Online linear discrepancy of partially ordered sets

Abstract

The linear discrepancy of a poset P is the least k for which th ere is a linear extension L of P such th at if x and yare incomparable in P, then |hL(x) hL(y)| ≤ k, where hL(X) is th e height of x in L. In this paper, we consider linear discrepancy in an online set t ing and devise an online algorit hm that constructs a linear extension L of a poset P so that |hL(x) - hL(y)| ≤ 3k - 1, when th e linear discr epancy of P is k. This inequality is best possible, even for the class of inte rval orders. Furthermore, if th e poset P is a semiorder, th en the inequality is improved to |hL(x) - hL(y)| ≤ 2k. Again, this result is best possible.