Direct dominance of points

Several transitive relations of geometrical objects (like inclusion of intervals on a line or polygons in the plain), which are important in VLSI design applications, can be translated into the dominance relation a dominates b iff (a ≠ b and a _j ≧ b _j for j = 1,…d) of points a = (a ₁,...,a _d ),b = (b ₁,…b _d ) in R ^d by representing the objects as points in a suitable way. If only the transitive reduction (see [7]) of the given relation is required and not all the implications by transitivity, one can restrict oneself to the direct dominances in the corresponding point set N; here a dominates b directly means that a dominates b and there is no—with respect to dominance—intermediate c in N (see [5]). To estimate the advantage of this restriction, information about the numbers of dominant and directly dominant pairs in a set of n points is required, both numbers essentially depending upon how the points are distributed in R ^d . In this paper we assume the n points in question to be identically and independently distributed; then we can expect q·n·(n–1) dominance pairs. For a certain class of distributions including the uniform distribution we prove the theorem, that the expected number of direct dominance pairs is asymptotically equal to 1/(d?1)! · n1n ^{d ? 1}(n). Hence algorithms which compute only the direct dominances instead of all dominances are worth to be considered.