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06-G-01. Optimal And Simpler

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06-G-01. Optimal And Simpler课程教案、知识点、字幕

对于刚才我们介绍的

persistent structure

我相信你

肯定会印象深刻

因为它的技巧

太高明了

当然高明的东西

也会显得更加的复杂

某种意义上讲

也未必实用

实际上情况也是如此

我们不禁要问

这样一个问题

有没有更简单的方法

有没有更

方便的简明的实现

同样能够在刚才所说的

那一系列最优的指标下

来完成point location这个问题呢

在这个问题的发展历史上

Kirkpatrick

是一个非常重要的人物

因为他提出的

一个专门的一个几何结构

就可以做到这样

我们将会看到

Kirkpatrick

所提出的这样的一个几何结构

是一个分层次的结构

它从上到下

由粗到精

描述了我们所要做点定位的

那个原始的subdivision

实际上查找过程也是如此

每次都是从顶上出发

然后

基于一个相对粗糙一点的模型

先做一个定位

将搜索的范围

锁定到一个相对小的范围

然后呢

在进一步的

进到下一个层次

再做一次搜索

又往深入的方向

再前进一步

将查找的范围

再次搜索到更小的一个范围

如此不断的下行 下行

直到最终

达到最精细的那个subdivision

从而给出正确的答案

所以这样的一个过程

整个从方法论的角度讲

我们也可以概括为

叫做由粗到细

不断的

依然是

简而治之

计算几何课程列表:

00. Introduction

-Before we start

--html

-Evaluation

--html

-Online Judge

--html

-Lecture notes

--html

-Discussion

--html

-A. History of This Course

--00-A. History of This Course

-B. What's Computational Geometry

--00-B. What's Computational Geometry

-B. What's Computational Geometry--作业

-C. How to Learn CG Better

--00-C. How to Learn CG Better

-C. How to Learn CG Better--作业

-D. Why English

--00-D. Why English

01. Convex Hull

-A. Convexity

--01-A-01. Why Convex Hull

--01-A-02. Nails In The Table

--01-A-03. Paint Blending

--01-A-04. Color Space

--01-A-05. Convex Hull

-A. Convexity--作业

-B. Extreme Points

--01-B-01. Extremity

--01-B-02. Strategy

--01-B-03. In-Triangle Test

--01-B-04. To-Left Test

--01-B-05. Determinant

-B. Extreme Points--作业

-C. Extreme Edges

--01-C-01. Definition

--01-C-02. Algorithm

--01-C-03. Demonstration

-C. Extreme Edges--作业

-D. Incremental Construction

--01-D-01. Decrease and Conquer

--01-D-02. In-Convex-Polygon Test

--01-D-03. Why Not Binary Search

--01-D-04. Support-Lines

--01-D-05. Pattern Of Turns

--01-D-06. Exterior/Interior

-D. Incremental Construction--作业

-E. Jarvis March

--01-E-01. Selectionsort

--01-E-02. Strategy

--01-E-03. Coherence

--01-E-04. To-Left Test

--01-E-05. Degeneracy

--01-E-06. Lowest-Then-Leftmost

--01-E-07. Implementation

--01-E-08. Output Sensitivity

-E. Jarvis March--作业

-F. Lower Bound

--01-F-01. Reduction

--01-F-02. CAO Chong's Methodology

--01-F-03. Transitivity

--01-F-04. Reduction: Input

--01-F-05. Reduction: Output

--01-F-06. Sorting ≤_N 2d-CH

-F. Lower Bound--作业

-G. Graham Scan: Algorithm

--01-G-01. Preprocessing

--01-G-02. Scan

--01-G-03. Simplest Cases

-G. Graham Scan: Algorithm--作业

-H. Graham Scan: Example

--01-H-01. Example (1/2)

--01-H-02. Example (2/2)

-H. Graham Scan: Example--作业

-I. Graham Scan: Correctness

--01-I-01. Left Turn

--01-I-02. Right Turn

--01-I-03. Presorting

-I. Graham Scan: Correctness--作业

-J. Graham Scan: Analysis

--01-J-01. Ω(n) Backtracks

--01-J-02. Planarity

--01-J-03. Amortization

--01-J-04. Simplification

-J. Graham Scan: Analysis--作业

-K. Divide-And-Conquer (1)

--01-K-01. Merge

--01-K-02. Common Kernel

--01-K-03. Interior

--01-K-04. Exterior

-K. Divide-And-Conquer (1)--作业

-L. Divide-And-Conquer (2)

--01-L-01. Preprocessing

--01-L-02. Common Tangents

--01-L-03. Topmost + Bottommost ?

--01-L-04. Stitch

--01-L-05. Zig-Zag

--01-L-06. Time Cost

--01-L-07. More Considerations

-L. Divide-And-Conquer (2)--作业

-M. Wrap-Up

--01-M. Wrap-Up

02. Geometric Intersection

-0. Introduction

--02-0. Introduction

-0. Introduction--作业

-A. Preliminary

--02-A-01. EU

--02-A-02. Min-Gap

--02-A-03. Max-Gap

--02-A-04. IEU

-A. Preliminary--作业

-B. Interval Intersection Detection

--02-B-01. Algorithm

--02-B-02. Lower Bound

-B. Interval Intersection Detection--作业

-C. Segment Intersection Reporting

--02-C-01. Brute-force

--02-C-02. Hardness

-C. Segment Intersection Reporting--作业

-D. BO Algorithm: Strategy

--02-D-01. Proximity & Separability

--02-D-02. Comparability & Ordering

--02-D-03. Data Structures

--02-D-04. Possible Cases

-D. BO Algorithm: Strategy--作业

-E. BO Algorithm: Implementation

--02-E-01. Degeneracy

--02-E-02. Event Queue

--02-E-03. Events & Operations

--02-E-04. Sweepline Status

-E. BO Algorithm: Implementation--作业

-F. BO Algorithm: Analysis

--02-F-01. Correctness

--02-F-02. Example

--02-F-03. Retesting

--02-F-04. Complexity of Event Queue

--02-F-05. Complexity of Status Structure

-F. BO Algorithm: Analysis--作业

-G. Convex Polygon Intersection Detection

--02-G-01. Problem Specification

--02-G-02. Monotone Partitioning

--02-G-03. Criterion

--02-G-04. Decrease-And-Conquer

--02-G-05. Example Cases

--02-G-06. Complexity

-G. Convex Polygon Intersection Detection--作业

-H. Edge Chasing

--02-H-01. Eliminating Sickles

--02-H-02. Example

--02-H-03. Analysis

-H. Edge Chasing--作业

-I. Plane Sweeping

--02-I. Plane Sweeping

-I. Plane Sweeping--作业

-J. Halfplane Intersection Construction

--02-J-01. The Problem

--02-J-02. Lower Bound

--02-J-03. Divide-And-Conquer

-J. Halfplane Intersection Construction--作业

03. Triangulation

-0. Methodology

--03-0. Methodology

-0. Methodology--作业

-A. Art Gallery Problem

--03-A-01. Definition

--03-A-02. Lower & Upper Bounds

--03-A-03. Hardness

--03-A-04. Approximation & Classification

-A. Art Gallery Problem--作业

-B. Art Gallery Theorem

--03-B-01. Necessity of floor(n/3)

--03-B-02. Sufficiency by Fan Decomposition

-B. Art Gallery Theorem--作业

-C. Fisk's Proof

--03-C-01. Triangulation

--03-C-02. 3-Coloring

--03-C-03. Domination

--03-C-04. Pigeon-Hole Principle

--03-C-05. Generalization

-C. Fisk's Proof--作业

-D. Orthogonal Polygons

--03-D-01. Necessity of floor(n/4)

--03-D-02. Sufficiency by Convex Quadrilateralization

--03-D-03. Generalization

-D. Orthogonal Polygons--作业

-E. Triangulation

--03-E-01. Existence

--03-E-02. Ear & Mouth

--03-E-03. Two-Ear Theorem

--03-E-04. Well-Order

--03-E-05. Ear Candidate

--03-E-06. Induction

--03-E-07. Well-Order (Again)

--03-E-08. Properties

-E. Triangulation--作业

-F. Triangulating Monotone Polygons

--03-F-01. Monotone Polygon

--03-F-02. Monotonicity Testing

--03-F-03. Strategy

--03-F-04. Stack-Chain Consistency

--03-F-05. Same Side + Reflex

--03-F-06. Same Side + Convex

--03-F-07. Opposite Side

--03-F-08. Example

--03-F-09. Analysis

-F. Triangulating Monotone Polygons--作业

-G. Monotone Decomposition

--03-G-01. Cusps

--03-G-02. Helper

--03-G-03. Helper Candidate

--03-G-04. Sweep-Line Status

--03-G-05. Possible Cases

--03-G-06. Example

--03-G-07. Analysis

-G. Monotone Decomposition--作业

-I. Tetrahedralization

--03-I-01. Polyhedron Decomposition

--03-I-02. Schonhardt's Polyhedron

--03-I-03. Seidel's Polygon

-I. Tetrahedralization--作业

04. Voronoi Diagram

-A. Introduction

--04-A-01. A First Glance

--04-A-02. Dining Halls on Campus

--04-A-03. More Analogies & Applications

--04-A-04. Voronoi

-A. Introduction--作业

-B. Terminologies

--04-B-01. Site & Cell

--04-B-02. Intersecting Halfspaces

--04-B-03. Voronoi Diagram

--04-B-04. Planar Voronoi Diagram

-B. Terminologies--作业

-C. Properties

--04-C-01. Non-Empty Cells

--04-C-02. Empty Disks

--04-C-03. Nearest = Concyclic

--04-C-04. Number of Nearest Sites = Degree

--04-C-05. Split & Merge

-C. Properties--作业

-D. Complexity

--04-D-01. Linearity

--04-D-02. Proof

-D. Complexity--作业

-E. Representation

--04-E-01. Subdivision

--04-E-02. Fary's Theorem

--04-E-03. Representing VD

-E. Representation--作业

-F. DCEL

--04-F-01. Twin Edges

--04-F-02. Half-Edge

--04-F-03. Vertex & Face

--04-F-04. Traversal

--04-F-05. True Or False

--04-F-06. Application

-F. DCEL--作业

-G. Hardness

--04-G-01. 1D Voronoi Diagram

--04-G-02. 2D Voronoi Diagram

--04-G-03. Voronoi Diagram In General Position

-G. Hardness--作业

-H. Sorted Sets

--04-H-01. Convex Hull Made Easier

--04-H-02. Convex Hull As A Combinatorial Structure

--04-H-03. Voronoi Diagram As A Geometric Structure

-H. Sorted Sets--作业

-I. VD_sorted

--04-I-01. ε-Closeness

--04-I-02. Lifting

--04-I-03. Projection

--04-I-04. Case A

--04-I-05. Case B

--04-I-06. Sorting Not Made Easier

-I. VD_sorted--作业

-J. Naive Construction

--J. Naive Construction

-J. Naive Construction--作业

-K. Incremental Construction

--04-K-01. Royal Garden

--04-K-02. Disjoint Union

--04-K-03. Complexity

-K. Incremental Construction--作业

-L. Divide-And-Conquer

--04-L-01. Strategy

--04-L-02. Solving Overlaps

--04-L-03. Contour

--04-L-04. Bisectors

--04-L-05. Y-Monotonicity

--04-L-06. Common Tangents

--04-L-07. Contour Length

--04-L-08. Clip & Stitch

--04-L-09. Intersecting with Cells

--04-L-10. Convexity

--04-L-11. Avoiding Rescans

-L. Divide-And-Conquer--作业

-M. Plane-Sweep

--04-M-01. A First Glance

--04-M-02. Backtracking

--04-M-03. Fortune's Trick

--04-M-04. Frozen Region

--04-M-05. Beach Line

--04-M-06. Lower Envelope

--04-M-07. Break Points

--04-M-08. Events

--04-M-09. Circle Event: What, When & Where

--04-M-10. Circle Event: Why

--04-M-11. Circle Event: How

--04-M-12. Site Event: What

--04-M-13. Site Event: How

-M. Plane-Sweep--作业

05. Delaunay Triangulation

-A. Point Set Triangulation

--05-A-01. Definition

--05-A-02. Edge Flipping

--05-A-03. Upper Bound

--05-A-04. Lower Bound

-A. Point Set Triangulation--作业

-B. Delaunay Triangulation

--05-B-01. Dual Graph

--05-B-02. Triangulation

--05-B-03. Hardness

--05-B-04. History

-B. Delaunay Triangulation--作业

-C. Properties

--05-C-01. Empty Circumcircle

--05-C-02. Empty Circle

--05-C-03. Nearest Neighbor

--05-C-04. Complexity

-C. Properties--作业

-D. Proximity Graph

--05-D-01. Gabriel Graph

--05-D-02. Relative Neighborhood Graph

--05-D-03. Lower Bounds

-D. Proximity Graph--作业

-E. Euclidean Minimum Spanning Tree

--05-E-01. Definition

--05-E-02. Construction

--05-E-03. Subgraph of RNG

--05-E-04. Example

-E. Euclidean Minimum Spanning Tree--作业

-F. Euclidean Traveling Salesman Problem

--05-F-01. Definition

--05-F-02. NP-Hardness

--05-F-03. Approximation

-G. Minimum Weighted Triangulation

--05-G-01. Definition

--05-G-02. Counter-Example

--05-G-03. Hardness

-G. Minimum Weighted Triangulation--作业

-H. Construction

--05-H-01. Subtended Arc

--05-H-02. Angle Vector

--05-H-03. Maximizing The Minimum Angle

--05-H-04. Evolution By Edge Flipping

--05-H-05. Strategies

-H. Construction--作业

-I. RIC With Example

--05-I-01. Idea

--05-I-02. Point Location

--05-I-03. In-Circle Test

--05-I-04. Edge Flipping

--05-I-05. Frontier

--05-I-06. Convergence

-I. RIC With Example--作业

-J. Randomized Incremental Construction

--05-J-01. Recursive Implementation

--05-J-02. Iterative Implementation

--05-J-03. In-Circle Test

--05-J-04. Point Location

-J. Randomized Incremental Construction--作业

-K. RIC Analysis

--05-K-01. Time Cost

--05-K-02. Backward Analysis

--05-K-03. Preconditions

--05-K-04. Types Of Edge Change

--05-K-05. Number Of Edge Changes

--05-K-06. Average Degree

--05-K-07. Number Of Rebucketings

--05-K-08. Probability For Rebucketing

--05-K-09. Expectation

--05-K-10. Further Consideration

06. Point Location

-0. Online/Offline Algorithms

--06-0. Online/Offline Algorithms

-0. Online/Offline Algorithms--作业

-A. Introduction

--06-A-01. Where Am I

--06-A-02. Point Location

--06-A-03. Assumptions For Clarity

--06-A-04. Input Size

--06-A-05. Performance Measurements

--06-A-06. A Global View

-A. Introduction--作业

-B. Slab Method

--06-B-01. Slab Decomposition

--06-B-02. Ordering Trapezoids

--06-B-03. Tree of Trees

--06-B-04. Example

--06-B-05. Query Time

--06-B-06. Preprocessing Time

--06-B-07. Storage Cost

--06-B-08. Worst Case

-B. Slab Method--作业

-C. Persistence

--06-C-01. Ephemeral Structure

--06-C-02. Persistent Structure

--06-C-03. Persistent Slabs

-C. Persistence--作业

-D. Path Copying

--06-D-01. Strategy

--06-D-02. X-Search

--06-D-03. Storage Optimization

-D. Path Copying--作业

-E. Node Copying

--06-E-01. O(1) Rotation

--06-E-02. Strategy

--06-E-03. Why Red-Black

--06-E-04. Linear Space

--06-E-05. Time Penalty

-E. Node Copying--作业

-F. Limited Node Copying

--06-F-01. Idea

--06-F-02. Split

--06-F-03. Complexity

--06-F-04. Recoloring

-G. Kirkpatrick Structure

--06-G-01. Optimal And Simpler

--06-G-02. Triangulation

--06-G-03. Example

--06-G-04. Hierarchy

--06-G-05. Independent Subset

--06-G-06. The More The Better

--06-G-07. The Fewer The Better

--06-G-08. Degree

--06-G-09. Existence Of Independent Subset

--06-G-10. Construction Of Independent Subset

--06-G-11. DAG

-G. Kirkpatrick Structure--作业

-H. Trapezoidal Map

--06-H-01. Ray Shooting

--06-H-02. Decomposition

--06-H-03. Properties & Complexity

--06-H-04. Search Structure: Example

--06-H-05. Search Structure: Nodes

--06-H-06. Search Structure: Performance

-H. Trapezoidal Map--作业

-I. Constructing Trapezoidal Map

--06-I-01. Initialization

--06-I-02. Iteration

--06-I-03. Challenges

--06-I-04. Case 1: Two Endpoints

--06-I-05. Case 2: One Endpoint

--06-I-06. Case 3: No Endpoints

--06-I-07. Example

-J. Performance Of Trapezoidal Map

--06-J-01. Randomization

--06-J-02. Expectation

--06-J-03. Number Of Ray Trimmed

--06-J-04. Number Of Trapezoidals Created (1)

--06-J-05. Number Of Trapezoidals Created (2)

--06-J-06. Time For Point Location

--06-J-07. Size Of Search Structure

--06-J-08. Fixed Query Point + Randomly Created Maps

--06-J-09. Each Single Step

--06-J-10. Probability Of Enclosing Trapezoid Changed

--06-J-11. Query Time

07. Geometric Range Search

-A. Range Query

--07-A-01. 1-Dimensional Range Query

--07-A-02. Brute-force

--07-A-03. Binary Search

--07-A-04. Output Sensitivity

--07-A-05. Planar Range Query

-A. Range Query--作业

-B. BBST

--07-B-01. Structure

--07-B-02. Lowest Common Ancestor

--07-B-03. Query Algorithm

--07-B-04. Complexity (1)

--07-B-05. Complexity (2)

-B. BBST--作业

-C. kd-Tree: Structure

--07-C-01. 2d-Tree

--07-C-02. Example

--07-C-03. Construction

--07-C-04. Example

--07-C-05. Canonical Subsets

-C. kd-Tree: Structure--作业

-D. kd-Tree: Algorithm

--07-D-01. Query

--07-D-02. Example

--07-D-03. Optimization

-D. kd-Tree: Algorithm--作业

-E. kd-Tree: Performance

--07-E-01. Preprocessing Time + Storage

--07-E-02. Query Time

--07-E-03. Beyond 2D

-E. kd-Tree: Performance--作业

-F. Range Tree: Structure

--07-F-01. x-Query + y-Query

--07-F-02. Worst Case

--07-F-03. x-Query * y-Queries

-F. Range Tree: Structure--作业

-G. Range Tree: Query

--07-G-01. Painters' Strategy

--07-G-02. X-Tree

--07-G-03. Y-Trees

--07-G-04. Algorithm

-G. Range Tree: Query--作业

-H. Range Tree: Performance

--07-H-01. Storage

--07-H-02. Preprocessing Time

--07-H-03. Query Time

--07-H-04. Beyond 2D

-H. Range Tree: Performance--作业

-I. Range Tree: Optimization

--07-I-01. Y-Lists

--07-I-02. Coherence

--07-I-03. Idea

--07-I-04. Fractional Cascading

--07-I-05. Complexity

08. Windowing Query

-A. Orthogonal Windowing Query

--08-A-01. Definition

--08-A-02. Classification

-A. Orthogonal Windowing Query--作业

-B. Stabbing Query

--08-B-01. 1D Windowing Query

--08-B-02. Stabbing Query

-C. Interval Tree: Construction

--08-C-01. Median

--08-C-02. Partitioning

--08-C-03. Balance

--08-C-04. Associative Lists

--08-C-05. Complexity

-C. Interval Tree: Construction--作业

-D. Interval Tree: Query

--08-D-01. Algorithm (1)

--08-D-02. Algorithm (2)

--08-D-03. Complexity

-D. Interval Tree: Query--作业

-E. Stabbing With A Segment

--08-E-01. Definition

--08-E-02. Interval Tree

--08-E-03. Query Algorithm (1)

--08-E-04. Query Algorithm (2)

--08-E-05. Overview

--08-E-06. Complexity

-F. Grounded Range Query

--08-F-01. O(n) Space

--08-F-02. 2D-GRQ

--08-F-03. 1D-GRQ Using Range Tree

--08-F-04. 1D-GRQ By Linear Scan

-G. 1D-GRQ Using Heap

--08-G-01. Heap

--08-G-02. Query

--08-G-03. Example

--08-G-04. Complexity

-G. 1D-GRQ Using Heap--作业

-H. Priority Search Tree

--08-H-01. PST = Heap + BBST

--08-H-02. Order Property

--08-H-03. Sibling Partitioning

--08-H-04. Construction

-H. Priority Search Tree--作业

-I. 2D-GRQ Using PST

--08-I-01. Algorithm (1/2)

--08-I-02. Algorithm (2/2)

--08-I-03. Example (1/3)

--08-I-04. Example (2/3)

--08-I-05. Example (3/3)

--08-I-06. Query Time (1/3)

--08-I-07. Query Time (2/3)

--08-I-08. Query Time (3/3)

-I. 2D-GRQ Using PST--作业

-J. Segment Tree

--08-J-01. General Windowing Query

--08-J-02. Elementary Interval

--08-J-03. Discretization

--08-J-04. Worst Case

--08-J-05. BBST

--08-J-06. Solving Stabbing Query

--08-J-07. Worst Case

--08-J-08. Common Ancestor

--08-J-09. Canonical Subsets

--08-J-10. O(nlogn) Space

--08-J-11. Constructing A Segment Tree

--08-J-12. Inserting A Segment (1)

--08-J-13. Inserting A Segment (2)

--08-J-14. Inserting A Segment (3)

--08-J-15. Query Algorithm

--08-J-16. Query Time

-K. Vertical Segment Stabbing Query

--08-K-01. Review

--08-K-02. X-Segment Tree

--08-K-03. Associative Structure

--08-K-04. Vertical Segment Stabbing Query

06-G-01. Optimal And Simpler笔记与讨论

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