The direct dominance problem

Given two points a=(a<subscrpt>1</subscrpt>,a<subscrpt>2</subscrpt>,…,a<subscrpt>d</subscrpt>) and b=(b<subscrpt>1</subscrpt>,b<subscrpt>2</subscrpt>,…,b<subscrpt>d</subscrpt>) in d-dimensional space, a dominates b if a≠b and for each i=1…d holds a<subscrpt>i</subscrpt>≥b<subscrpt>i</subscrpt>. The direct dominance problem consists of computing a relation of minimal size on a given set of n points such that the transitive closure of the relation gives all the dominances in the set. We present an &Ogr;(n log n +k) time and &Ogr;(n) space algorithm for the problem of two-dimensional space. The algorithm is optimal within a constant factor. Further, we show that the three-dimensional problem can be solved in &Ogr;((n+k) log<supscrpt>2</supscrpt>n) time and space, or alternatively in &Ogr;((n+k) log<supscrpt>3</supscrpt>n) time and &Ogr;((n+k)log n) space.