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From Proximity to Utility: A Voronoi Partition of Pareto Optima

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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.

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Notes

  1. 1.

    Unless you are feeling adventurous enough that day to eat the frozen mystery food stuck to the back of the freezer, which we strongly discourage you from doing.

  2. 2.

    There is of course a lot of other work on Pareto optimal points, from connections to Nash equilibrium to scheduling. We resisted the temptation of including many such references which are not directly related to our paper.

  3. 3.

    The lifting of the sites to the paraboloid \(z = -(x^2+y^2)\) is done so that the definition of the k-level coincide with the standard definition.

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Acknowledgments

The authors would like to thank Pankaj Agarwal, Ken Clarkson, Nirman Kumar, and Raimund Seidel for useful discussions related to this work. We are also grateful to the anonymous SoCG reviewers for their helpful comments. Work on this paper was partially supported by NSF AF award CCF-1421231, and CCF-1217462. A preliminary version of the paper appeared in the 31st International Symposium on Computational Geometry (SoCG 2015) [12].

Author information

Correspondence to Benjamin Raichel.

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Editor in Charge: Kenneth Clarkson

Appendix: An Integral Calculation

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
figure4

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

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

\(\square \)

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Chang, H., Har-Peled, S. & Raichel, B. From Proximity to Utility: A Voronoi Partition of Pareto Optima. Discrete Comput Geom 56, 631–656 (2016). https://doi.org/10.1007/s00454-016-9808-0

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Keywords

  • Voronoi diagrams
  • Expected complexity
  • Backward analysis
  • Pareto optima
  • Candidate diagram
  • Clarkson-Shor technique