Voronoi Diagram Of Line Segments Research Articles

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions

AbstractWe 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 $$\mathbb {S}^{d-1}$$ S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is $$O(\min \{k,n-k\}n^{d-1})$$ O ( min { k , n - k } n d - 1 ) , which is tight for $$n-k=O(1)$$ n - k = O ( 1 ) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its $$(d-1)$$ ( 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 $$n \ge 2$$ n ≥ 2 lines in general position has exactly $$n(n-1)$$ n ( n - 1 ) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in $$O(n^{d-1} \alpha (n))$$ O ( n d - 1 α ( n ) ) time, for $$d\ge 4$$ d ≥ 4 , while if $$d=3$$ d = 3 , the time drops to worst-case optimal $$\Theta (n^2)$$ Θ ( n 2 ) . We extend the obtained results to bounded polyhedra and clusters of points as sites.

Convex-Straight-Skeleton Voronoi Diagrams for Segments and Convex Polygons

In this paper, we study the convex-straight-skeleton Voronoi diagrams of line segments and convex polygons. We explore the combinatorial complexity of these diagrams, and provide efficient algorithms for computing compact representations of them.

Layout pattern analysis using the Voronoi diagram of line segments

Early identification of problematic patterns in very large scale integration (VLSI) designs is of great value as the lithographic simulation tools face significant timing challenges. To reduce the processing time, such a tool selects only a fraction of possible patterns which have a probable area of failure, with the risk of missing some problematic patterns. We introduce a fast method to automatically extract patterns based on their structure and context, using the Voronoi diagram of line-segments as derived from the edges of VLSI design shapes. Designers put line segments around the problematic locations in patterns called “gauges,” along which the critical distance is measured. The gauge center is the midpoint of a gauge. We first use the Voronoi diagram of VLSI shapes to identify possible problematic locations, represented as gauge centers. Then we use the derived locations to extract windows containing the problematic patterns from the design layout. The problematic locations are prioritized by the shape and proximity information of the design polygons. We perform experiments for pattern selection in a portion of a 22-nm random logic design layout. The design layout had 38,584 design polygons (consisting of 199,946 line segments) on layer Mx, and 7079 markers generated by an optical rule checker (ORC) tool. The optical rules specify requirements for printing circuits with minimum dimension. Markers are the locations of some optical rule violations in the layout. We verify our approach by comparing the coverage of our extracted patterns to the ORC-generated markers. We further derive a similarity measure between patterns and between layouts. The similarity measure helps to identify a set of representative gauges that reduces the number of patterns for analysis.

The Higher-Order Voronoi Diagram of Line Segments

The order- $$k$$ Voronoi diagram of line segments has properties surprisingly different from its counterpart for points. For example, a single order- $$k$$ Voronoi region may consist of $$\varOmega (n)$$ disjoint faces. In this paper, we analyze the structural properties of this diagram and show that its combinatorial complexity is $$O(k(n-k))$$ , for $$n$$ non-crossing line segments, despite the presence of disconnected regions. The same bound holds for $$n$$ intersecting line segments, for $$k\ge n/2$$ . We also consider the order- $$k$$ Voronoi diagram of line segments that form a planar straight-line graph, and augment the definition of an order- $$k$$ diagram to cover sites that are not disjoint. On the algorithmic side, we extend the iterative approach to construct this diagram, handling complications caused by the presence of disconnected regions. All bounds are valid in the general $$L_p$$ metric, $$1\le p\le \infty $$ . For non-crossing segments in the $$L_\infty $$ and $$L_1$$ metrics, we show a tighter $$O((n-k)^2)$$ bound for $$k>n/2$$ .

Tight bound and improved algorithm for farthest-color Voronoi diagrams of line segments

We establish a tight bound on the worst-case combinatorial complexity of the farthest-color Voronoi diagram of line segments in the plane. More precisely, given k sets of n total line segments, the combinatorial complexity of the farthest-color Voronoi diagram is shown to be Θ(kn+h) in the worst case, under any Lp metric with 1≤p≤∞, where h is the number of crossings between the n line segments. We then present an improved analysis on Huttenlocher et al.'s algorithm [9] to show that the diagram can be computed in O((kn+h)logn) time under the L1 or L∞ metric, or in O((kn+h)(α(k)logk+logn)) time under the Lp metric for any 1<p<∞, where α(⋅) denotes the inverse of the Ackermann function.

ON THE FARTHEST LINE-SEGMENT VORONOI DIAGRAM

The farthest line-segment Voronoi diagram illustrates properties surprisingly different from its counterpart for points: Voronoi regions may be disconnected and they are not characterized by convex-hull properties. In this paper we introduce the farthest hull and its Gaussian map as a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram, and derive tighter bounds on the (linear) size of this diagram. With the purpose of unifying construction algorithms for farthest-point and farthest line-segment Voronoi diagrams, we adapt standard techniques to construct a convex hull and compute the farthest hull in O(n log n) or output sensitive O(n log h) time, where n is the number of line-segments and h is the number of faces in the corresponding farthest Voronoi diagram. As a result, the farthest line-segment Voronoi diagram can be constructed in output sensitive O(n log h) time. Our algorithms are given in the Euclidean plane but they hold also in the general Lp metric, 1 ≤ p ≤ ∞.

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Abstract. We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probability using O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of work done since the sequential time bound for this problem is Ω(n log n) . Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm, given by Goodrich, O'Dunlaing, and Yap, which runs in O( log 2 n) time using O(n) processors. We obtain this result by using a new ``two-stage'' random sampling technique. By choosing large samples in the first stage of the algorithm, we avoid the hurdle of problem-size ``blow-up'' that is typical in recursive parallel geometric algorithms. We combine the two-stage sampling technique with efficient search and merge procedures to obtain an optimal algorithm. This technique gives an alternative optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use the transformation to three-dimensional half-space intersection).

A Randomized Algorithm for the Voronoi Diagram of Line Segments on Coarse-Grained Multiprocessors

We present a randomized algorithm for computing the Voronoi diagram of line segments using coarse-grained parallel machines. Operating on P processors, for any input of n line segments, this algorithm performs O((n log n)/P) local operations per processor, O(n/P) messages per processor, and O(1) communication phases, with high probability for n=Ω(P 3+e ) .

Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments and Motion Planning in the Plane

The motion planning problem of an object with two degrees of freedom moving in the plane can be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional mobile object B with two degrees of freedom, determine if it is possible to move B from a start position to a final position while avoiding the obstacles. If so, plan a path for such a motion. Techniques from computational geometry have been used to develop exact algorithms for this fundamental case of motion planning. In this paper we obtain optimal mesh implementations of two different methods for planning motion in the plane. We do this by first presenting optimal mesh algorithms for some geometric problems that, in addition to being important substeps in motion planning, have numerous independent applications in computational geometry. In particular, we first show that the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane can be constructed in O(√ n) time on a √ n × √ n mesh, which is optimal for the mesh. Consequently, we obtain an optimal mesh implementation of the sequential motion planning algorithm described by Ó′Dúlainy and Yap (J. Algorithms 6 (1985), 104-111); in other words, given a disc B and a polygonal obstacle set of size n, we can plan a path (if it exists) for the motion of B from a start position to a final position in O(√ n) time on a mesh of size n. We also show that the shortest path motion between a start position and a final position for a convex object B (of constant size) moving among convex polygonal obstacles of total size n can be found in O(n) time on an n × n mesh, which is worst-case optimal.

Applications of random sampling to on-line algorithms in computational geometry

This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power of the technique is illustrated by new efficient on-line algorithms for constructing convex hulls and Voronoi diagrams in any dimension, Voronoi diagrams of line segments in the plane, arrangements of curves in the plane, and others.