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A $n$-item list can be verified as sorted by comparing every item to its neighbor. In my application, I will not be able to compare every item with its neighbor: instead, the comparisons will sometimes be between distant elements. Given that the list contains more than three items and also that comparison is the only supported operation, does there ever exist a "network" of comparisons that will prove that the list is sorted, but is missing at least one direct neighbor-to-neighbor comparison?

Formally, for a sequence of elements $e_i$, I have a set of pairs of indices $(j,k)$ for which I know whether $e_j > e_k$, $e_j = e_k$, or $e_j < e_k$. There exists a pair $(l,l+1)$ that is missing from the set of comparisons. Is it ever possible, then, to prove that the sequence is sorted?

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It is impossible. Suppose that you have the result of all comparisons except for the pair $(i,i+1)$. Then you wouldn't be able to distinguish between the following two cases: $$ 1,2,\ldots,i-1,i,i+1,i+2,\ldots,n \\ 1,2,\ldots,i-1,i+1,i,i+2,\ldots,n $$

Yuval Filmus
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