Computing the maximum using $(\min, +)$ formulas

We study computation by formulas over (min,+). We consider the computation of max{x1, . . . , xn} over N as a difference of (min,+) formulas, and show that size n+ n log n is sufficient and necessary. Our proof also shows that any (min,+) formula computing the minimum of all sums of n − 1 out of n variables must have n log n leaves; this too is tight. Our proofs use a complexity measure for (min,+) functions based on minterm-like behaviour and on the entropy of an associated graph.