A Planar Straight-line Grid Drawing Algorithm for High Degree General Trees with User-Specified Angular Coefficient

2008 ◽  
Author(s):  
Chu Yao ◽  
Adrian Rusu ◽  
Andrew Crowell
Keyword(s):  
Angular Coefficient ◽  
Grid Drawing ◽  
Straight Line ◽  
High Degree

CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1031-1060 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Outer Face ◽  
Plane Graphs ◽  
Line Segments ◽  
Grid Drawing ◽  
Straight Line ◽  
Grid Points

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

3D straight-line grid drawing of 4-colorable graphs

1997 ◽  
Vol 63 (2) ◽  
pp. 97-102 ◽  
Author(s):  
Tiziana Calamoneri ◽  
Andrea Sterbini
Keyword(s):  
Grid Drawing ◽  
Straight Line

scholarly journals Analytical Expressions of risks Involved in General Insurance

1958 ◽  
Vol 1 (1) ◽  
pp. 32-41
Author(s):  
C. Philipson
Keyword(s):  
New York ◽  
Generating Functions ◽  
Convex Curve ◽  
Angular Coefficient ◽  
General Insurance ◽  
Maximum Value ◽  
Straight Line ◽  
In Trans ◽  
Aggregate Amount ◽  
Analytical Expressions

The following comments on the papers written in English, French and German under Subject IVA and published in Trans. XVth. International Congress of Actuaries 1957, formed the basis of an address by Mr. C. Philipson, Sweden, on Thursday 17th October, to the members of Astin in New York.Let sn be the maximum value of si and tn the minimum value of the angular coefficient of the straight line connecting the point (si, Ai) with point (1, 1), when i takes the values 1, 2 … n and the point (si, Ai) is taken to mean the vertex of the convex curve representing the generating function of the ith favourable risk. In a portfolio of m + n independent risks—where m are unfavourable and n favourable—Franckx finds that the superior limit of the probability that the aggregate amount of claims shall not exceed k can be written, where Im is taken to mean the product of the generating functions for the m unfavourable risks.Speaking of the theory of risk as first given by Filip Lundberg and developed by Cramér and the Scandinavian School he writes:“Cela n'empêche qu'il y a encore un monde entre la théorie et la pratique et que les actuaires n'auront la partie gagnée que dans la mesure ou leurs travaux seront suffissamment simples que pour qu'ils soient et compréhensibles et directement utilisables”.

Reliability and Generality of Measures of Acceleration, Planned Agility, and Reactive Agility

2009 ◽  
Vol 4 (3) ◽  
pp. 345-354 ◽  
Author(s):  
Jon L. Oliver ◽  
Robert W. Meyers
Keyword(s):  
Light Stimulus ◽  
Systematic Bias ◽  
Physically Active ◽  
Change Of Direction ◽  
Straight Line ◽  
Coefficients Of Variation ◽  
Reactive Agility ◽  
Physical Abilities ◽  
High Degree ◽  
Visual Light

Purpose:The purpose of the current study was to assess the reliability of a new protocol that examines different components of agility using commercially available timing gates.Methods:Seventeen physically active males completed four trials of a new protocol, which consisted of a number of 10-m sprints. Sprints were completed in a straight line or with a change of direction after 5 m. The change of direction was either planned or reactive, with participants reacting to a visual light stimulus.Results:There was no systematic bias in any of the measures, although random variation was reduced in the straight acceleration and planned agility when considering only the fnal pair of trials, with mean coefficients of variation (CV) of 1.6% (95%CI, 1.2% to 2.4%) and 1.1% (0.8% to 1.7%), respectively. Reliability of reactive agility remained consistent throughout with mean CVs of approximately 3%. Analyses revealed a high degree of common variance between acceleration times and both planned (r2 = .93) and reactive (r2 = .83) agility, as well as between the two agility protocols (r2 = .87).Conclusion:Both planned and reactive agility could be measured reliably. Protocol design and use of a light stimulus in the reactive test emphasize physical abilities comparable with other test measures. Therefore, inclusion of a reactive light stimulus does not appear to require any additional perceptual qualities.

CONVEX DRAWINGS OF INTERNALLY TRICONNECTED PLANE GRAPHS ON O(n2) GRIDS

2010 ◽  
Vol 02 (03) ◽  
pp. 347-362 ◽  
Author(s):  
XIAO ZHOU ◽  
TAKAO NISHIZEKI
Keyword(s):  
Linear Time ◽  
Grid Point ◽  
Plane Graph ◽  
Straight Line Segment ◽  
Constant Factor ◽  
Decomposition Tree ◽  
Grid Drawing ◽  
Component Decomposition ◽  
Straight Line ◽  
Convex Drawing

In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n3) to O(n2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n × 4n grid. We also present an algorithm to find such a drawing in linear time.

ON OPEN RECTANGLE-OF-INFLUENCE AND RECTANGULAR DUAL DRAWINGS OF PLANE GRAPHS

2009 ◽  
Vol 01 (03) ◽  
pp. 319-333 ◽  
Author(s):  
HUAMING ZHANG ◽  
MILIND VAIDYA
Keyword(s):  
High Probability ◽  
Linear Time ◽  
Time Algorithm ◽  
Grid Size ◽  
Tight Bound ◽  
Plane Graphs ◽  
Additive Error ◽  
Grid Drawing ◽  
Straight Line ◽  
The One

Irreducible triangulations are plane graphs with a quadrangular exterior face, triangular interior faces and no separating triangles. Fusy proposed a straight-line grid drawing algorithm for irreducible triangulations, whose grid size is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of [Formula: see text]. Later on, Fusy generalized the idea to quadrangulations and obtained a straight-line grid drawing, whose grid size is asymptotically with high probability 13n/27 × 13n/27 up to an additive error of [Formula: see text]. In this paper, we first prove that the above two straight-line grid drawing algorithms for irreducible triangulations and quadrangulations actually produce open rectangle-of-influence drawings for them respectively. Therefore, the above mentioned straight-line grid drawing size bounds also hold for the open rectangle-of-influence drawings. These results improve previous known drawing sizes. In the second part of the paper, we present another application of the results obtained by Fusy. We present a linear time algorithm for constructing a rectangular dual for a randomly generated irreducible triangulation with n vertices, one of its dimensions equals [Formula: see text] asymptotically with high probability, up to an additive error of [Formula: see text]. In addition, we prove that the one dimension tight bound for a rectangular dual of any irreducible triangulations with n vertices is (n + 1)/2.

AREA-EFFICIENT ORDER-PRESERVING PLANAR STRAIGHT-LINE DRAWINGS OF ORDERED TREES

2003 ◽  
Vol 13 (06) ◽  
pp. 487-505 ◽  
Author(s):  
ASHIM GARG ◽  
ADRIAN RUSU
Keyword(s):  
Aspect Ratio ◽  
Binary Tree ◽  
Binary Trees ◽  
Line Drawings ◽  
Ordered Trees ◽  
Grid Drawing ◽  
Aspect Ratios ◽  
Straight Line ◽  
Ordered Tree ◽  
Area Efficient

Ordered trees are generally drawn using order-preserving planar straight-line grid drawings. We investigate the area-requirements of such drawings and present several results. Let T be an ordered tree with n nodes. We show that: • T admits an order-preserving planar straight-line grid drawing with O(n log n) area. • If T is a binary tree, then T admits an order-preserving planar straight-line grid drawing with O(n log log n) area. • If T is a binary tree, then T admits an order-preserving upward planar straight-line grid drawing with optimalO(n log n) area. We also study the problem of drawing binary trees with user-specified aspect ratios. We show that an ordered binary tree T with n nodes admits an order-preserving planar straight-line grid drawing with area O(n log n), and any user-specified aspect ratio in the range [1,n/ log n]. All the drawings mentioned above can be constructed in O(n) time.

CANONICAL DECOMPOSITION, REALIZER, SCHNYDER LABELING AND ORDERLY SPANNING TREES OF PLANE GRAPHS

2005 ◽  
Vol 16 (01) ◽  
pp. 117-141 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
MACHIKO AZUMA ◽  
TAKAO NISHIZEKI
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Linear Time ◽  
Plane Graph ◽  
Canonical Decomposition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Grid Drawing ◽  
Straight Line ◽  
Necessary And Sufficient

A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph encoding, etc. It is known that the triconnectivity is a sufficient condition for their existence, but no necessary and sufficient condition has been known. In this paper, we present a necessary and sufficient condition for their existence, and show that a canonical decomposition, a realizer, a Schnyder labeling, an orderly spanning tree, and an outer triangular convex grid drawing are notions equivalent with each other. We also show that they can be found in linear time whenever a plane graph satisfies the condition.

scholarly journals Lower Bounds on the Area Requirements of Series-Parallel Graphs

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
Fabrizio Frati
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Line Drawing ◽  
Grid Drawing ◽  
Bounding Box ◽  
Straight Line ◽  
International Audience

Graphs and Algorithms International audience We show that there exist series-parallel graphs requiring Omega(n2(root log n)) area in any straight-line or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K(2,n), one side of the bounding box has length Omega(n), thus answering two questions posed by Biedl et al. Second, we show a family of series-parallel graphs requiring Omega(2(root log n)) width and Omega(2(root log n)) height in any straight-line or poly-line grid drawing. Combining the two results, the Omega(n2(root log n)) area lower bound is achieved.

scholarly journals Performance of models to determine flow rate using orifice plates

2021 ◽  
Vol 25 (1) ◽  
pp. 10-16
Author(s):  
Nicolas D. Cano ◽  
Antonio P. de Camargo ◽  
Gustavo L. Muniz ◽  
Jonesmar de Oliveira ◽  
José G. Dalfré Filho ◽  
...  
Keyword(s):  
Measurement Errors ◽  
Discharge Coefficient ◽  
Measurement Range ◽  
Orifice Plate ◽  
Equation Method ◽  
Flow Section ◽  
Angular Coefficient ◽  
Flow Meters ◽  
Flow Estimation ◽  
Straight Line

ABSTRACT This study aimed to evaluate three methodologies for orifice-plate water-flow estimation by quantifying errors in the flow determinations to propose an appropriate measurement range for each evaluated condition. Two orifice-plate models (nominal diameters of 100 and 150 mm) with 50% restriction in the flow section were evaluated. In the theoretical equations, the discharge coefficient was obtained using the Reader-Harris/Gallagher equation (Method 1) and approximated from experimental data using the angular coefficient of a zero-intercept straight line (Method 2). The recommended measurement ranges for errors that were lower than 5% for the 100 and 150 mm plates were 30 to 65 m3 h-1 and 70 to 130 m3 h-1 using the theoretical equation and 20 to 65 m3 h-1 and 40 to 130 m3 h-1 using the empirical equation, respectively. The Reader-Harris/Gallagher equation (Method 1) adequately estimated the discharge coefficient of the orifice plates; however, the use of empirical equations (Method 3) demonstrated smaller measurement errors and greater rangeability of the evaluated flow meters.

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