Hausdorff Voronoi Diagram Research Articles

Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended

This paper applies the randomized incremental construction (RIC) framework to computing the Hausdorff Voronoi diagram of a family of k clusters of points in the plane. The total number of points is n. The diagram is a generalization of Voronoi diagrams based on the Hausdorff distance function. The combinatorial complexity of the Hausdorff Voronoi diagram is O(n+m), where m is the total number of crossings between pairs of clusters. For non-crossing clusters (m=0), our algorithm works in expected O(n log n + k log n log k) time and deterministic O(n) space. For arbitrary clusters (m=O(n^2)), the algorithm runs in expected O((m+n log k) log n) time and O(m +n log k) space. When clusters cross, bisectors are disconnected curves resulting in disconnected Voronoi regions that challenge the incremental construction. This paper applies the RIC paradigm to a Voronoi diagram with disconnected regions and disconnected bisectors, for the first time.

Preface-S.I.: LATIN 2014

This special issue contains the journal versions of a selection of papers from the 11th Latin American Symposium on Theoretical Informatics (LATIN), held on March 31–April 4 2014, in Montevideo, Uruguay. The conference covered a broad range of topics in theoretical computer science, including algorithms, analytic and enumerative combinatorics, analysis of algorithms, approximation algorithms, automata theory, combinatorics and graph theory, complexity theory, computability, computational algebra, computational geometry, data structures, graph drawing, and random structures. Eight papers were selected for this special issue, all of which went through the thorough reviewing process that is the standard for Algorithmica. The selection was based on two criteria: the quality of the manuscripts and the desire to give readers a sense of the breadth and depth of the topics covered by LATIN. The overview that follows underscores these points In A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters, Cheilaris, Khramtcova, Langerman and Papadopoulou present a construction based on point location, that computes this diagram in expected O(n log2 n) time and expected O(n) space. Their techniques efficiently handle nonstandard characteristics of generalizedVoronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. The conjugacy problem asks whether two words over generators of a fixed group G are conjugated. In Conjugacy in Baumslag’s group, generic case complexity,

A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters

In the Hausdorff Voronoi diagram of a family of clusters of points in the plane, the distance between a point t and a cluster P is measured as the maximum distance between t and any point in P, and the diagram is defined in a nearest-neighbor sense for the input clusters. In this paper we consider non-crossing clusters in the plane, for which the combinatorial complexity of the Hausdorff Voronoi diagram is linear in the total number of points, n, on the convex hulls of all clusters. We present a randomized incremental construction, based on point location, that computes this diagram in expected $$O(n\log ^2{n})$$O(nlog2n) time and expected O(n) space. Our techniques efficiently handle non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. The diagram finds direct applications in VLSI computer-aided design.

The L∞ Hausdorff Voronoi Diagram Revisited

We revisit the [Formula: see text] Hausdorff Voronoi diagram of clusters of points in the plane and present a simple two-pass plane sweep algorithm to construct it. This problem is motivated by applications in the semiconductor industry, in particular, critical area analysis and yield prediction in VLSI design. We show that the structural complexity of this diagram is [Formula: see text], where [Formula: see text] is the number of given clusters and [Formula: see text] is a number of specially crossing clusters, called essential. Our algorithm runs in [Formula: see text] time and [Formula: see text] space, where [Formula: see text] reflects a slight superset of essential crossings, [Formula: see text], and [Formula: see text] is the total number of crossing clusters. For non-crossing clusters ([Formula: see text]) or clusters with only a small number of crossings ([Formula: see text]) the algorithm is optimal. The latter is the case of interest in the motivating application, where [Formula: see text]. This is achieved by augmenting the wavefront data structure of the plane sweep, and a preprocessing step, based on point dominance, which is interesting in its own right.

Computing the Map of Geometric Minimal Cuts

In this paper we consider the following problem of computing a map of geometric minimal cuts (called MGMC problem): Given a graph G=(V,E) and a planar rectilinear embedding of a subgraph H=(V H ,E H ) of G, compute the map of geometric minimal cuts induced by axis-aligned rectangles in the embedding plane. The MGMC problem is motivated by the critical area extraction problem in VLSI designs and finds applications in several other fields. In this paper, we propose a novel approach based on a mix of geometric and graph algorithm techniques for the MGMC problem. Our approach first shows that unlike the classic min-cut problem on graphs, the number of all rectilinear geometric minimal cuts is bounded by a low polynomial, O(n 3). Our algorithm for identifying geometric minimal cuts runs in O(n 3logn(loglogn)3) expected time which can be reduced to O(nlogn(loglogn)3) when the maximum size of the cut is bounded by a constant, where n=|V H |. Once geometric minimal cuts are identified we show that the problem can be reduced to computing the L ∞ Hausdorff Voronoi diagram of axis aligned rectangles. We present the first output-sensitive algorithm to compute this diagram which runs in O((N+K)log2 NloglogN) time and O(Nlog2 N) space, where N is the number of rectangles and K is the complexity of the Hausdorff Voronoi diagram. Our approach settles several open problems regarding the MGMC problem.

THE HAUSDORFF VORONOI DIAGRAM OF POLYGONAL OBJECTS: A DIVIDE AND CONQUER APPROACH

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conquer algorithm for the construction of this diagram that improves upon previous results. As a byproduct we introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI Layout, a measure reflecting the sensitivity of a VLSI design to random manufacturing defects, described in a companion paper.13

The Hausdorff Voronoi Diagram of Point Clusters in the Plane

We study the Hausdorff Voronoi diagram of point clusters in the plane, a generalization ofVoronoi diagrams based on the Hausdorff distance function. We derive a tight combinatorial bound on the structural complexity of this diagram and present a plane sweep algorithm for its construction. In particular, we show that the size of the Hausdorff Voronoi diagram is Θ(n + m), where n is the number of points on the convex hulls of the given clusters, and m is the number of crucial supporting segments between pairs of crossing clusters. The plane sweep algorithm generalizes the standard plane sweep paradigm for the construction of Voronoi diagrams with the ability to handle disconnected Hausdorff Voronoi regions. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI layout, a measure reflecting the sensitivity of the VLSI design to spot defects during manufacturing.