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.

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⌋.

scholarly journals Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

scholarly journals The XL-mHG test for gene set enrichment

2017 ◽  
Author(s):  
Florian Wagner
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
P Value ◽  
Gene Set Enrichment ◽  
Gene Set ◽  
Efficient Calculation ◽  
Reference Gene Expression ◽  
Quantitative Manner ◽  
The One ◽  
Ks Test

The nonparametric minimum hypergeometric (mHG) test is a popular alternative to Kolmogorov-Smirnov (KS)-type tests for determining gene set enrichment. However, these approaches have not been compared to each other in a quantitative manner. Here, I first perform a simulation study to show that the mHG test is significantly more powerful than the one-sided KS test for detecting gene set enrichment. I then illustrate a shortcoming of the mHG test, which has motivated a semiparametric generalization of the test, termed the XL-mHG test. I describe an improved quadratic-time algorithm for the efficient calculation of exact XL-mHG p-values, as well as a linear-time algorithm for calculating a tighter upper bound for the p-value. Finally, I demonstrate that the XL-mHG test outperforms the one-sided KS test when applied to a reference gene expression study, and discuss general principles for analyzing gene set enrichment using the XL-mHG test. An efficient open-source Python/Cython implementation of the XL-mHG test is provided in the xlmhg package, available from PyPI and GitHub (https://github.com/flo-compbio/xlmhg) under an OSI-approved license.

scholarly journals AN AVERAGE LINEAR TIME ALGORITHM FOR WEB USAGE MINING

2004 ◽  
Vol 03 (02) ◽  
pp. 307-319 ◽  
Author(s):  
JOSÉ BORGES ◽  
MARK LEVENE
Keyword(s):  
Markov Chain ◽  
High Probability ◽  
Linear Time ◽  
Time Algorithm ◽  
Web Pages ◽  
Data Mining Algorithm ◽  
Web Navigation ◽  
Navigation Data ◽  
The Web ◽  
Average Behaviour

In this paper, we study the complexity of a data mining algorithm for extracting patterns from user web navigation data that was proposed in previous work.3 The user web navigation sessions are inferred from log data and modeled as a Markov chain. The chain's higher probability trails correspond to the preferred trails on the web site. The algorithm implements a depth-first search that scans the Markov chain for the high probability trails. We show that the average behaviour of the algorithm is linear time in the number of web pages accessed.

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.

scholarly journals Convex Grid Drawings of 3-Connected Planar Graphs

1997 ◽  
Vol 07 (03) ◽  
pp. 211-223 ◽  
Author(s):  
Marek Chrobak ◽  
Goos Kant
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Linear Time Algorithm ◽  
Convex Polygons ◽  
Line Segments ◽  
Polynomial Size ◽  
Straight Line

We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.

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