Computing on-line the lattice of maximal antichains of posets

Abstract

We consider the on-line computation of the lattice of maximal antichains of a finite poset$\tilde P$. This on-line computation satisfies what we call the “linear extension hypothesis”: the new incoming vertex is always maximal in the current subposet of$\tilde P$. In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of$\mathcal{O}((\left| P \right| + \omega ^2 (P))\omega (P)\left| {MA(P)} \right|),$, where |P| is the number of elements of the poset,$\tilde P$, |MA(P)| is the number of maximal antichains of$\tilde P$ and ω(P) is the width of$\tilde P$. This is more efficient than the best off-line algorithms known so far.