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04-B-03. Voronoi Diagram在线视频

04-B-03. Voronoi Diagram

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04-B-03. Voronoi Diagram课程教案、知识点、字幕

这样的话 我们就顺利的得到了

特定的一个site

所对应的cell

当然从数学上我们可以如法炮制

也就是说

我们确实可以比较轻松的得到

除了i之外的

任何一个site

所对应的cell

我们都可以按照刚才那种求交的方法

在数学上把它们得到

而且也确实可以知道

它们是凸的

当然这里还有一条

所有这些cell中

有一些像这样是封闭的

也会有一些是无界的

这背后的原因也非常简单

我们说如果点集的规模是有限的话

那么它们所对应的cell中

必然至少有一个是无界的

鸽巢原理 没错

整个平面是无界的

将一个无界的对象切分成有限块

根据鸽巢原理

其中至少有一块依然是无界的

那我们这里所讨论的Voronoi图呢

显然在计算机中

我们并不方便去保存

所有的这其中的这些元素

我们只需要保存

它们之间的边界特征就可以了

什么呢

也就是我们会把所有的这些cell

给它忽略掉

因为它们需要记忆的东西太多了

而本质上讲

我们只需要记录它们之间的

这些边界线就够了

确实如此

所谓的Voronoi图

在我们这一章接下来要讨论的

无论是证明它的性质

还是来构造它的算法

我们所讨论的其实都是

将所有的这些cell

移掉之后所剩下的那个部分

这些部分才是计算机中

可以有效的存储

表示和利用的部分

同样的因为时间的限制

在这一章中

我们主要讨论的还是二维的

也就是平面上的Voronoi图

它的性质 它的原理和相关的算法

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

04-B-03. Voronoi Diagram笔记与讨论

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