LAYOUT OF AN ARBITRARY PERMUTATION IN A MINIMAL RIGHT TRIANGLE AREA

2007 ◽  
Vol 08 (02) ◽  
pp. 101-118
Author(s):  
MARIA ARTISHCHEV-ZAPOLOTSKY ◽  
YEFIM DINITZ ◽  
SHIMON EVEN ◽  
VLADIMIR YANOVSKY
Keyword(s):  
Interconnection Networks ◽  
Minimum Area ◽  
Worst Case ◽  
Perpendicular Line ◽  
Restricted Area ◽  
Vlsi Layout ◽  
Arbitrary Permutation ◽  
Layout Area ◽  
Number Of Bends ◽  
Optimal Area

In VLSI layout of interconnection networks, routing two-point nets in some restricted area is one of the central operations. The main aim is usually minimization of the layout area, while reducing the number of wire bends is also very useful. In this paper, we consider connecting a set of N inputs on a line to a set of N outputs on a perpendicular line inside a right triangle shaped area, where the order of the outputs is a given permutation of the order of the corresponding inputs. Such triangles were used, for example, by Dinitz, Even, and Artishchev-Zapolotsky for an optimal layout of the Butterfly network. That layout was of a particular permutation, while here we solve the problem for an arbitrary permutation case. We show two layouts in an optimal area of ½ N2 + o(N2), with O (N) bends each. We prove that the first layout requires the absolutely minimum area and yields the irreducible number of bends, while containing knock-knees. The second one eliminates knock-knees, still keeping a constant number, 7, of bends per connection. As well, we prove a lower bound of 3N - o(N) for the number of bends in the worst case layout in an optimal area of ½ N2 + o(N2).

A Parallel Algorithm for Planar Orthogonal Grid Drawings

2000 ◽  
Vol 10 (01) ◽  
pp. 141-150
Author(s):  
ROBERTO TAMASSIA ◽  
IOANNIS G. TOLLIS ◽  
JEFFREY SCOTT VITTER
Keyword(s):  
Graph Drawing ◽  
Edge Length ◽  
Optimal Time ◽  
Worst Case ◽  
Vlsi Layout ◽  
Orthogonal Grid ◽  
Graph Layouts ◽  
Crew Pram ◽  
Number Of Bends ◽  
Simply Connected

In this paper we consider the problem of constructing planar orthogonal grid drawings (or more simply, layouts) of graphs, with the goal of minimizing the number of bends along the edges. We present optimal parallel algorithms that construct graph layouts with O(n) maximum edge length, O(n2) area, and at most 2n+4 bends (for biconnected graphs) and 2.4n+2 bends (for simply connected graphs). All three of these quality measures for the layouts are optimal in the worst case for biconnected graphs. The algorithm runs on a CREW PRAM in O( log n) time with n/ log n processors, thus achieving optimal time and processor utilization. Applications include VLSI layout, graph drawing, and wireless communication.

PLANAR UPWARD TREE DRAWINGS WITH OPTIMAL AREA

1996 ◽  
Vol 06 (03) ◽  
pp. 333-356 ◽  
Author(s):  
ASHIM GARG ◽  
MICHAEL T. GOODRICH ◽  
ROBERTO TAMASSIA
Keyword(s):  
Linear Time ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Bounded Degree ◽  
Asymptotically Optimal ◽  
Grid Drawing ◽  
Optimal Area ◽  
Linear Time Algorithms

Rooted trees are usually drawn planar and upward, i.e., without crossings and with-out any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree rooted tree with N nodes. Our results are summarized as follows: • We show that T admits a planar polyline upward grid drawing with area O(N), and with width O(Nα) for any prespecified constant a such that 0<α<1. • If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O (N log log N). • We show that if T is ordered, it admits an O(N log N)-area planar upward grid drawing that preserves the left-to-right ordering of the children of each node. • We show that all of the above area bounds are asymptotically optimal in the worst case. • We present O(N)-time algorithms for constructing each of the above types of drawings of T with asymptotically optimal area. • We report on the experimentation of our algorithm for constructing planar polyline upward grid drawings, performed on trees with up to 24 million nodes.

scholarly journals Robust Optimal Design of Beams Subject to Uncertain Loads

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Ismail Kucuk ◽  
Sarp Adali ◽  
Ibrahim Sadek
Keyword(s):  
Optimal Design ◽  
Nonlinear Differential Equations ◽  
A Priori ◽  
Optimal Designs ◽  
Transverse Load ◽  
Cross Sectional ◽  
Worst Case ◽  
Operational Conditions ◽  
Robust Optimal Design ◽  
Optimal Area

Optimality conditions are derived for the robust optimal design of beams subject to a combination of uncertain and deterministic transverse and boundary loads using a variational min-max approach. The potential energy of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross-sectional shape which results in coupled nonlinear differential equations for the unknown functions except for the case of a variable width beam. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to an norm constraint. Similarly the optimal area function is subject to a volume constraint leading to an isoperimetric variational problem. The min-max approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The solution methodology is illustrated for the variable width beam by obtaining analytical results for several cases. The efficiency of the optimal designs is computed with respect to a uniform beam under worst case loading taking the maximum deflection as the quantity for comparison. It is observed that the optimal shapes are more than 70% efficient for the examples given in this study.

Machine Learning Prediction and Reliability Analysis Applied to Subsea Spool and Jumper Design

2021 ◽  
Author(s):  
Mengdi Song ◽  
Massyl Gheroufella ◽  
Paul Chartier
Keyword(s):  
Machine Learning ◽  
Reliability Analysis ◽  
Design Process ◽  
Predictive Modelling ◽  
Machine Learning Algorithms ◽  
Case Scenario ◽  
Worst Case ◽  
Deterministic Solution ◽  
Number Of Bends ◽  
Subsea Pipelines

Abstract In subsea pipelines projects, the design of rigid spool and jumper can be a challenging and time-consuming task. The selected spool layout for connecting the pipelines to the subsea structures, including the number of bends and leg lengths, must offer the flexibility to accommodate the pipeline thermal expansion, the pipe-lay target box and misalignments associated with the post-lay survey metrology and spool fabrication. The analysis results are considerably affected by many uncertainties involved. Consequently, a very large amount of calculations is required to assess the full combination of uncertainties and to capture the worst-case scenario. Rather than applying the deterministic solution, this paper uses machine learning prediction to significantly improve the efficiency of the design process. In addition, thanks to the fast predictive model using machine learning algorithms, the uncertainty quantification and propagation analysis using probabilistic statistical method becomes feasible in terms of CPU time and can be incorporated into the design process to evaluate the reliability of the outputs. The latter allows us to perform a systematic probabilistic design by considering a certain level of acceptance on the probability of failure, for example as per DNVGL design code. The machine learning predictive modelling and the reliability analysis based upon the probability distribution of the uncertainties are introduced and explained in this paper. Some project examples are shown to highlight the method’s comprehensive nature and efficient characteristics.

A PARALLEL ALGORITHM FOR MINIMUM DUAL-COVER WITH APPLICATION TO CMOS LAYOUT

1991 ◽  
Vol 01 (02) ◽  
pp. 177-204 ◽  
Author(s):  
Y.M. HUANG ◽  
M. SARRAFZADEH
Keyword(s):  
Parallel Algorithm ◽  
Planar Graphs ◽  
Dual Graph ◽  
Minimum Area ◽  
Minimum Cardinality ◽  
Vlsi Layout ◽  
Model Based ◽  
Pram Model ◽  
Better Than

In a pair of planar graphs (G, Gd), with Gd being the dual graph of G, a sequence of distinct edges is a dual-Euler trail if it is a trail both in G and in Gd. A set of disjoint dual-Euler trails that simultaneously cover G and Gd is called a dual-cover. We present an O( log n) time and O(n) processors algorithm, in PRAM model, based on the graph separator theory, for obtaining a minimum cardinality dual-cover in a pair of series-parallel graphs (G, Gd), where n is the total number of edges. We employ the proposed algorithm to obtain a minimum-area VLSI layout of CMOS functional cells. Our algorithm, when implemented in a serial environment performs better than previous algorithms and produces more compact layouts.

OPTIMAL LAYOUT OF TRIVALENT CAYLEY INTERCONNECTION NETWORKS

1999 ◽  
Vol 10 (03) ◽  
pp. 277-287 ◽  
Author(s):  
TIZIANA CALAMONERI ◽  
ROSSELLA PETRESCHI
Keyword(s):  
Lower Bound ◽  
Interconnection Networks ◽  
Equal Length ◽  
Optimal Layout ◽  
Layout Area

In this paper we deal with the layout of Trivalent Cayley Interconnection Networks. Namely, we prove that a lower bound on their layout area is Ω(2n-1× 2n-1) and we exhibit some methods to lay these networks out in O(2n-1× 2n-1). We piont out that these layout methods work for other networks decomposable into equal length cycles connected by following fixed rules.

scholarly journals ON EMBEDDING A GRAPH ON TWO SETS OF POINTS

2006 ◽  
Vol 17 (05) ◽  
pp. 1071-1094 ◽  
Author(s):  
EMILIO DI GIACOMO ◽  
GIUSEPPE LIOTTA ◽  
FRANCESCO TROTTA
Keyword(s):  
Planar Graph ◽  
Worst Case ◽  
Blue Vertex ◽  
Polygonal Curve ◽  
Point Set ◽  
Number Of Bends ◽  
Necessary And Sufficient

Let R and B be two sets of points such that the points of R are colored red and the points of B are colored blue. Let G be a planar graph such that |R| vertices of G are red and |B| vertices of G are blue. A bichromatic point-set embedding of G on R ∪ B is a crossing-free drawing of G such that each blue vertex is mapped to a point of B, each red vertex is mapped to a point of R, and each edge is a polygonal curve. We study the curve complexity of bichromatic point-set embeddings; i.e., the number of bends per edge that are necessary and sufficient to compute such drawings. We show that O(n) bends are sometimes necessary. We also prove that two bends per edge suffice if G is a 2-colored caterpillar and that for properly 2-colored caterpillars, properly 2-colored wreaths, 2-colored paths, and 2-colored cycles the number of bends per edge can be reduced to one, which is worst-case optimal.

Mosaic Mapping: A Means of Tiling Images to Shorten Acquisition Speed for Lower - Magnification SEM Images and Wavelength Dispersive Spectrometers (WDS) X-Ray Maps

1996 ◽  
Vol 54 ◽  
pp. 438-439
Author(s):  
J.D. Geller ◽  
C.R. Herrington
Keyword(s):  
Limiting Factors ◽  
X Rays ◽  
Angular Range ◽  
Acceptance Angle ◽  
Sem Images ◽  
Worst Case ◽  
X Ray ◽  
Focusing Distance ◽  
Layered Crystals ◽  
Working Distance

The minimum magnification for which an image can be acquired is determined by the design and implementation of the electron optical column and the scanning and display electronics. It is also a function of the working distance and, possibly, the accelerating voltage. For secondary and backscattered electron images there are usually no other limiting factors. However, for x-ray maps there are further considerations. The energy-dispersive x-ray spectrometers (EDS) have a much larger solid angle of detection that for WDS. They also do not suffer from Bragg’s Law focusing effects which limit the angular range and focusing distance from the diffracting crystal. In practical terms EDS maps can be acquired at the lowest magnification of the SEM, assuming the collimator does not cutoff the x-ray signal. For WDS the focusing properties of the crystal limits the angular range of acceptance of the incident x-radiation. The range is dependent upon the 2d spacing of the crystal, with the acceptance angle increasing with 2d spacing. The natural line width of the x-ray also plays a role. For the metal layered crystals used to diffract soft x-rays, such as Be - O, the minimum magnification is approximately 100X. In the worst case, for the LEF crystal which diffracts Ti - Zn, ˜1000X is the minimum.

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