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Let $S$ be a commutative semiring and $T\subset S$, how many semiring operations are required to compute the following $$\prod_{a,b\in T} a+b$$?

This problem can be solved for commutative rings in $O(n \log n)$ ring operations using multipoint polynomial evaluation from a related problem. However, the algorithm requires there is an inverse to addition.

Actually I'm mostly interested in a similar problem. We have $M = \{A_1,\ldots,A_k\}$ such that $A_i\subset S$. We want to compute

$$\prod_{A\in M} \prod_{A\neq B\in M} \prod_{a\in A,b\in B} a+b$$

with least number of semiring operations.

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Not an answer to your exact question, but may be useful if you happen to have a semiring with cancellation: if you have an algorithm which works in any ring, you can apply that algorithm to the difference ring $S-S$ consisting of subtraction pairs $s - t$. The result will have the form $s - t$ where $s,t \in S$, and the answer will be $d \in S$ s.t. $s = d + t$. Thus, in addition to semiring operations, you'd need to perform a single cancellation step at the end.

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    $\begingroup$ A difference ring can be defined only for a semiring with the cancellation property with respect to addition (x+z=y+z ⇒ x=y). For example, the (min, +) semiring on ℕ∪{∞} does not have the cancellation property. Therefore, I do not think that this trick works. $\endgroup$ – Tsuyoshi Ito Feb 15 '13 at 21:51

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