Rectilinear Regions Research Articles

Two-point L1 shortest path queries in the plane

Let $P$ be a set of $h$ pairwise-disjoint polygonal obstacles with a total of $n$ vertices in the plane. We consider the problem of building a data structure that can quickly compute an $L_1$ shortest obstacle-avoiding path between any two query points $s$ and $t$. Previously, a data structure of size $O(n^2\log n)$ was constructed in $O(n^2\log^2 n)$ time that answers each two-point query in $O(\log^2 n+k)$ time, i.e., the shortest path length is reported in $O(\log^2 n)$ time and an actual path is reported in additional $O(k)$ time, where $k$ is the number of edges of the output path. In this paper, we build a new data structure of size $O(n+h^2 \log h 4^{\sqrt{\log h}})$ in $O(n+h^2 \log^{2}h 4^{\sqrt{\log h}})$ time that answers each query in $O(\log n+k)$ time. (In contrast, for the Euclidean version of this two-point query problem, the best known algorithm uses $O(n^{11})$ space to achieve an $O(\log n+k)$ query time.) Further, we extend our techniques to the weighted rectilinear version in which the ``obstacles of $P$ are rectilinear regions with ``weights and allow $L_1$ paths to travel through them with weighted costs. Previously, a data structure of size $O(n^2\log^2 n)$ was built in $O(n^2\log^2 n)$ time that answers each query in $O(\log^2 n+k)$ time. Our new algorithm answers each query in $O(\log n+k)$ time with a data structure of size $O(n^2 \log n 4^{\sqrt{\log n}})$ that is built in $O(n^2 \log^2 n 4^{\sqrt{\log n}})$ time.

Deformation of a shock wave under strong compression of nonspherical bubbles

The deformation of a radial convergent shock wave appearing inside a spheroidal cavitation bubble under strong compression is studied. It is shown that the shock wave in the course of its convergence can be deformed strongly. In particular, initially the most convex parts of the contour of the surface of the shock wave in its axial cross section can first become rectilinear and then be transformed into concave regions. Along with the formation of concave regions in the contour, sharpenings appear in it. Further small rectilinear regions can be formed instead of sharpenings. In the considered cases of initially elongated and flattened bubbles, the closure of the shock wave begins in the center of the bubble, just as under the closure of a spherical shock wave in an ideal spherical bubble.

Shock wave structures ahead of nonuniform fan cascades

We study shock wave structures (SWS), consisting of shock waves and expansion waves between them, that occur in supersonic flow past nonuniform fan cascades when the velocity component normal to their front (“axial” component) is subsonic. The cascade nonuniformity is due to the scatter in the setting angles of identical blades, either sharp or blunt. A result of the uniformity is the generation of combined noise, whose frequencies are much smaller than the fundamental frequency of the uniform cascade, and slower nonlinear SWS attenuation. The accurate and fast “simple wave method” and “nonlinear acoustics approximation”, together with numerical algorithms for integrating Euler equations on overlapping grids (in calculating flow past blunt edges) and on SWS-adapted grids, are applied to determine the “guiding” action of nonuniform cascades and to describe the SWS evolution. The application of the Fourier analysis gives the sound field spectrum. The use of blades with rectilinear initial regions of the “backs” for reducing supersonic fan blade noise is efficient only at small (less than 0.25°) scatter in the setting angles. The shock wave structures attenuate more rapidly ahead of nonuniform cascades composed of blunt blades than ahead of those with sharp blades. For uniform cascades the blade bluntness effect is not large.

On rectilinear duals for vertex-weighted plane graphs

Let G = ( V , E ) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into | V | simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E . A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph G admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. Furthermore, such a rectilinear cartogram can be constructed in O ( n log n ) time where n = | V | .

Solidification and downstream meniscus prediction in the planar-flow spin casting process

A mathematical model describing the fluid flow, heat transfer and solidification in planar-flow spin casting is introduced. Asymptotic solutions are derived, including expressions for key physical quantities such as the solid/liquid and liquid/gas interfaces. The time-dependent fluid flow in the turning and nearly rectilinear regions is investigated numerically. A recirculation region, in which vorticity accumulates over time, is shown to form immediately downstream of the nozzle. A significant part of this region is periodically washed downstream. As its vorticity diffuses, it eventually fills a large portion of the available gap. This large recirculation region, which restricts the mass flow by almost entirely blocking the gap, is proposed as the physical mechanism for the detachment of the downstream meniscus.

Rectilinear workspace partitioning for parallel coverage using multiple unmanned aerial vehicles

Given a planar workpiece R, the objective of region coverage is to find an ordered list of waypoints and the geometry of paths between consecutive waypoints along which the centroid of a sensor footprint can be moved to efficiently trace a minimal superset of R. We consider the problem of maximally parallelizing the coverage of a contiguous rectilinear region represented by P ∈ R by dividing P amongst η unmanned aerial vehicles (UAVs). The rates at which the UAVs can perform region coverage may differ. The optimum solution to this problem comprises a decomposition of P into η parts such that the cost C i of covering a part using UAV i is proportional to the relative rate at which the UAV can coverage. This problem, however, is NP-hard and we instead give a polynomial time algorithm for solving a closely related problem: divide P into η parts that are each rectilinear, contiguous and whose areas are in the ratio of relative capabilities of the UAVs. Following the division of P, each part is covered by the corresponding UAV. We prove that our algorithm runs in O(N log N + ηN) time. To the best of our knowledge, this is the first time that an algorithm for solving our constrained partitioning problem has been reported.

The study of the volume expansion of aluminum during porous oxide formation at galvanostatic regime

The volume expansion factor of porous alumina, formed by through anodizing of an Al foil of thickness 11.5μm in the range of current densities of 4–35mAcm−2 in oxalic and sulfuric acid at 18–24°C has been studied. The microstructure of anodizing samples has been observed using scanning electron microscopy. The thickness of obtained porous alumina films was measured by a mechanical profilometer with a computer signal-processing. The volume expansion factor of porous alumina varied from 1.35 to 1.65. Linear dependences were obtained for the volume expansion factor of porous alumina versus the anodizing voltage and the ionic current-density logarithm versus the inverse volume expansion factor. Unlike oxide formation in sulfuric acid, these dependences have two subsequential rectilinear regions in oxalic acid. This peculiarity of the dependences in oxalic acid was explained by formation of a region of the immobile negative space charge in the barrier Al oxide layer and its influence on the ionic transport.

A Model Problem of the Design and Aerodynamical Calculation of the Gliding Wing Airfoil of a Wing-in-Ground Vehicle

The problem of the design and aerodynamical calculation of a wing airfoil whose trailing edge glides over a plane horizontal surface is formulated and solved. The known airfoil undersurface is a rectilinear segment forming a given angle with the ground surface, while the upper surface is sought on the basis of a preassigned velocity distribution. This distribution is taken from a class of hydrodynamically advantageous distributions ensuring separationless flow around the airfoil within the framework of the mathematical flow model adopted. The problem is reduced to a mixed boundary value problem in a half-plane and solved analytically. For calculating the lift coefficient the assumption of a thin jetlet flowing between the horizontal region of the airfoil contour and the ground surface is introduced. The effect of the law of pressure decrease along the jetlet, from the stagnation to the exit value, on the lift coefficient is studied. On the basis of the calculations performed a conclusion is drawn concerning the influence of the rectilinear region slope on the airfoil shape; it is also shown how the slope and the value of the maximum velocity on the airfoil affect its shape and the lift coefficient.

Numerical study of a DRAKON-type closed magnetic configuration with a spatial axis

Results are presented from numerical studies of the magnetic field lines and the charged-particle trajectories in the magnetic system of the DRAKON device and of its curvilinear element—the KREL with magnetic mirrors. For the KREL, mirror ratio values are found that do not worsen the particle-drift compensation. The dimensions of the input region for the electrons injected into the KREL to create the beam-plasma discharge are calculated. Calculations show that, in the paraxial approximation, after multiple passes around the device, the magnetic field lines and trajectories of individual transit particles form a system of embedded toroidal surfaces with circular cross sections. When symmetrically changing the current distributions in the coils of the device, these surfaces shift with respect to their previous positions, but their shape remains the same. For the DRAKON device with helical KRELs, the shift of the drift surfaces with respect to the magnetic surfaces, as well as the flow of the longitudinal diamagnetic currents from the KRELs into the rectilinear regions, is found as a function of the pitch angle θ of the KREL helix.

The asymptotic determinant of the discrete Laplacian

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.

Microhardness of chromium steels ground by different methods

1. The electrochemical method provides a higher microhardness of the surface layer than conventional grinding that is closer to the initial level. 2. Whatever the grinding method, the microhardness in processing the grooves of die sections on curvilinear regions is 60–80H lower than on rectilinear regions.

Implementation and evaluation of decision trees with range and region splitting

We propose an extension of an entropy-based heuristic for constructing a decision tree from a large database with many numeric attributes. When it comes to handling numeric attributes, conventional methods are inefficient if any numeric attributes are strongly correlated. Our approach offers one solution to this problem. For each pair of numeric attributes with strong correlation, we compute a two-dimensional association rule with respect to these attributes and the objective attribute of the decision tree. In particular, we consider a family {\cal R} of grid-regions in the plane associated with the pair of attributes. For R \in {\cal R}, the data can be split into two classes: data inside R and data outside R. We compute the region R_{opt} \in {\cal R} that minimizes the entropy of the splitting, and add the splitting associated with R_{opt} (for each pair of strongly correlated attributes) to the set of candidate tests in an entropy-based heuristic. We give efficient algorithms for cases in which {\cal R} is (1) x-monotone connected regions, (2) based-monotone regions, (3) rectangles, and (4) rectilinear convex regions. The algorithm has been implemented as a subsystem of SONAR (System for Optimized Numeric Association Rules) developed by the authors. We have confirmed that we can compute the optimal region efficiently. And diverse experiments show that our approach can create compact trees whose accuracy is comparable with or better than that of conventional trees. More importantly, we can grasp non-linear correlation among numeric attributes which could not be found without our region splitting.

On the integration of partitioning and global routing for rectilinear placement problems

Rectilinear placement is an important process for mixed macro block/standard cells design due to frequent presence of preplaced macro blocks. A prevailing solution to this problem is to divide the rectilinear region into rectangular sub-regions and partition cells into these sub-regions before applying standard placement algorithms. For such a strategy to work, the partitioning objective must closely correlate with the global routing objective. In this paper, we propose a new method to integrate global routing with partitioning. The salient features of our approach consist of (1) a new edge cost model and its computation using an incremental assignment technique; (2) a resistive network analogy for net cost modeling and a randomized algorithm for resistance computation; (3) a robust partitioning framework that combines the ideas of (1) and (2). Also, a dynamic strategy is proposed to evaluate the performance of the algorithm. Our method has been tested on several benchmark circuits and displayed excellent results in terms of both wire length and routability.

General interior Hermite collocation methods for second-order elliptic partial differential equations

The method of collocation based on C 1 piecewise polynomials has been shown (by Houstis et al. (1985)) to be an efficient approach to solving general elliptic partial differential equations (PDEs). The iterative solution of the discrete collocation equations is a very challenging problem and only recently it has been fully resolved by the authors (1992) in the case of rectangular domains. For the case of general PDE domains, the iterative solution of the corresponding discrete equations is still an open problem. In this paper we generalize the method of interior collocation for PDEs defined on rectilinear regions, study the structure of these equations under different ordering schemes, and apply AOR and CG-type iterative solvers to the discrete equations. One of the ordering schemes introduced here has been successfully applied to iterative solvers for the general discrete collocation equations. A number of numerical experiments are reported that indicate the applicability and effectiveness of the AOR and CG iterative schemes. Moreover, we have identified experimentally the appropriate range of the semi-optimal acceleration parameters and some effective preconditioning matrices. These preliminary results indicate that iterative approaches are efficient in solving the general Hermite collocation equations on general domains.

On hearing the shape of rectilinear regions

From complete knowledge of the eigenvalues of the negative Laplacian on a bounded domain, one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function. Explicit calculations are performed for an equilateral triangular domain with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, some applications to eigenvalue problems for an equilateral triangular prism are dealt with, including the solid vertex terms.

Circuit placement on arbitrarily shaped regions using the self-organization principle

An algorithm, called self-organization assisted placement (SOAP), for circuit placement in arbitrarily shaped regions, including two-dimensional (2-D) rectilinear regions, nonplanar surfaces of three-dimensional objects, and three-dimensional volumes, is presented. SOAP is based on the self-organizing which adjusts the weight of synapses connected to neurons such that topologically close neurons become sensitive to inputs that are physically similar. A standard cell placement method and a global placement method of macro cells using SOAP algorithm are described. Several examples showing the circuit placement on rectilinear regions, nonplanar surfaces, and 3-D volumes are shown. Experimental results on benchmark circuits show that the SOAP algorithm is competitive with the state-of-the-art algorithms even for the case of placement in a rectangular region, which is a special case of a 2-D rectilinear region. >

Switch box routing: a retrospective

Detailed routing is a crucial task in physical design of VLSI chips. Many routing algorithms have been proposed and most of them, besides the maze and line search, are oriented towards handling specified detailed routing tasks. In this paper, we discuss routing strategies for switch box problems. Switch boxes are closed, rectilinear regions with terminals on all of their sides. They create the most difficult detailed routing tasks. Here, we discuss problems which make switch box routing so difficult and give an overview of representative heuristics for solving it.

Adaptive cluster growth: a new algorithm for circuit placement in rectilinear regions

A new algorithm called adaptive cluster growth (ACG) for circuit packing (or detailed placement) in any rectilinear region is described; it is an analogy to the growth of a low-stress crystal in a cavity of any given shape. The algorithm ACG is suitable for the packing of circuit modules, either standard-cell or macrocell, in a rectilinear region by the refinement of a result of global placement obtained using such techniques as force-directed relaxation or force-and-cut placement. The overlaps among modules or overlaps between any module and chip boundary are removed in the ACG algorithm in such a way that the estimation of the total routing length is kept as low as possible. An experiment with standard-cell circuits using benchmark data has shown that the ACG algorithm outperforms, or performs close to, other packing techniques that are applicable only to rectangular regions, even in the rectangular-region case. Examples are given to demonstrate the packing of standard-cell circuits in several arbitrarily shaped rectilinear regions. An efficient tiling scheme for representing the rectilinear-shaped cluster and region boundary is also described. The program is written in c, and the vax 11/750 CPU time for the detailed placement of 752-cell circuits is about 70 s.

Switchbox routing with rerouting capabilities in VLSI design

The macrocell approach to the design of VLSI integrated circuits needs a tool for the solution of the switchbox problem, where terminals lie on all the edges of a generic rectilinear region. In the paper, such a problem is thoroughly analysed and a two-layer symbolic router is proposed, which is able to successfully route very dense examples of switchboxes. The program applies a suitable algorithm which tries to connect nets like an expert designer. Connections are placed without affecting the wireability of further connections, when this is possible. In crowded situations, blocking nets are identified and other solutions are tried by a rip-up and rerouting step. In this way, it is possible to notably extend the space in which to search for a solution.

Stresses and strains in a rectilinear region of a cryoturbogenerator field winding

A theoretical study is described of stresses and strains in a rectilinear region of a cryoturbogenerator field winding under the thermomechanical and ponderomotive forces and those forces encounterd during the winding assembly. Windings with a number of transverse cooling channels are considered. The model permits us to choose the winding parameters which will ensure the required cooling conditions and to calculate the strength of the winding.