Evanthia Papadopoulou

The all-pairs quickest path problem

In this paper we discuss the quickest path problem and proposed an θ(λn 2 ) space data structure which allows to answer quickest path queries in O(log r) time and whose construction requires θ(min {mn 2 , rn 2 log n+rnm}) time

Critical area computation via Voronoi diagrams

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 O(n log n) time, where n is the size of the input. The method is presented for rectilinear layouts and layouts containing edges of slope /spl plusmn/1. As a byproduct, we briefly sketch how to speed up the grid method of Wagner and Koren [1995].

THE L∞ VORONOI DIAGRAM OF SEGMENTS AND VLSI APPLICATIONS

In this paper we address the L∞ Voronoi diagram of polygonal objects and present application in VLSI layout and manufacturing. We show that L∞ 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 L∞ Voronoi diagram can provide a better alternative. Using the L∞ 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.

Critical area computation for missing material defects in VLSI circuits

We address the problem of computing critical area for missing material defects in a circuit layout. The extraction of critical area is the main computational problem in very large scale integration yield prediction. Missing material defects cause open circuits and are classified into breaks and via blocks. Our approach is based on the L/sub /spl infin// medial axis of polygons and the weighted L/sub /spl infin// Voronoi diagram of segments. We also introduce the min-max Voronoi diagram of rectangles, a combinatorial structure of independent interest. The critical area problem for breaks and via blocks is reduced to variations of weighted L/sub /spl infin// Voronoi diagram of segments. Plane sweep algorithms to compute the appropriate Voronoi diagrams for each case are presented. As a result, the critical area for breaks and via blocks on a single layer can be computed accurately in one pass of the layout. The time complexity is O(n log n) in the case of breaks and O((n+K)log n) in the case of via blocks, where n is the size of the input and K is upper-bounded by the number of interacting vias (in practice K is small). The critical area computation assumes square defects and reflects all possible defect sizes following the D(r)=r/sub 0//sup 2//r/sup 3/ defect size distribution. The method is presented for rectilinear layouts.

Net-Aware Critical Area Extraction for Opens in VLSI Circuits Via Higher-Order Voronoi Diagrams

We address the problem of computing critical area for open faults (opens) in a circuit layout in the presence of multilayer loops and redundant interconnects. The extraction of critical area is the main computational bottleneck in predicting the yield loss of a very large scale integrated design due to random manufacturing defects. We first model the problem as a geometric graph problem and we solve it efficiently by exploiting its geometric nature. To model open faults, we formulate a new geometric version of the classic min-cut problem in graphs, termed the geometric min-cut problem. Then the critical area extraction problem gets reduced to the construction of a generalized Voronoi diagram for open faults, based on concepts of higher order Voronoi diagrams. The approach expands the Voronoi critical area computation paradigm with the ability to accurately compute critical area for missing material defects even in the presence of loops and redundant interconnects spanning over multiple layers. The generalized Voronoi diagrams used in the solution are combinatorial structures of independent interest.

A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted Polygonal Domains

Abstract. We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divide-and-conquer algorithm. The time complexity is O((n+k) log(n+k)) where n is the number of vertices and k is the number of points, improving upon previously known bounds. Space is O(n+k) . Other polygonal domains where our method is applicable include (among others) a polygonal domain of parallel disjoint line segments and a polygonal domain of rectangles in the L1 metric.

The Hausdorff Voronoi Diagram of Point Clusters in the Plane

Abstract 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.

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

Voronoi diagrams for direction-sensitive distances

Most computational geometry research on planar problems assumes that the underlying plane is perfectly ‘flat’, in the sense that movement between any two points cm the plane always takes the same cost as long as the Euclidean distance between the two points is the same. In real environments, distances may depend on the direction one moves along [10], or even may be influenced by local properties of the plane [8]. These situations sometimes can be modeled by considering a piecewiselinear surface as the underlying ‘plane’, and measuring distances therein; see e.g. [7]. In fact, many distance problems on non-flat planes are hard to deal with from the computational geometry point of view. We study distance problems for the basic case of a ‘tilted’ plane in three-space. In this model, the cost of moving depends not only on the Euclidean distance but also on how much upwards or downwards the movement has to travel, simulating the situation when driving a vehicle on the tilted plane. Direction-sensitive distances and, in particular, their induced Voronoi di-

SKEW VORONOI DIAGRAMS

On a tilted plane T in three-space, skew distances are defined as the Euclidean distance plus a multiple of the signed difference in height. Skew distances may model realistic environments more closely than the Euclidean distance. Voronoi diagrams and related problems under this kind of distances are investigated. A relationship to convex distance functions and to Euclidean Voronoi diagrams for planar circles is shown, and is exploited for a geometric analysisis and a plane-sweep construction of Voronoi diagrams on T. An output-sensitive algorithm running in time O(n log h) is developed, where n and h are the numbers of sites and non-empty Voronoi regions, respectively. The all nearest neighbors problem for skew distances, which has certain features different from its Euclidean counterpart, is solved in O(n log n) time.