enoone's blog

By enoone, history, 3 years ago, In English

I am now trying to solve problems from IZHO 2018. I have already done 3 problems, but now I am stuck on problem 2 from the second day called "nice sequence". I don't have any idea and can't find the solution anywhere. Can you help me?

P.S. You can find the problem here.

 
 
 
 
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3 years ago, # |
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Hint1
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    4 months ago, # ^ |
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    can u prove that Length of the sequence is n + m - gcd(n, m) - 1 ?

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      4 months ago, # ^ |
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      It's similar to Periodicity Lemma. The difference is the edges turn from undirected to directed.

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      4 months ago, # ^ |
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      The previos is fake proof.. I can only prove $$$n+m-1$$$ now.

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      4 months ago, # ^ |
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      I have completed the proof. Maybe better to say learning the proof from arc127f. Orz maroonrk!

      Assume $$$A,B$$$ are coprime. If we can transition from $$$x$$$ to $$$x+A$$$ by $$$+A,-B$$$, add an edge $$$(x,x+A)\bmod B$$$. If $$$n>=A+B$$$, it forms a single loop. We only need to prove the case when $$$n=A+B-1$$$, that there must exists a valid solution.

      We can find that all nodes except $$$A$$$ has in-degree, and all except $$$B$$$ has out-degree. So the graph looks like a chain. We can easily construct a topological sequence by the chain.

      In fact we have proved the Periodicity Lemma..

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4 months ago, # |
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yeah pretty nice