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Do you really understand Connected Components DP?

Revision en2, by TheScrasse, 2022-11-21 18:09:58

[title inspired by this blog]

Hello everyone,

today, during a NEERC virtual contest, I found an unintended solution for problem 1089I - Interval-Free Permutations. I've checked all the official submissions and no one of them uses my solution, so I think it's worth sharing it.

Abridged statement: count the permutations of [1,,n] such that there are no subarrays of length between 2 and n1 where all the values are contiguous. For example, the permutation [2,8,4,6,3,5,1,7] is bad because it contains [4,6,3,5] as a subarray.

My solution:

  • Let's use PIE (inclusion-exclusion principle) on minimal bad subarrays.
  • Let's use Connected Components DP, somehow keeping track of minimal bad subarrays.

  • Let dpi,j,k be the number of ordered sets of j connected components with total length i, and k= parity of minimal bad subarrays. Then, the number of good permutations of length i is dpi,1,0dpi,1,1.
    Instead of adding elements one at a time to the permutation, let's consider two cases:
    - We add only one element (using the standard Connected Components DP transitions);
    - We add a minimal bad subarray of length 2li1 (the transitions are similar, but using dpil,*,k1 instead of dpi1,*,k. Note that the number of ways to add a minimal bad subarray of length l is equal to the number of good permutations of length l.
  • When you calculate dpi,,, assume that dpj,1,=0 (j<i), because the corresponding elements are good as arrays but bad as subarrays.

This solution is actually wrong: in most cases, it produces the correct output ±2! It turns out it's enough to add 2(1)n to the result, for n3. (AC code: 181878668)

So my questions are:

  • Why is the initial solution wrong?
Hint
  • Why is the solution with 2(1)n correct? Actually I don't know, I've just found the formula using the samples.
  • Can this solution be generalized to solve harder problems? For example,
    "An array is weird if the local minimums are bitonic (i.e., decreasing, then increasing). Count the weird permutations of [1,,n] such that there are no weird subarrays of length between 2 and n1 where all the values are contiguous."
Tags dp, dumb_experiments

History

 
 
 
 
Revisions
 
 
  Rev. Lang. By When Δ Comment
en6 English TheScrasse 2022-11-21 18:50:05 14
en5 English TheScrasse 2022-11-21 18:48:58 37 Tiny change: 'he answer for all $' -> 'he answer (modulo a prime, given in the input) for all $'
en4 English TheScrasse 2022-11-21 18:47:37 47
en3 English TheScrasse 2022-11-21 18:30:18 8
en2 English TheScrasse 2022-11-21 18:09:58 12 Tiny change: ' inspired from [this](ht' -> ' inspired by [this](ht'
en1 English TheScrasse 2022-11-21 18:03:42 2781 Initial revision (published)