Papers by Evanthia Papadopoulou
Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewis... more Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewiselinear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it. We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex ...
We revisit the k-nearest-neighbor (k-NN) Voronoi diagram and present a new paradigm for its const... more We revisit the k-nearest-neighbor (k-NN) Voronoi diagram and present a new paradigm for its construction. We introduce the k-NN Delaunay graph, which is the graph-theoretic dual of the k-NN Voronoi diagram, and use it as a base to directly compute this diagram in R2. We implemented our paradigm in the L1 and L1 metrics, using segment-dragging queries, resulting in the rst output sensitive, O((n+m) log n)-time algorithm to compute the k-NN Voronoi diagram of n points in the plane, where m is the structural complexity (size) of this diagram. We also show that the structural complexity of the k-NN Voronoi diagram in the L1 (equiv. L1) metric is O(minfk(n k); (n k)2g). Ecient implementation of our paradigm in the L2 (resp. Lp, 1 < p <1) metric remains an open problem.
In this paper we address the L1 Voronoi diagram of polygonal objects and present applications in ... more In this paper we address the L1 Voronoi diagram of polygonal objects and present applications in VLSI layout and manufacturing. We show that the L1 Voronoi diagram of polygonal objects consists of straight line segments and thus it is much simpler to compute than its Euclidean counterpart; the degree of the computation is significantly lower. Moreover, it has a natural interpretation. In applications where Euclidean precision is not essential the L1 Voronoi diagram can provide a better alternative. Using the L1 Voronoi diagram of polygons we address the problem of calculating the critical area for shorts in a VLSI layout. The critical area computation is the main computational bottleneck in VLSI yield prediction.
Lecture Notes in Computer Science, 2017
The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi d... more The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi diagrams based on the Hausdorff distance function. Its combinatorial complexity is \(O(n+m)\), where n is the total number of points and \(m\) is the number of crossings between the input clusters (\(m=O(n^2)\)); the number of clusters is k. We present efficient algorithms to construct this diagram via the randomized incremental construction (RIC) framework [Clarkson et al. 89,93]. For non-crossing clusters (\(m=0\)), our algorithm runs in expected \(O(n\log {n} + k\log n \log k)\) time and deterministic O(n) space. For arbitrary clusters the algorithm runs in expected \(O((m+n\log {k})\log {n})\) time and \(O(m+n\log {k})\) space. The two algorithms can be combined in a crossing-oblivious scheme within the same bounds. We show how to apply the RIC framework efficiently to handle non-standard characteristics of generalized Voronoi diagrams, including sites (and bisectors) of non-constant complexity, sites that are not enclosed in their Voronoi regions, empty Voronoi regions, and finally, disconnected bisectors and Voronoi regions. The diagram finds direct applications in VLSI CAD.
Discrete Applied Mathematics, 2021
Abstract We present a generalization of a combinatorial result by Aggarwal et al. (1989) on a lin... more Abstract We present a generalization of a combinatorial result by Aggarwal et al. (1989) on a linear-time algorithm that selects a constant fraction of leaves, with pairwise disjoint neighborhoods, from a binary tree embedded in the plane. This result of Aggarwal et al. (1989) is essential to the linear-time framework, which they also introduced, that computes certain Voronoi diagrams of points with a tree structure in linear time. An example is the diagram computed while updating the Voronoi diagram of points after deletion of one site. Our generalization allows that only a fraction of the tree leaves is considered, and it is motivated by research on linear time construction algorithms for Voronoi diagrams of non-point sites. We are given a plane tree T of n leaves, m of which have been marked, and each marked leaf is associated with a neighborhood (a subtree of T ) such that any two topologically consecutive marked leaves have disjoint neighborhoods. We show how to select in linear time a constant fraction of the marked leaves having pairwise disjoint neighborhoods.
We present a generalization of a combinatorial result from Aggarwal, Guibas, Saxe and Shor [1] on... more We present a generalization of a combinatorial result from Aggarwal, Guibas, Saxe and Shor [1] on selecting a fraction of leaves, with pairwise disjoint neighborhoods, in a tree embedded in the plane. This result has been used by linear-time algorithms to compute certain tree-like Voronoi diagrams, such as the Voronoi diagram of points in convex position. Our generalization allows that only a fraction of the tree leaves is considered: Given is a plane tree T of n leaves, m of which have been marked. Each marked leaf is associated with a neighborhood (a subtree of T) and any topologically consecutive marked leaves have disjoint neighborhoods. We show how to select in linear time a constant fraction of the marked leaves that have pairwise disjoint neighborhoods.
In this paper, we present a new approach for computing the critical area for shorts in a circuit ... more In this paper, we present a new approach for computing the critical area for shorts in a circuit layout. The critical area calculation is the main computational problem in very large scale integration yield prediction. The method is based on the concept of Voronoi diagrams and computes the critical area for shorts (for all possible defect radii, assuming square defects) accurately in time, where is the size of the input. The method is presented for rectilinear layouts and layouts containing edges of slope 1. As a byproduct, we briefly sketch how to speed up the grid method of Wagner and Koren (18).
We address the problem of computing critical area for opens in a circuit layout in the presence o... more We address the problem of computing critical area for opens in a circuit layout in the presence of loops and redundant interconnects. The extraction of critical area is the main computational problem in VLSI yield prediction for random manufacturing defects. Our approach first models the problem as a graph problem and solves it efficiently by exploiting its geometric nature. The approach expands the Voronoi critical area computation paradigm [10, 7] with the ability to accurately compute critical area for missing material defects in a net-aware fashion. Generalized Voronoi diagrams used in the solution are combinatorial structures of independent interest.
We show that the number of 3-dimensional cells in the farthest-site Voronoi diagram of n segments... more We show that the number of 3-dimensional cells in the farthest-site Voronoi diagram of n segments (or lines) in R is Θ(n) in the worst case, and that the diagram can be computed in O(k log n) time, where k is the complexity of the diagram, using O(k) space. In R, the number of d-dimensional cells in the diagram is Θ(nd−1) in the worst case.
We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line ... more We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions Sd−1. We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k, n−k}nd−1), which is tight for n−k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d−1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n2−n three-dimensional cells, when n ≥ 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(nd−1α(n)) time, while if d = 3, the time drops to wors...
Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open... more Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem for a long time. Similarly for various concrete Voronoi diagrams of generalized sites, other than points. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion. We introduce the concept of a Voronoi-like diagram, a relaxed version of a Voronoi construct that has a structure similar to an abstract Voronoi diagram, without however being one. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under an insertion operation, thus, enabling its use in incremental constructions.
In this paper we address the L1 Voronoi diagram of polygonal objects and present applications in ... more In this paper we address the L1 Voronoi diagram of polygonal objects and present applications in VLSI layout and manufacturing. We show that the L1 Voronoi diagram of polygonal objects consists of straight line segments and thus it is much simpler to compute than its Euclidean counterpart; the degree of the computation is signiicantly lower. Moreover, it has a natural interpretation. In applications where Euclidean precision is not essential the L1 Voronoi diagram can provide a better alternative. Using the L1 Voronoi diagram of polygons we address the problem of calculating the critical area for shorts in a VLSI layout. The critical area computation is the main computational bottleneck in VLSI yield prediction.
We introduce the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are ... more We introduce the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays, and the distance function is the counterclockwise angular distance between a point and a ray-site. This novel Voronoi diagram is motivated by illumination and coverage problems, where a domain has to be covered by floodlights (wedges) of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the ray Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle. This last algorithm improves upon previous results, settling an interesting open problem.
Computer Graphics Forum, 2016
Let X = { f 1 , . . ., fn} be a set of scalar functions of the form f i : R 2 → R which satisfy s... more Let X = { f 1 , . . ., fn} be a set of scalar functions of the form f i : R 2 → R which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε-isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi-algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.
Given a set A of n points in R with weight function w : A → R>0, the Fermat distance function ... more Given a set A of n points in R with weight function w : A → R>0, the Fermat distance function is φ(x) := ∑ a∈A w(a)∥x − a∥. A classic problem in facility location dating back to 1643, is to find the Fermat point x∗, the point that minimizes the function φ. We consider the problem of computing a point x̃ that is an ε-approximation of x∗ in the sense that ∥x̃ − x∗∥ < ε. The algorithmic literature has so far used a different notion based on ε-approximation of the value φ(x∗). We devise a certified subdivision algorithm for computing x̃, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x∗, turning it into an ε-approximate Fermat point algorithm. Our second problem is the certified construction of ε-isotopic approximations of n-ellipses. These are the level sets φ−1(r) for r > φ(x∗) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as ...
Journal of Combinatorial Optimization
The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi d... more The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi diagrams based on the Hausdorff distance function. Its combinatorial complexity is O(n + m), where n is the total number of points and m is the number of crossings between the input clusters (m = O(n 2)); the number of clusters is k. We present efficient algorithms to construct this diagram following the randomized incremental construction (RIC) framework [Clarkson et al. 89, 93]. Our algorithm for non-crossing clusters (m = 0) runs in expected O(n log n + k log n log k) time and deterministic O(n) space. The algorithm for arbitrary clusters runs in expected O((m + n log k) log n) time and O(m + n log k) space. The two algorithms can be combined in a crossing-oblivious scheme within the same bounds. We show how to apply the RIC framework to handle non-standard characteristics of generalized Voronoi diagrams, including sites (and bisectors) of non-constant complexity, sites that are not enclosed in their Voronoi regions, empty Voronoi regions, and finally, disconnected bisectors and disconnected Voronoi regions. The Hausdorff Voronoi diagram finds direct applications in VLSI CAD.
Computational Geometry, 2016
Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regi... more Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance. In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn 1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O(k 2 n log n) and O(n 2 2 α(n) log n) time, respectively, where α(•) is the inverse of the Ackermann function. Our approach directly implies an O(kn 1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the L p norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the abstract setting and to non-point sites.
International Journal of Computational Geometry and Applications, 2001
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Papers by Evanthia Papadopoulou