On the Complexity of Randomly Weighted Voronoi Diagrams

Raichel

2016

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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. /~bar150630. 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.

“…The proof of the following lemma is similar in spirit to the argument of Har-Peled and Raichel [HR14].…”

Section: Putting It All Togethermentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From Proximity to Utility: A Voronoi Partition of Pareto Optima

Chang

Raichel

2016

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Union of Random Minkowski Sums and Network Vulnerability Analysis

Agarwal

Kaplan

et al. 2014

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Let C = {C1, . . . , Cn} be a set of n pairwise-disjoint convex sgons, for some constant s, and let π be a probability density function (pdf) over the non-negative reals. For each i, let Ki be the Minkowski sum of Ci with a disk of radius ri, where each ri is a random non-negative number drawn independently from the distribution determined by π. We show that the expected complexity of the union of K1, . . . , Kn is O(n log n), for any pdf π; the constant of proportionality depends on s, but not on the pdf.Next, we consider the following problem that arises in analyzing the vulnerability of a network under a physical attack. Let G = (V, E) be a planar geometric graph where E is a set of n line segments with pairwise-disjoint relative interiors. Let ϕ : R ≥0 → [0, 1] be an edge failure probability function, where a physical attack at a location x ∈ R 2 causes an edge e of E at distance r from x to fail with probability ϕ(r); we assume that ϕ is of the form 1 − Π(x), where Π is a cumulative distribution function on the non-negative reals. The goal is to compute the most vulnerable location for G, i.e., the location of the attack that maximizes the expected number of failing edges of G. Using our bound on the complexity of the union of random Minkowski sums, we present a near-linear Monte-Carlo algorithm for computing a location that is an approximately most vulnerable location of attack for G. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems-Geometrical problems and computations General TermsAlgorithms, Theory Keywords Arrangements, union complexity, network analysis Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

On the Complexity of Randomly Weighted Multiplicative Voronoi Diagrams

Raichel

2015

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We provide an O(n polylog n) bound on the expected complexity of the randomly weighted multiplicative Voronoi diagram of a set of n sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal et al.[AHKS14] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems.Generalizations of Voronoi diagrams. In the additive weighted Voronoi diagram, the distance to a Voronoi site is the regular Euclidean distance plus some constant (which depends on the site). Additive Voronoi diagrams have linear descriptive complexity in the plane, as their cells are star shaped (and thus simply connected), as can be easily verified. This holds even if the sites are arbitrary convex sets. In the multiplicative weighted Voronoi diagram, for each site one multiplies the Euclidean distance by a constant (again, that depends on the site). However, unlike the additive case, the worst case complexity for multiplicative weighted Voronoi diagrams is Θ(n 2 ) [AE84], even in the plane. In the weighted case, the cells are not necessarily connected, and a bisector of two sites is either a line or an (Apollonius) circle.In the Power diagram, each site s i has an associated radius r i , and the distance of a point p to this site is s i − p 2 − r 2 i ; that is, the squared length of the tangent from p to the disk of radius r i centered at s i . As such, Power diagrams allow including weight in the distance function, while still having bisectors that are straight lines and having linear combinatorial complexity overall.Klein [Kle88] introduced (and this was further refined by Klein et al. [KLN09]) the notion of abstract Voronoi diagrams to help unify the ever growing list of variants of Voronoi diagrams which have been considered. Specifically, a simple set of axioms was identified, focusing on the bisectors and the regions they define, which classifies a large class of Voronoi diagrams with linear complexity (hence such axioms are not intended to model, for example, multiplicative diagrams).Randomization and Expected Complexity. In many cases, there is a big discrepancy between the worst case analysis of a structure (or an algorithm) and its average case behavior. This suggests that in practice, the worst case is seldom encountered. For example, recently, Agarwal et al. [AKS13, AHKS14], showed that the expected union complexity of a set of randomly expanded disjoint segments is O(n log n), while in the worst case the union complexity can be quadratic. In other words, Agarwal et al. bounded the expected complexity of a level set of the randomly weighted Voronoi diagram of disjoint segments.If the sites are placed...