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You are given an array of length $n$. Each element of the array belongs to one of $K$ classes. You are supposed to rearrange the array using minimum number of swap operations so that all elements from the same class are always grouped together, that is they form a contiguous subarray.
For example: $$ \begin{align*} &[2, 1, 3, 3, 2, 2] \longrightarrow [2, 2, 2, 1, 3, 3], \text{ or} \\ &[2, 1, 3, 3, 2, 2] \longrightarrow [1, 2, 2, 2, 3, 3], \text{ or} \\ &[2, 1, 3, 3, 2, 2] \longrightarrow [3, 3, 2, 2, 2, 1]. \end{align*} $$ Three other valid arrangements remain.

What is this problem called in literature? Is there an efficient algorithm for it?

D.W.
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Marko Bukal
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    I'm not sure this problem has a name, though it's certainly possible. Not all conceivable problems have names. – Yuval Filmus Oct 10 '17 at 10:35
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    In practice, this would be called grouping. I'm not aware of terminology in classical algorithmics. (It's certainly an interesting, and potentially hard problem! Minimizing the number of swaps has the feeling of "find the best permutation of groups", which in turns feels NP-hard-ish.) – Raphael Oct 10 '17 at 11:01
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    Well guys, thank you for now. Of course I am interested in solution to the problem, but thought that it was already studied so was asking for a reference. – Marko Bukal Oct 10 '17 at 11:20

2 Answers2

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Note: It is a hardness proof, and I think there are practical algorithms like integer programming, etc.

Given a BIN_PACKING instance where you want to pack $K$ numbers $n_1,\ldots,n_K$ into $L$ bins of size $m_1,\ldots,m_L$, and it is ensured that $\sum n_i=\sum m_j=N$, then we could design a instance of your problem as follows:

  • There are $K+(N+1)(L-1)$ classes;
  • The first $K$ classes have size $n_1,\ldots,n_K$ respectively, and each of the rest classes have size $N+1$;;
  • The array is partitioned into slots of size: $$m_1,(N+1)^2,m_2,(N+1)^2,m_3,\ldots,(N+1)^2,m_L$$ where each slot of size $(N+1)^2$ is packed with $N+1$ classes, arranged contiguously, and the rest are arbitrarily arranged.

Now a key observation is that it is meaningless to keep at least one class in a $(N+1)^2$ slot unmoved and move other ones (because it won't change the size of a 'bin'). So the original bin packing is available if and only if the minimum number of swaps is no larger than $N$. Since BIN-PACKING is known to be strongly NP-complete, your problem is NP-hard.

Wei Zhan
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    This is beautiful! In case others also struggled: (1) $N$ swaps is always enough to create any "bin packing" that we want in the slots of size $m_1, \dots, m_L$, since a single swap is enough to move one element to its final location without disturbing already-placed elements. (2) There are only 2 possible things we could try to do with the length-$(N+1)^2$ "buffer zones" between "bins": move an entire length-$(N+1)$ class at either end somewhere else (but this costs $N+1$ swaps already, so we can't), or "slide" the whole thing one position left or right (but this entails "sliding" each ... – j_random_hacker Oct 10 '17 at 21:41
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    ... class in the buffer left or right one position, and although we can do that with a single swap per class, there are $N+1$ classes in the zone, so at least $N+1$ swaps are needed, so once again: impossible). This point is needed for arguing that a solution to the OP's problem with cost $\le N$ implies a valid bin packing. (3) Because bin packing is strongly NP-complete, the usual "no-no" of creating a number of gadgets (here, array elements) proportional to numbers encoded in the input doesn't apply here :) – j_random_hacker Oct 10 '17 at 21:47
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I also suspect this is NP-hard, but in the absence of an idea for a proof, here are a couple of quickly computable lower bounds that might be useful for checking optimality of a heuristic solution, or pruning a branch-and-bound search.

Let class $i$ contain $n_i$ elements. In any valid solution, class $i$ must begin at some position $j$. Thus we can compute a lower bound $L_i$ on the cost of "fixing" class $i$ by trying all possible starting positions $j$, counting the number of non-$i$ elements in the length-$n_i$ block beginning at position $j$ (each of these positions will require a swap), and taking the minimum. This $L_i$ can be computed for any $i$ in $O(n)$ time using a sliding window approach, for $O(Kn)$ time overall. Two overall lower bounds are then:

  1. Take the maximum over all $L_i$. Tight for $K=2$, probably very weak for large $K$.
  2. Sum all $L_i$ and divide by 2, rounding up. This is valid because any swap can fix at most 2 incorrect positions.

In your example these bounds both give 1 (0.5 can be rounded up in the latter case), which is of course loose.

j_random_hacker
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