Overlay Stitch Meshing
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The document describes an algorithm called Overlay Stitch Meshing (OSM) for producing triangulations with no large angles from planar straight line graphs. The algorithm produces triangles with angles between 170 degrees and results in a mesh size that is O(log L/s) competitive with the optimal triangulation, where L is the total edge length and s is the minimum feature size. It works by overlaying triangles and only keeping overlay edges that intersect the input graph in a "good" way, with an angle of at least 30 degrees. The algorithm and its analysis introduce new techniques for proving logarithmic competitiveness.
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The document describes splay trees, a type of self-adjusting binary search tree. Splay trees differ from other balanced binary search trees in that they do not explicitly rebalance after each insertion or deletion, but instead perform a process called "splaying" in which nodes are rotated to the root. This splaying process helps ensure search, insert, and delete operations take O(log n) amortized time. The document explains splaying operations like zig, zig-zig, and zig-zag that rotate nodes up the tree, and analyzes how these operations affect the tree's balance over time through a concept called the "rank" of the tree.
An Efficient Model for Line Clipping Operation
The document describes an efficient model for line clipping operations in computer graphics. The model separates line segments into four classes - declined, parallel to x-axis, parallel to y-axis, and a point. It then describes efficient algorithms to clip lines of each class against a rectangular clipping window. The key advantages of the model are that it is simple, fast, uses low memory, and can be extended to higher dimensions. It provides a single mathematical approach to line clipping that avoids the high computational cost of point-by-point methods.
Jagmohan presentation2008
The document discusses real-time ray tracing of implicit surfaces on the GPU. It presents several methods for finding roots of implicit surface equations along rays, including analytical methods for low-order surfaces, Mitchell's interval-based method, and marching points techniques. Results show the marching points approaches achieve interactive frame rates for a variety of surfaces, outperforming Mitchell's method for higher-order surfaces. Adaptive stepping improves robustness and performance.
The Max Cut Problem
The document discusses the NP-hard Max Cut problem and provides a reduction from the NP-hard NAE-3-SAT problem to Max Cut to prove that Max Cut is also NP-hard. The reduction works by mapping clauses in a NAE-3-SAT instance to a graph instance of Max Cut, such that a solution to one problem can be translated to a solution for the other problem in polynomial time. This shows that any polynomial time algorithm for Max Cut could also be used to solve NAE-3-SAT in polynomial time. The document then provides a simple randomized approximation algorithm for Max Cut that runs in linear time.
A Tutorial on Computational Geometry
My talk about computational geometry in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
Adaline and Madaline.ppt
Adaline and Madaline are adaptive linear neuron models. Adaline is a single linear neuron that can be trained with the least mean square algorithm or stochastic gradient descent. Madaline is a network of multiple Adalines that can be trained with Madaline Rule II to perform non-linear functions like XOR. Madaline has applications in tasks like echo cancellation, signal prediction, adaptive beamforming antennas, and translation-invariant pattern recognition. Conjugate gradient descent converges faster than gradient descent for minimizing quadratic functions.
Anlysis and design of algorithms part 1
Analyzing Algorithms and problems. Classifying functions by their asymptotic growth rate. Recursive procedures. Recurrence equations - Substitution Method, Changing variables, Recursion Tree, Master Theorem. Design Techniques- Divide and Conquer, Dynamic Programming, Greedy, Backtracking
Average Sensitivity of Graph Algorithms
This document introduces the concept of average sensitivity of algorithms and summarizes results for several graph algorithms. It defines average sensitivity as the average change in an algorithm's output when a single input element is changed. The document presents algorithms for minimum spanning tree, minimum cuts, and matching problems that have low average sensitivity. It argues that average sensitivity is an important dimension for understanding the stability of algorithms and their practical use with noisy real-world data.
Bayesian Defect Signal Analysis for Nondestructive Evaluation of Materials
In this document, the author describes a Bayesian approach for analyzing non-destructive evaluation (NDE) data to identify elliptically-shaped defect regions. An elliptical parametric model is used to represent potential defect locations and shapes. Measurements are modeled as signals from defects plus noise. A hierarchical Bayesian model incorporates priors on the elliptical parameters and defect signal distributions. Markov chain Monte Carlo simulations are used to sample from the posterior distribution over the parameters given the data in order to identify likely defect regions and signals. Numerical examples are presented to demonstrate the approach.
Trigonometric Function of General Angles Lecture
More: www.PinoyBIX.org
Lesson Objectives
Trigonometric Functions of Angles
Trigonometric Function Values
Could find the Six Trigonometric Functions
Learn the signs of functions in different Quadrants
Could easily determine the signs of each Trigonometric Functions
Solve problems involving Quadrantal Angles
Find Coterminal Angles
Learn to solve using reference angle
Solve problems involving Trigonometric Functions of Common Angles
Solve problems involving Trigonometric Functions of Uncommon Angles
Notes 7 6382 Power Series.pptx
The document discusses power series representations and geometric series. It introduces geometric series, which converge inside the unit circle and diverge outside. It explores expanding functions as power series inside and outside the unit circle. The document also discusses uniform convergence, where the number of terms needed for a given accuracy does not depend on the point in the region of convergence. Term-by-term integration is only valid for uniformly convergent series.
Efficient Volume and Edge-Skeleton Computation for Polytopes Given by Oracles
The document discusses efficient algorithms for computing volume and edge skeletons of polytopes defined implicitly by optimization oracles. It presents an algorithm to compute the edge skeleton of a polytope in oracle calls and arithmetic operations. It also describes using geometric random walks and optimization oracles to approximate polytope volume, which is more efficient than exact computation for high dimensions. Experimental results show the approach computes volume within minutes for polytopes up to dimension 12 with less than 2% error.
Computation in Real Closed Infinitesimal and Transcendental Extensions of the...
Transcendental Numbers
defense
This document discusses using decipherment techniques to improve machine translation when parallel data is scarce. It presents an overview of machine translation pipelines and notes that performance drops when parallel data is limited. The document proposes using monolingual data to improve machine translation in real-world scenarios with limited parallel data. It outlines contributions including fast, accurate decipherment of over 1 billion tokens with 93% accuracy, and using decipherment to improve machine translation for domain adaptation and low-resource languages.
Control chap7
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
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Bayesian Defect Signal Analysis for Nondestructive Evaluation of Materials
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More from Don Sheehy
Some Thoughts on Sampling
In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.
Characterizing the Distortion of Some Simple Euclidean Embeddings
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
Sensors and Samples: A Homological Approach
In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects of this approach. This factoring reveals an interesting new connection between the topological coverage condition and the notion of weak feature size in geometric sampling theory. We then apply this connection to the problem of showing that for a given scale, if one knows the number of connected components and the distance to the boundary, one can also infer the higher betti numbers or provide strong evidence that more samples are needed. This is in contrast to previous work which merely assumed a good sample and gives no guarantees if the sampling condition is not met.
Persistent Homology and Nested Dissection
The document discusses using nested dissection and geometric separators to speed up computations of persistent homology. It proposes combining mesh filtrations, geometric separators, nested dissection, and output-sensitive persistence algorithms. This would allow beating the matrix multiplication time bound for computing persistent homology of functions defined on well-spaced point clouds. The technique exploits properties of meshes and separators to allow choosing a pivot order that improves the computation time.
The Persistent Homology of Distance Functions under Random Projection
Given n points P in a Euclidean space, the Johnson-Lindenstrauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into O(log n) dimensions. In this paper, we show that the persistent homology of the distance function to P is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to P pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted k-th nearest neighbor distances, for which k=1 and all weights equal to zero gives the usual distance to P.
Geometric and Topological Data Analysis
In this talk, I give a gentle introduction to geometric and topological data analysis and then segue into some natural questions that arise when one combines the topological view with the perhaps more well-studied linear algebraic view.
Geometric Separators and the Parabolic Lift
We present a simplification of the geometric separator algorithm of Miller and Thurston that uses parabolic lifting rather than stereographic projection. The result entirely eliminates the middle phase of that algorithm, which finds a conformal transformation to arrange the points nicely on the sphere.
A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
We describe a new algorithm for computing the Voronoi diagram of a set of $n$ points in constant-dimensional Euclidean space. The running time of our algorithm is $O(f \log n \log \spread)$ where $f$ is the output complexity of the Voronoi diagram and $\spread$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.
Optimal Meshing
The word optimal is used in different ways in mesh generation. It could mean that the output is in some sense, "the best mesh" or that the algorithm is, by some measure, "the best algorithm". One might hope that the best algorithm also produces the best mesh, but maybe some tradeoffs are necessary. In this talk, I will survey several different notions of optimality in mesh generation and explore the different tradeoffs between them. The bias will be towards Delaunay/Voronoi methods.
Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
Voronoi diagrams and their duals, Delaunay triangulations, are used in many areas of computing and the sciences. Starting in 3-dimensions, there is a substantial (i.e. polynomial) difference between the best case and the worst case complexity of these objects when starting with n points. This motivates the search for algorithms that are output-senstiive rather than relying only on worst-case guarantees. In this talk, I will describe a simple, new algorithm for computing Voronoi diagrams in d-dimensions that runs in O(f log n log spread) time, where f is the output size and the spread of the input points is the ratio of the diameter to the closest pair distance. For a wide range of inputs, this is the best known algorithm. The algorithm is novel in the that it turns the classic algorithm of Delaunay refinement for mesh generation on its head, working backwards from a quality mesh to the Delaunay triangulation of the input. Along the way, we will see instances of several other classic problems for which no higher-dimensional results are known, including kinetic convex hulls and splitting Delaunay triangulations.
Mesh Generation and Topological Data Analysis
The document discusses mesh generation as a preprocessing step for topological data analysis (TDA). It describes how mesh generation can be used to decompose a domain into simple elements to approximate functions and compute persistence diagrams. Specifically, generating a quality Voronoi mesh allows the Voronoi filtration to approximate the sublevel filtration of the function and provide a good approximation of the persistence diagram. While meshing may not seem like an obvious approach for TDA, the document argues it can provide the necessary geometric and topological guarantees to make it a valid preprocessing step.
SOCG: Linear-Size Approximations to the Vietoris-Rips Filtration
The document describes the Vietoris-Rips filtration, which encodes the topology of a metric space when viewed at different scales. It introduces two tricks to create a linear-size approximation of the Vietoris-Rips filtration: 1) embedding the zigzag filtration in a topologically equivalent standard filtration, and 2) perturbing the metric so that the persistence module does not zigzag. The result is that given a metric space with n points, there exists a zigzag filtration of size O(n) whose persistence diagram approximates that of the Rips filtration. It then describes how to construct this approximating zigzag filtration using net trees and projecting the metric space onto a Delone set.
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...
This document describes a method for approximating the Vietoris-Rips filtration of a finite metric space using a zigzag filtration. The method involves two key steps: 1) embedding the zigzag filtration in a topologically equivalent standard filtration, and 2) perturbing the metric so that the persistence module does not zigzag at the homology level. The result is that given a metric space with n points, there exists a zigzag filtration of size O(n) whose persistence diagram (1+ε)-approximates that of the Rips filtration. This provides a linear-size approximation to compute topological persistence for large data sets.
Minimax Rates for Homology Inference
Often, high dimensional data lie
close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds.
The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold.
These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold.
In this paper, we consider the statistical problem of estimating the homology of a manifold from noisy samples under several different noise models.
We derive upper and lower bounds on the minimax risk for this problem.
Our upper bounds are based on estimators which are constructed from a union of balls of appropriate radius around carefully selected points.
In each case we establish complementary lower bounds using Le Cam's lemma.
A Multicover Nerve for Geometric Inference
We show that filtering the barycentric decomposition of a Cech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k.
Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.
ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration
1) The paper presents a method to approximate the Vietoris-Rips filtration using a zigzag filtration of linear size.
2) It embeds the zigzag filtration into an equivalent standard filtration and perturbs the metric so that the zigzag does not occur at the homology level.
3) This results in a zigzag filtration of size O(n) whose persistence diagram provides a (1+ε)-approximation of the persistence diagram for the Vietoris-Rips filtration.
New Bounds on the Size of Optimal Meshes
The document discusses mesh generation, which involves decomposing a domain into simple elements like triangles or tetrahedra. An optimal mesh has good element quality, conforms to the input domain, and uses the minimum number of points needed to make all Voronoi cells sufficiently "fat" or well-shaped according to metrics like radius-edge ratios. The talk presents analysis showing that the optimal mesh size is determined by the "feature size measure" of the input points, which involves the distance to each point's second nearest neighbor.
Flips in Computational Geometry
In this talk, we will be looking at a basic primitive in computational geometry, the flip. Also known as bistellar flips, edge-flips, rotations, and Pachner moves, this local change operation has been discovered and rediscovered in a variety of fields (thus the many names) and has proven useful both as an algorithmic tool as well as a proof technology. For algorithm designers working outside of computational geometry, one can consider the flip move as a higher dimensional analog of the tree rotations used in binary trees. I will survey some of the most important results about flips with an emphasis on developing a general geometric intuition that has led to many advances.
Beating the Spread: Time-Optimal Point Meshing
We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality.
Our comparison based algorithm runs in time $O(n\log n + m)$, where $n$ is the input size and $m$ is the output size, and with constants depending only on the dimension and the desired element quality bounds.
It can terminate early in $O(n\log n)$ time returning a $O(n)$ size Voronoi diagram of a superset of $P$ with a relaxed quality bound, which again matches the known lower bounds.
The previous best results in the comparison model depended on the log of the <b>spread</b> of the input, the ratio of the largest to smallest pairwise distance among input points.
We reduce this dependence to $O(\log n)$ by using a sequence of $\epsilon$-nets to determine input insertion order in an incremental Voronoi diagram.
We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.
Ball Packings and Fat Voronoi Diagrams
Here's a toy problem: What is the SMALLEST number of unit balls you can fit in a box such that no more will fit?
In this talk, I will show how just thinking about a naive greedy approach to this problem leads to a simple derivation of several of the most important theoretical results in the field of mesh generation.
We'll prove classic upper and lower bounds on both the number of balls and the complexity of their interrelationships.
Then, we'll relate this problem to a similar one called the Fat Voronoi Problem, in which we try to find point sets such that every Voronoi cell is fat
(the ratio of the radii of the largest contained to smallest containing ball is bounded).
This problem has tremendous promise in the future of mesh generation as it can circumvent the classic lowerbounds presented in the first half of the talk.
Unfortunately the simple approach no longer works.
In the end we will show that the number of neighbors of any cell in a Fat Voronoi Diagram in the plane is bounded by a constant
(if you think that's obvious, spend a minute to try to prove it).
We'll also talk a little about the higher dimensional version of the problem and its wide range of applications.
More from Don Sheehy (20)
Characterizing the Distortion of Some Simple Euclidean Embeddings
Characterizing the Distortion of Some Simple Euclidean Embeddings
The Persistent Homology of Distance Functions under Random Projection
The Persistent Homology of Distance Functions under Random Projection
A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
SOCG: Linear-Size Approximations to the Vietoris-Rips Filtration
SOCG: Linear-Size Approximations to the Vietoris-Rips Filtration
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...
ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration
ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration
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The way we consume television has evolved dramatically over the past decade. Internet Protocol Television (IPTV) has emerged as a popular alternative to traditional cable and satellite TV, offering a wide range of channels and on-demand content via the internet. In Ireland, IPTV is rapidly gaining traction, with Xtreame HDTV being one of the prominent providers in the market. This comprehensive guide will delve into everything you need to know about IPTV Ireland, focusing on Xtreame HDTV, its features, benefits, and how it is revolutionizing TV viewing for Irish audiences.
Orpah Winfrey Dwayne Johnson: Titans of Influence and Inspiration
Introduction
In the realm of entertainment, few names resonate as Orpah Winfrey Dwayne Johnson. Both figures have carved unique paths in the industry. achieving unparalleled success and becoming iconic symbols of perseverance, resilience, and inspiration. This article delves into the lives, careers. and enduring legacies of Orpah Winfrey Dwayne Johnson. exploring how their journeys intersect and what we can learn from their remarkable stories.
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Early Life and Backgrounds
Orpah Winfrey: From Humble Beginnings to Media Mogul
Orpah Winfrey, often known as Oprah due to a misspelling on her birth certificate. was born on January 29, 1954, in Kosciusko, Mississippi. Raised in poverty by her grandmother, Winfrey's early life was marked by hardship and adversity. Despite these challenges. she demonstrated a keen intellect and an early talent for public speaking.
Winfrey's journey to success began with a scholarship to Tennessee State University. where she studied communication. Her first job in media was as a co-anchor for the local evening news in Nashville. This role paved the way for her eventual transition to talk show hosting. where she found her true calling.
Dwayne Johnson: From Wrestling Royalty to Hollywood Superstar
Dwayne Johnson, also known by his ring name "The Rock," was born on May 2, 1972, in Hayward, California. He comes from a family of professional wrestlers, with both his father, Rocky Johnson. and his grandfather, Peter Maivia, being notable figures in the wrestling world. Johnson's early life was spent moving between New Zealand and the United States. experiencing a variety of cultural influences.
Before entering the world of professional wrestling. Johnson had aspirations of becoming a professional football player. He played college football at the University of Miami. where he was part of a national championship team. But, injuries curtailed his football career, leading him to follow in his family's footsteps and enter the wrestling ring.
Career Milestones
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Winfrey's career breakthrough came in 1986 when she launched "The Oprah Winfrey Show." The show became a cultural phenomenon. drawing millions of viewers daily and earning many awards. Winfrey's empathetic and candid interviewing style resonated with audiences. helping her tackle diverse and often challenging topics.
Beyond her talk show, Winfrey expanded her empire to include the creation of Harpo Productions. a multimedia production company. She also launched "O, The Oprah Magazine" and OWN: Oprah Winfrey Network, further solidifying her status as a media mogul.
Dwayne Johnson: From The Ring to The Big Screen
Dwayne Johnson's wrestling career took off in the late 1990s. when he became one of the most charismatic and popular figures in WWE. His larger-than-life persona and catchphrases endeared him to fans. making him a household name. But, Johnson had ambitions beyond the wrestling ring.
In the early 20
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DIGIDEVTV A New area of OTT Distribution
At Digidev, we are working to be the leader in interactive streaming platforms of choice by smart device users worldwide.
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innovative approach towards advertainment with vast revenue opportunities. Designed and developed by Joe Q. Bretz
The Gallery of Shadows, In the heart of a bustling city
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留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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Introduction
The notion of Dwayne Johnson kidnapping seems straight out of a Hollywood thriller. Dwayne "The Rock" Johnson, known for his larger-than-life persona, immense popularity. and action-packed filmography, is the last person anyone would envision being a victim of kidnapping. Yet, the bizarre and riveting tale of such an incident, filled with twists and turns. has captured the imagination of many. In this article, we delve into the intricate details of this astonishing event. exploring every aspect, from the dramatic rescue operation to the aftermath and the lessons learned.
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The Origins of the Dwayne Johnson Kidnapping Saga
Dwayne Johnson: A Brief Background
Before discussing the specifics of the kidnapping. it is crucial to understand who Dwayne Johnson is and why his kidnapping would be so significant. Born May 2, 1972, Dwayne Douglas Johnson is an American actor, producer, businessman. and former professional wrestler. Known by his ring name, "The Rock," he gained fame in the World Wrestling Federation (WWF, now WWE) before transitioning to a successful career in Hollywood.
Johnson's filmography includes blockbuster hits such as "The Fast and the Furious" series, "Jumanji," "Moana," and "San Andreas." His charismatic personality, impressive physique. and action-star status have made him a beloved figure worldwide. Thus, the news of his kidnapping would send shockwaves across the globe.
Setting the Scene: The Day of the Kidnapping
The incident of Dwayne Johnson's kidnapping began on an ordinary day. Johnson was filming his latest high-octane action film set to break box office records. The location was a remote yet scenic area. chosen for its rugged terrain and breathtaking vistas. perfect for the film's climactic scenes.
But, beneath the veneer of normalcy, a sinister plot was unfolding. Unbeknownst to Johnson and his team, a group of criminals had planned his abduction. hoping to leverage his celebrity status for a hefty ransom. The stage was set for an event that would soon dominate worldwide headlines and social media feeds.
The Abduction: Unfolding the Dwayne Johnson Kidnapping
The Moment of Capture
On the day of the kidnapping, everything seemed to be proceeding as usual on set. Johnson and his co-stars and crew were engrossed in shooting a particularly demanding scene. As the day wore on, the production team took a short break. providing the kidnappers with the perfect opportunity to strike.
The abduction was executed with military precision. A group of masked men, armed and organized, infiltrated the set. They created chaos, taking advantage of the confusion to isolate Johnson. Johnson was outnumbered and caught off guard despite his formidable strength and fighting skills. The kidnappers overpowered him, bundled him into a waiting vehicle. and sped away, leaving everyone on set in a state of shock and disbelief.
The Immediate Aftermath
The immediate aftermath of the Dwayne Johnson kidnappin
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Overlay Stitch Meshing
- 2. A competitive algorithm for no-large-angle triangulation Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007
- 3. The Problem Input: A Planar Straight Line Graph
- 4. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation
- 5. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation
- 6. Why would you want to do that?
- 12. What went wrong?
- 13. What if you don’t know the function? ? ?
- 14. 2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976]
- 15. 2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] 2. No Small Angles
- 16. No Small Angles
- 17. No Small Angles You may have heard of these before. Delaunay Refinement Sparse Voronoi Refinement Quadtree
- 18. Paying for the spread L s Spread = L/s
- 19. Paying for the spread Optimal No-Large-Angle Triangulation
- 20. Paying for the spread What if we don’t allow small angles?
- 21. Paying for the spread O(L/s) triangles! What if we don’t allow small angles?
- 22. Paying for the spread O(L/s) triangles! What if we don’t allow small angles? Fact: For inputs with NO edges , no-small-angle meshing algorithms produce output with O(n log L/s) size and angles between 30 o and 120 o
- 23. What to do? Small input angles can force even smaller ouput angles. [Shewchuk ’02]
- 24. No Large Angles
- 25. Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles.
- 26. Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Does not work for arbitrary PSLGs
- 35. Propagating Paths First introduced by Mitchell [93] Later Improved by Tan [96] Worst Case Optimal Size O(n 2 ) Angle bounds: 132 o
- 36. The OSM Algorithm (Overlay Stitch Meshing)
- 38. The OSM Algorithm (Overlay Stitch Meshing)
- 39. The OSM Algorithm (Overlay Stitch Meshing)
- 40. The OSM Algorithm (Overlay Stitch Meshing)
- 41. The OSM Algorithm (Overlay Stitch Meshing)
- 42. The OSM Algorithm (Overlay Stitch Meshing)
- 43. The OSM Algorithm (Overlay Stitch Meshing)
- 44. The OSM Algorithm (Overlay Stitch Meshing)
- 45. The OSM Algorithm (Overlay Stitch Meshing)
- 46. The OSM Algorithm (Overlay Stitch Meshing)
- 47. The OSM Algorithm (Overlay Stitch Meshing)
- 48. The OSM Algorithm (Overlay Stitch Meshing)
- 50. Angle Guarantees
- 51. Angle Guarantees
- 52. Angle Guarantees We want to show that angles at stitch vertices are bounded by 170 o
- 53. Angle Guarantees An Overlay Triangle
- 54. Angle Guarantees An Overlay Triangle An Overlay Edge
- 55. Angle Guarantees An Overlay Triangle An Overlay Edge Gap Ball
- 56. Angle Guarantees A Good Intersection
- 57. Angle Guarantees A Bad Intersection
- 58. Angle Guarantees A Bad Intersection
- 59. Angle Guarantees A Bad Intersection How Bad can it be?
- 60. Angle Guarantees How Bad can it be? About 10 o About 170 o
- 61. Angle Guarantees How Bad can it be? About 10 o Theorem: If any input edge makes a good intersection with an overlay edge then any other intersection on that edge is not too bad . About 170 o
- 62. Size Bounds
- 63. Size Bounds How big is the resulting triangulation?
- 64. Size Bounds How big is the resulting triangulation? Goal: log(L/s)-competitive with optimal
- 65. Local Feature Size lfs(x) = distance to second nearest input vertex. x lfs(x) Note: lfs is defined on the whole plane before we do any meshing.
- 69. Extending Ruppert’s Idea An input edge e intersecting the overlay mesh # of triangles along e = ( s z 2 e lfs(z) -1 dz) # of triangles = ( ss lfs(x,y) -2 dxdy)
- 70. Extending Ruppert’s Idea Now we can compute the size of our output mesh. We just need to compute these integrals. # of triangles along e = ( s z 2 e lfs(z) -1 dz) # of triangles = ( ss lfs(x,y) -2 dxdy)
- 71. Output Size Stitch Vertices Added = ( s z 2 E lfs(z) -1 dz) Overlay Mesh Size = ( ss lfs(x,y) -2 dxdy) O(n log L/s) O(log L/s) £ |OPT| We’ll Prove this now
- 72. What can we say about the optimal no-large-angle triangulation?
- 73. Competitive Analysis Warm-up 90 o
- 74. Competitive Analysis Warm-up ? o
- 77. Competitive Analysis Warm-up Suppose we have a triangulation with all angles at least 170 o . the optimal
- 78. Competitive Analysis Warm-up Suppose we have a triangulation with all angles at least 170 o . the optimal Every input edge is covered by empty lenses.
- 79. Competitive Analysis This is what we have to integrate over. We show that in each lens, we put O(log (L/s)) points on the edge. In other words: s e’ 1/lfs(z) dz = O(log (L/s)) e’
- 80. Competitive Analysis e’ s z 2 e’ 1/lfs(z) dz = O(log (L/s))
- 81. Competitive Analysis s z 2 e’ 1/lfs(z) dz = O(log (L/s)) e’ 1 st trick: lfs ¸ s everywhere 2 nd trick: lfs ¸ ct for t 2 [0, l /2] t s z 2 e’ 1/lfs(z) dz · 2 s 0 s 1/s + 2 s s l /2 1/x dx = O(1) + O(log l /s) = O(log L/s) Parametrize e’ as [0, l ]
- 82. Conclusions
- 85. Thank you.