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Quintomania Div 3 hard version
By i_am_not_special, history, 8 days ago, In English

You are given an array a of integers with n elements. You are allowed to rearrange the elements of the array in any order. Your task is to determine if it is possible to reorder the array such that for every i from 0 to n-2 (i.e., for all adjacent elements), the absolute difference between consecutive elements is either 5 or 7. That is, for all i such that 0 <= i < n — 1, you need to check if: abs(v[i] — v[i + 1]) = 5 or abs(v[i] — v[i+ 1]) = 7.

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8 days ago, # |
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Auto comment: topic has been updated by i_am_not_special (previous revision, new revision, compare).

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8 days ago, # |
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What is the expected time complexity?

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8 days ago, # |
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what are the constraints on a[i]?

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8 days ago, # |
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I think I solved it. Lets discuss in the dm. But damn this was such a good question. You should have offered it as a problem in a contest. Please ping in dm.

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    8 days ago, # ^ |
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    Why dont you post it here so everybody can ser

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      8 days ago, # ^ |
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      First let me be sure if my approach is correct or not :)

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        8 days ago, # ^ |
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        even i dont know the solution

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        8 days ago, # ^ |
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        Yea well I am curious about the solution, share it in my DMs if you are comfortable sharing here i guess? Thanks

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          8 days ago, # ^ |
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          i have some solution but i dont know is it correct or not. i am thinking that first sort the array then define the range as l = v[0] and r = v[0] then select the next element and insert the element on either side of range and then update thhe range.

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            8 days ago, # ^ |
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            And how will you determine which side of the range to insert it on?

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              8 days ago, # ^ |
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              i am thiking first on right side if it is not possible on right side then on left side.

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8 days ago, # |
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Nice problem

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8 days ago, # |
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ignore plz

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8 days ago, # |
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Just writing whatever comes into my faulty memory at this point, but I think I faced a similar problem in an OA once and later found out that such problems are solved my building a graph out of it and determining the maximum flow or something similar on that line, I am not educated on the algorithm yet.

I dont believe theres a simpler solution but I'd be happy if someone figures it out.

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    8 days ago, # ^ |
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    I tried it out and initially thought it would work, bu then found myself a testcase where it would fail

    The approach was like store all the elements in a multiset, and choose an elenment that is smallest and then find its neighbouring elements, that are either situated at a distance of +5 or +7. If you find it, put it adjacent to that element and erase them from the multiset and continue till the multiset is empty.

    Ex: let us take a simpler testcase. {3 5 8 10}, smallest is 3 so now answer looks like

    ...3...

    search for possible adjacent elements, ie 8 and 10, we find here both, so the array looks like

    ...8 3 10... now for 8, neighbours could be 3,13,15, we find none.

    So search for neighbours of 10 we find 5. so the array now is 8 3 10 5, and our multiset is now empty.

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      8 days ago, # ^ |
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      i also thiking of the same approach.

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      8 days ago, # ^ |
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      But this fails miserably on a testcase like {10 3 10 17 22 15 8} Here, comes the problem, the neighbours could be 8 3 10 or 10 3 10

      Unfortunately, if we take neighbours to be 8 3 10, we cant always empty the multiset.

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        8 days ago, # ^ |
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        for this, if you take 8 3 10 15 22, you are stuck, now no edge element have corresponding neighbour in set :(

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      8 days ago, # ^ |
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      Well again, I am sorry but that is a wrong greedy.

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    8 days ago, # ^ |
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    If you construct a graph such that there exists an edge $$$(u,v)$$$ if and only if $$$|a_u - a_v| = 5$$$ or $$$|a_u - a_v| = 7$$$, then the problem is a Hamiltonian Path Problem, and cannot be solved in poly time.

    I bet this problem cannot be solved better than $$$\mathcal O(2^n \text{poly}(n))$$$.

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      8 days ago, # ^ |
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      Right, this problem is indeed NP-Complete.

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      8 days ago, # ^ |
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      This is a clever observation. Well done !

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