From Proximity to Utility: A Voronoi Partition of Pareto Optima

Abstract

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

Appendix: An Integral Calculation

Lemma 28

Let $F_{d}\left( {\varDelta }\right)$ be the total measure of the points $\mathsf {p}= (\mathsf {p}_1,\ldots , \mathsf {p}_d)$ in the hypercube $[0,1]^d$, such that $\mathrm {pv}\left( {\mathsf {p}}\right) = \mathsf {p}_1 \mathsf {p}_2 \ldots \mathsf {p}_d \le \varDelta$. That is, $F_{d}\left( {\varDelta }\right)$ is the measure of all points in hypercube with point volume at most $\varDelta$. Then

$$\begin{aligned} \Big . F_{d}\left( {\varDelta }\right) = \sum _{i=0}^{d-1} \frac{\varDelta }{i!} { \ln ^{i} \frac{1}{\varDelta } }. \end{aligned}$$

Proof

The claim follows by tedious but relatively standard calculations. As such, the proof is included for the sake of completeness (Fig. 4).

Fig. 4

The $d=2$ case: the region with point volume at most $\varDelta$ is decomposed into two pieces, the shaded rectangle and the area below the intersection of the unit square with the curve $xy=\varDelta$

The case for $d=1$ is trivial. Consider the $d=2$ case. Here the points whose point volume equals $\varDelta$ are defined by the curve $xy = \varDelta$. This curve intersects the unit square at the point $(\varDelta ,1)$. As $F_{d}\left( {\varDelta }\right)$ is the total volume under this curve in the unit square we have that

$$\begin{aligned} F_{2}\left( {\varDelta }\right) = \varDelta + \int _{x=\varDelta }^1 \frac{\varDelta }{x} \mathrm {d}x = \varDelta + \varDelta \ln \frac{1}{\varDelta }. \end{aligned}$$

In general, we have

$$\begin{aligned} \frac{1}{(d-1)!} \int _{x=\varDelta }^1 \frac{\varDelta }{x} \ln ^{d-1} \frac{x}{\varDelta } \mathrm {d}x = \frac{\varDelta }{(d-1)!} \Big [{ \frac{1}{d} \ln ^{d} \frac{x}{\varDelta } }\Big ]_{x=\varDelta }^1 = \frac{\varDelta }{d!} \ln ^{d} \frac{1}{\varDelta }. \end{aligned}$$

Now assume inductively that

$$\begin{aligned} F_{d-1}\left( {\varDelta }\right) = \sum _{i=0}^{d-2} \frac{1}{i!} \varDelta \ln ^{i} \frac{1}{\varDelta }, \end{aligned}$$

then we have

$$\begin{aligned} F_{d}\left( {\varDelta }\right)&= \varDelta + \int _{x_d=\varDelta }^1 F_{d-1}\big ({\frac{\varDelta }{x_d}}\big )\mathrm {d}x_d = \varDelta + \int _{x_d=\varDelta }^1 \Big ({ \sum _{i=0}^{d-2} \frac{ \varDelta }{i! x_d} \ln ^{i} \frac{x_d}{\varDelta } }\Big ) \mathrm {d}x_d\\&= \varDelta + \sum _{i=0}^{d-2} \frac{1}{i!} \Big ({ \int _{x_d=\varDelta }^1 \frac{\varDelta }{x_d} \ln ^{i} \frac{x_d}{\varDelta } \mathrm {d}x_d}\Big ) = \varDelta + \sum _{i=1}^{d-1} \frac{\varDelta }{i!} { \ln ^{i} \frac{1}{\varDelta } } = \sum _{i=0}^{d-1} \frac{\varDelta }{i!} { \ln ^{i} \frac{1}{\varDelta }}. \end{aligned}$$

$\square$