Elena Khramtcova

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

Searching Edges in the Overlap of Two Plane Graphs

O(n\log ^2{n})

Dynamic Smooth Compressed Quadtrees

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.

Randomized incremental construction for the hausdorff voronoi diagram revisited and extended

We present linear-time algorithms to construct tree-like Voronoi diagrams with disconnected regions after the sequence of their faces along an enclosing boundary (or at infinity) is known. We focus on the farthest-segment Voronoi diagram, however, our techniques are also applicable to constructing the order-\((k{+}1)\) subdivision within an order-k Voronoi region of segments and updating a nearest-neighbor Voronoi diagram of segments after deletion of one site. Although tree-structured, these diagrams illustrate properties surprisingly different from their counterparts for points. The sequence of their faces along the relevant boundary forms a Davenport-Schinzel sequence of order \(\ge 2\). Once this sequence is known, we show how to compute the corresponding Voronoi diagram in linear time, expected or deterministic, augmenting the existing linear-time frameworks for points in convex position with the ability to handle non-point sites and multiple Voronoi faces.

Dynamic Stabbing Queries with Sub-logarithmic Local Updates for Overlapping Intervals

Consider a pair of plane straight-line graphs whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a \(O(n\log {n})\)-time O(n)-space technique to preprocess such a pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in \(O(n\log {n})\) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains, one of which is convex, in \(O(n\log {n})\) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in \(O((n+m)\log ^3{n})\) time and \(O(n+m)\) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n disjoint axis-aligned rectangles in \(O(n\log ^2{n})\) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal \(O(n\log {n})\) time. All these results are new or improve on the best known algorithms.

Dynamic stabbing queries with sub-logarithmic local updates for overlapping intervals

In the Hausdorff Voronoi diagram of a set of point-clusters 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 reveals the nearest cluster to t. This diagram finds direct applications in VLSI computer-aided design. In this paper, we consider “non-crossing” clusters, for which the combinatorial complexity of the diagram is linear in the total number n of points on the convex hulls of all clusters. We present a randomized incremental construction, based on point-location, to compute the diagram in expected O(nlog2 n) time and expected O(n) space, which considerably improves previous results. Our technique efficiently handles 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.

Dynamic stabbing queries with sub-logarithmic local updates for overlapping intervals : Proc. 12th International Computer Science Symposium in Russia

We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in R^d can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2^{O(d log d)} time, except for the point location part in the second version which has a lower bound of Omega(log n); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.

Stabbing Circles for Sets of Segments in the Plane

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.