if exist).

Here is my implementation, which imitates Elegia's implementations for other problems.

(but it's just as fast as a brute force lol)

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    22 months ago, # ^ |
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    Do you have this report as a PDF?

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      22 months ago, # ^ |
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      Here (sorry for Chinese only)

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        22 months ago, # ^ |
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        Do you know how to find ref. [5] (罗煜翔。(2020)。浅谈 Nimber 和多项式算法。IOI2020 中国国家集训队论⽂集。) ? I wanna see how to do semi-relaxed multiplication in $$$ \frac{n\log ^2 n}{\log \log n} $$$(if that's what it says). And what nimbers have to do with it xd.

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          22 months ago, # ^ |
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          Let's discuss about it here, actually in that report, most contents are just putting the general algorithm into the nimber framework. Nimber does not play an important role in the relaxed convolution.

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22 months ago, # |
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"The array may large" in E made me realise how I don't trust my gut feeling at all.

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22 months ago, # |
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Where does the log(divs(m)) of the complexity in question E come from Isn't it only need O(divs(m)⋅z(m)) to calculate the dp array?

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    22 months ago, # ^ |
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    oh i get it,because you can't find the element within O(1)

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    22 months ago, # ^ |
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    then why not use hash

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      18 months ago, # ^ |
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      constant factor is worse, I converted hash -> binary search + vector and it ran within time limit

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22 months ago, # |
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can somebody explain why in D if we use lower_bound ans EITHER result or the previous one? why not just result? thanks in advance

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    22 months ago, # ^ |
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    It depends on the value immediately after the longest common prefix. While searching for $$$p$$$, you find some inverse $$$q_1, q_2, \dots, q_m$$$ that starts with $$$p_1, p_2, \dots, p_k$$$ for some $$$k$$$. The next value is different. If $$$q_{k+1} > p_{k+1}$$$, then lower_bound will point at $$$q$$$ (or one of inverses with such prefix if there are multiple). However, if $$$q_{k+1} < p_{k+1}$$$, then $$$q$$$ will be smaller than $$$p$$$, and lower_bound will jump over it. The next inverse has to be greater than $$$p$$$, so you only have to look one step behind.

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22 months ago, # |
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For D, I thought while $$$n=5\times 10^4$$$, the number of subsets of $$$\{1,...,m\}$$$ is $$$10^3$$$.

So I constructed a map $$$M$$$ mapping each subset $$$L\subset \{1,...,m\}$$$ to $$$\{(a_i[p_1], ..., a_i[p_t])\ |\ i\in \{1,...,n\},\ \{p_1<...<p_t\}=L\}$$$. Then I iterated over every permutation and every $$$k$$$ to see whether the corresponding $$$k$$$ locations have the desired values.

For constructing $$$M$$$, it seemed that I can iterate over all subsets of $$$\{1,...,m\}$$$ and for each subset iterate over all permutations. This 2-layered loop is about $$$5\times 10^7$$$. But my actual implementation requires $$$O(2^m\times n\times m\times \log(n))\approx 8\times 10^9$$$ and thus got TLE. Not sure whether this is optimizable.

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    22 months ago, # ^ |
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    I thought of the same thing during the contest but midway realized that its gonna give TLE :(

    Edit: This video has a similar approach

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    22 months ago, # ^ |
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    You can implement it with Trie in $$$O(2^m\times n \times m)$$$. I'm not sure whether it is fast enough.

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    21 month(s) ago, # ^ |
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    It is not necessary to store all 2^m subsets. Let a permutation be 3 2 4 1. For this we need to store 0 0 0 1, 0 2 0 1, 3 2 0 1, 3 2 4 1. So the complexity is O(n*m)

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      20 months ago, # ^ |
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      Why don't we activate the numbers randomly, something like '0 2 0 0'?

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22 months ago, # |
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I have some questions in problem F1

  1. Why iterate k — the number of vertices whick are in the same 'red' component as 1 but not iterate the blue component, is this because you are counting the blue component?

  2. What about the edges between vertices in the same componet as 1 and the rest vertices, are they must be the same color? how to proof?

Hope someone can help me, thx a lot

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    22 months ago, # ^ |
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    1. We are considering the case when the whole graph is blue-connected, so there's no need to iterate on the blue component. The case when the graph is red-connected is symmetric to it.
    2. It's easy to see that all these edges are blue, since any red edge between any vertex from the "red" component of vertex 1 and any vertex outside this component means that we haven't picked the whole component
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22 months ago, # |
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Can someone give a test case on which my code is failing? I couldn't view the 156th item of 2nd test case. Is there any way to view it?

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    22 months ago, # ^ |
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    Click the "Click to see detail" button in the bottom of the page to see the detailed test cases

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22 months ago, # |
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has anyone solved problem C with binary search ?

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    20 months ago, # ^ |
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    Code
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22 months ago, # |
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NEED PROVEMENT FOR PROBLEM F

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22 months ago, # |
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F1's solution is flawed I guess. Consider case with n=3, when A1=B1=1, A2=2, B2=1, you get B3=4 which is incorrect.

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    22 months ago, # ^ |
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    So what's wrong with the $$$B_3=4$$$?

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      22 months ago, # ^ |
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      A3 which is the answer for n=3 is 6 in the pretest case, but surely 6 is not 4 * 2 :P

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        22 months ago, # ^ |
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        Well, maybe the tutorial forget something or i just didn't see.

        But don't forget the limitation 1 and 2.

        The $$$A_3=8$$$ contains the case that all blue and all red.

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          22 months ago, # ^ |
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          I noticed that too. I'm just saying that define "An" as "the answer for n" is not very strict and could lead to someone's misunderstanding(like me)

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          22 months ago, # ^ |
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          另外看兄弟id八成也是国人，直接说汉语就好了吧:)

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    22 months ago, # ^ |
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    Sorry, I initially stated the problem without the constraints "at least one edge should be red" and "at least one edge should be blue". I wrote the editorial for that version, but then decided to introduce these constraints. So, the actual answer is $$$A_n-2$$$ since we need to discard the case "all edges are red" and the case "all edges are blue".

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19 months ago, # |
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Time for problem E should be more strict. The following brute force solution passes:

Generate an array of increasing divisors. Make a copy and maintain it in std::set<long long>. For each divisor $$$d_i$$$ in the original array, find the number of divisors that has $$$d_i$$$ as its first row, i.e. start from divs_set.lower_bound(d_i * d_i) and iterate until the value is greater than $$$d_i * n$$$, counting the number of divisors that are a multiple of $$$d_i$$$, and remove them from the set afterwards.

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18 months ago, # |
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Can someone give some intuition why way 2 in E works

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13 months ago, # |
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In editorial of problem E,it is mentioned that " There is also a way to get rid of extra log(divs(m)) factor if you iterate through dp in a smart way ".

How to do so? Could not find anything in comments and figure out neither.

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9 months ago, # |
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is it possible to solve problem 'D' using trie?

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    8 months ago, # ^ |
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    Yes, it is

    I just solved it using Trie: 251308798

    you insert in the $$$trie$$$ the inverse of arrays $$$a_i$$$

    and then for every array you look for the longest common prefix.

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4 months ago, # |
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D can be solved with string hashing...

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