Peter Widmayer

On some distance problems in fixed orientations

In VLSI design, technology requirements often dictate the use of only two orthogonal orientations, determining both the shape of objects and the distance function, the

A faster approximation algorithm for the Steiner problem in graphs

L_1

Twin grid files: space optimizing access schemes

-metric, to be used for wiring objects. More recent VLSI fabrication technology is capable of creating edges and wires in both the orthogonal and diagonal orientations.We generalize the distance concept to the case where any fixed set of orientations is allowed, and introduce a family of naturally induced metrics, and the subsequent generalization of geometrical concepts. A shortest connection between two points is in this case a path composed of line segments with only the given orientations. We derive optimal solutions for various basic planar distance problems in this setting, such as the computation of a Voronoi diagram, a minimum spanning tree, and the (minimum and maximum) distance between two convex polygons. Many other theoretically interesting and practically relevant problems remain to be solved. In particular, the new famil...

On translating a set of line segments

SummaryWe present an algorithm for finding a Steiner tree for a connected, undirected distance graph with a specified subset S of the set of vertices V. The set V-S is traditionally denoted as Steiner vertices. The total distance on all edges of this Steiner tree is at most 2(1−1/l) times that of a Steiner minimal tree, where l is the minimum number of leaves in any Steiner minimal tree for the given graph. The algorithm runs in O(¦E¦log¦V¦) time in the worst case, where E is the set of all edges and V the set of all vertices in the graph. It improves dramatically on the best previously known bound of O(¦S¦¦V¦2), unless the graph is very dense and most vertices are Steiner vertices. The essence of our algorithm is to find a generalized minimum spanning tree of a graph in one coherent phase as opposed to the previous multiple steps approach.

A fast algorithm for the Boolean masking problem

Storage access schemes for points, supporting spatial searching, usually suffer from an undesirably low storage space utilization. We show how a given set of points can be distributed among two grid files in such a way that storage space utilization is optimal. The optimal twin grid file can be built practically as fast as a standard grid file, i.e., the storage space optimality is obtained at almost no extra cost. We compare the performances of the standard grid file, the optimal static twin grid file, and an efficient dynamic twin grid file, where insertions and deletions trigger the redistribution of points among the two grid files. Twin grid files utilize storage space at roughly 90%, as compared with the 69% of the standard grid file. Typical range queries - the most important spatial search operations - can be answered in twin grid files at least as fast as in the standard grid file.

An optimal algorithm for the maximum alignment of terminals

Abstract An optimal one-pass algorithm is presented for the computation of an ordering for a set of line segments in the plane according to which the segments can be moved to the right by an arbitrary fixed distance, such that during the move no segment meets another.

Relaxed Balancing in Search Trees

Abstract An algorithm is presented for the calculation of Boolean combinations between layers of a VLSI circuit layout. Each layer is assumed to contain only polygons, which are specified by their edges; the output is also polygonal. The algorithm runs in O((n + k)(r + log n)) time and O(nr) space, where n is the total number of edges on all layers, k is the number of edge intersections, and r is the number of layers. Also a number of restrictions on the general problem are discussed which lead to substantial improvements in the time bounds. It is proved that when the polygons are presented using a hierarchical description language the problem becomes NP-hard. Finally how this approach can be used to solve the i-contour problem of computational geometry and the hidden-line-elimination problem of computer graphics is discussed.

Distance problems in computational geometry with fixed orientations

Abstract We consider the problem of finding a maximum subset of a given set of wires connecting two rows of terminals with fixed positions, such that no wires in the subset cross. We derive an algorithm that runs in O(p + (n − p) l g(p + 1)) time, where n is the number of wires given and p is the maximum number of noncrossing wires; in many practically relevant cases, e.g., when p is very high, it needs only linear time. We show how an extension of the algorithm solves the more general problem, where the positions of some terminals have some flexibility, within the same time bound.

An Approximation Algorithms for Steiner's Problem in Graphs

We consider search trees within the scheme that rebalancing transformations need not be connected with updates but may be delayed. Moreover, the various rebalancing operations are made to be composed of small local steps. This scheme of maintenance of search trees is called relaxed balancing in contrast to standard strict balancing (each update includes the required operations for keeping the tree in balance). With relaxed balancing we can solve the concurrency control problem of search trees efficiently: the maintenance of search trees is divided into operations each of which needs to lock only a small constant number of nodes at a time. Relaxed balancing is important for efficiency even in sequential applications where updates occur in bursts.

A worst-case efficient algorithm for hidden-line elimination †

In computational geometry, problems involving only rectilinear objects with edges parallel to the x -and y-axes have attracted great attention. They are often easier to solve than the same problems for arbitrary objects, and solutions are of high practical value, for instance in VLSI design. This is because in VLSI design technology requirements often dictate the use of only two orthogonal orientations for the boundary edges of objects as well as wires. The restriction on the boundary edges motivates the study of rectilinear objects, while the restriction on wires brings the focus on the well-known L1-metric (the Manhattan distance). In short, given the two orthogonal orientations, both the shape of objects and the distance function are determined in a natural way. More recent VLSI fabrication technology is capable of creating edges and wires in both the orthogonal and diagonal orientations. This motivates us to study more general polygons, and to generalize the distance concept to the case where any fixed set of orientations is allowed. We introduce a family of naturally induced metrics, and the subsequent generalization of geometrical concepts. A shortest connection between two points is in this case a path composed of line segments with only the given orientations. We derive optimal solutions for various basic planar distance problems in this setting, such as the computation of a Voronoi diagram, a minimum spanning tree, and the (minimum and maximum) distance between two convex polygons. Many other theoretically interesting and practically relevant problems remain to be solved. In particular, the new family of metrics may help bridge the gap between the L1- and the L2-metrics, as those are the limiting cases for two and infinitely many regularly distributed orientations.