From Proximity to Utility : 1 A Voronoi Partition of Pareto

5 We present an extension of Voronoi diagrams where when considering which site a client 6 is going to use, in addition to the site distances, other site attributes are also considered (for 7 example, prices or weights). A cell in this diagram is then the locus of all clients that consider 8 the same set of sites to be relevant. In particular, the precise site a client might use from this 9 candidate set depends on parameters that might change between usages, and the candidate 10 set lists all of the relevant sites. The resulting diagram is significantly more expressive than 11 Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might 12 be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same 13 distribution (note that the locations are fixed), then the expected complexity of the candidate 14 diagram is near linear. 15 To this end, we derive several new technical results, which are of independent interest. In 16 particular, we provide a high-probability, asymptotically optimal bound on the number of Pareto 17 optima points in a point set uniformly sampled from the d-dimensional hypercube. To do so 18 we revisit the classical backward analysis technique, both simplifying and improving relevant 19 results in order to achieve the high-probability bounds. 20