spookywooky's blog

https://atcoder.jp/contests/abc226

By spookywooky, history, 2 years ago, In English

Apparently there is no official announcement, so I hijack this. Ask questions here.

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2 years ago, # |
← Rev. 2   Vote: I like it +8 Vote: I do not like it

F Editorial says S(p) is the LCM of the loop lengths. Why this?

I would expect that it is the max loop length.

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    2 years ago, # ^ |
      Vote: I like it +10 Vote: I do not like it

    For something like 2 3 1 5 4 the max loop from the first 3 is 3, but if you simulate it through 3 iterations, the first 3 are ok but the last 2 are in the 'odd' phase of their 2-cycle.

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    2 years ago, # ^ |
    ← Rev. 2   Vote: I like it +9 Vote: I do not like it

    Consider $$$P = (2, 3, 1, 5, 4)$$$. The ball wills move like this:

    • 2, 3, 1, 5, 4
    • 3, 1, 2, 4, 5
    • 1, 2, 3, 5, 4

    After Snuke shouts three times, Person 4 has Ball 5, so 3 (the maximum loop length) is not an appropriate answer.

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      2 years ago, # ^ |
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      Each time Snuke screams, every Person i such that $$$i \neq p_i$$$ gives their ball to Person $$$p_i$$$ simultaneously.

      Doesn't that mean, that after the first exchange, $$$4th$$$ and $$$5th$$$ men have to stop exchanging? That's misunderstanding in statement. I counted max loop lengthes.

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        2 years ago, # ^ |
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        $$$p_i$$$ don't change after people pass balls. They are elements of a fixed permutation $$$P$$$.

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        2 years ago, # ^ |
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        I mixed that up, too. But I think the statement is clear. Especially allways all persons give their ball to p[i]. It is just that i!=p[i], because if i==p[i] that ball would never move at all.

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    2 years ago, # ^ |
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    Can anyone please explain the formula given in the editorial for getting the count of each partitions ? It would be of great help.

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

      Consider counting the number of permutations of such $$$(1, \ldots, N)$$$ that it consist of $$$k_1, \ldots, k_m$$$ cyclic permutations. These consist of two factors:

      What "cyclic permutation group" does each element belong?

      This is equivalent to determining such sequence $$$a_1, a_2, \ldots, a_n$$$ such that:

      • Each element $$$a_i$$$ is an integer between $$$0$$$ and $$$m$$$;
      • For each $$$j = 1, \ldots, m$$$, there are exactly $$$k_j$$$ elements such that $$$a_i = j$$$.

      So this means that element $$$i$$$ belongs a cyclic permutation group $$$a_i$$$, or does not belong to any cyclic permutation if $$$a_i = 0$$$.

      How many such sequences are there? Consider a sequence B = (($$$N - \sum k_j$$$ copies of $$$0$$$), ($$$k_1$$$ copies of $$$1$$$), ($$$k_2$$$ copies of $$$2$$$), $$$\ldots$$$, ($$$k_m$$$ copies of $$$m$$$)). This is an $$$n$$$-element sequence, and each permutation of $$$B$$$ correspond to the aforementioned $$$(a_i)$$$ one-to-one. Therefore, the number can be found as:

      $$$\displaystyle \frac{N!}{\prod_j k_j!}.$$$

      What does each cyclic permutation look like?

      Given a set of elements that forms a cyclic permutation, we have to determine the actual permutation. For simplicity, we consider a cyclic permutation of $$$\lbrace 1, 2, \ldots, Q \rbrace$$$, of length $$$Q$$$.

      What should $$$1$$$ map to? It should not map to $$$1$$$ itself, since it will never result in a cyclic permutation of length $$$Q$$$. Therefore there are $$$Q-1$$$ choices. Say $$$1$$$ maps to $$$p_1 \neq 1$$$. What should $$$p_1$$$ map to? It should not map to $$$1$$$, because it becomes a cyclic group of length $$$2$$$; nor should it map to $$$p_1$$$ itself. So there are $$$Q-2$$$ choices. The next element has $$$Q-3$$$ choices for its next mapping direction, the next has $$$Q-4$$$, ... and so on, before determine the destination of the last element, which should now go back to $$$1$$$ (no other option).

      Therefore, there are $$$(Q-1)!$$$ options.

      But wait, there's more!

      To sum up the last two sections, we obtain the following number:

      $$$\displaystyle \frac{N!}{\prod_j k_j!} \cdot \prod_j (k_j - 1)!.$$$

      However, in the first chapter we distinguished the cyclic group with the same length. So we have to divide by an additional duplicate-remover.

      To do this, let's convert the sequence $$$k_1, \ldots, k_m$$$ into the sequence of occurrences: $$$F_1$$$ elements of $$$k_1, \ldots, k_m$$$ are equal to $$$K_1$$$, $$$F_2$$$ elements are equal to $$$K_2$$$, and so on. Using this notation, the last expression can be re-written as follows:

      $$$\displaystyle \frac{N!}{\prod_{F_a, K_a} (K_a!)^{F_a}} \cdot \prod_{F_a, K_a} ((K_a - 1)!)^{F_a}.$$$

      Now we can divide by the freedom of permutation of same-length cycles:

      $$$\displaystyle \frac{N!}{\prod_{F_a, K_a} (K_a!)^{F_a}} \cdot \prod_{F_a, K_a} \frac{((K_a - 1)!)^{F_a}}{F_a!}.$$$

      This is equivalent to the expression in the editorial.

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    2 years ago, # ^ |
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    If you know some group theory, you will notice that $$$S(p)$$$ is the "order" of the permutation. If you google this term, you'll probably find a proof for why $$$S(p)$$$ is the LCM of the lengths of the disjoint cycles (what you call loop lengths).

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2 years ago, # |
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Great.Someone explain me the reason of the solution in the editorial of problem E.Why same number of edges and nodes guarantee that there are exactly two possible ways of directing a particular connected component?

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    2 years ago, # ^ |
      Vote: I like it +14 Vote: I do not like it

    The editorial seems to be updated.

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    2 years ago, # ^ |
    ← Rev. 2   Vote: I like it +20 Vote: I do not like it

    A Connected graph with 'n' nodes and 'n' edges will definitely have a simple cycle. You can think of it as a ring graph.

    Since it is a directed graph, their are 2 ways — clockwise and counter-clockwise.

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      2 years ago, # ^ |
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      Not a simple cycle, but a simple cycle with paths attached to it.

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    2 years ago, # ^ |
    ← Rev. 2   Vote: I like it +8 Vote: I do not like it

    The last sample is what helped things click for me. It showed a disconnected case, but 2 different arrangements that worked... started with asking if the connected parts had to have a cycle (yes), and then testing what happened with variations based on the part that had a 3-cycle with a vert-and-edge pointing into it. Not a proof, and still had to whip up the brute force sim, but was enough to convince myself... edit: also helped to narrow down range of ok e vs. v relationships, testing out (eliminating) various other graphs like hourglass, 7-segment digital display, flow thingies (start, n mids, end), etc...

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      2 years ago, # ^ |
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      I used a different approach instead of e == v relationship. I started from leaf nodes and removed all the leaf nodes sequentially. After this, all the remaining nodes in the graph must have a degree of 2.

      Somewhat inspired from this of a recent Div 3 contest.

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2 years ago, # |
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Can someone help me with [D].I can't understand how for case-> 3 (5,2),(7,2),(10,2) the answer is coming as 2 (https://atcoder.jp/contests/abc226/tasks/abc226_d)

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2 years ago, # |
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In B I made a vector of strings,sorted them and calculated the answer.It gave WA and TLE on some tc.But when I made a vector of vectors, sorted them and then calculated the answer, it gave AC. Can someone plz explain this?

WA->(https://atcoder.jp/contests/abc226/submissions/27120194)

AC->(https://atcoder.jp/contests/abc226/submissions/27119995)

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

    You basically make two mistakes:

    Step 1: try this
    Step 2: try this
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      2 years ago, # ^ |
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      Thanks for the reply ,understood my mistake! char wont take complete no ‘︿’

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2 years ago, # |
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An "User Editorial" has been provided for problem F. Can someone explain how we are able to iterate through all possible LCMs without TLE?

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    2 years ago, # ^ |
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    Let's fix N = 50, and we want to estimate what the maximum LCM could be. This is equivalent to, if we partition N to be $$$N = a_{1} + a_{2} + ... + a_{m}$$$，the LCM would be $$$LCM = lcm(a_{1}, a_{2}, ..., a_{m})$$$. Apparently we should make every $$$a_{i}$$$ to be coprime with others, to avoid loss of LCM. So an optimal solution to this would be: $$$a_{1} = 1, a_{2} = 4, a_{3} = 5, a_{4} = 7, a_{5} = 9, a_{6} = 11, a_{7} = 13, LCM = 180180$$$. 180180 is a loose upper bound, considered that some numbers in [1, 180180] won't be a valid LCM(some large prime, etc.). Actually for N=50, there are only 1056 valid LCM. So enumerating all possible LCMs would be easily fit in the time limit.

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