lecture intersection search(1)

8/12/2019 Lecture Intersection Search(1)

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Intersection Search

Input: A set Sof geometric objects(segments, polygons, disks, solid models, etc)

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Intersection Search

Versions of the problem: DETECT: Answer yes/no: Are there any

intersections among the objects S?

(if yes, then we may insist on returning a witness) REPORT: Output all pairs of objects that intersect

COMPUTE: Compute the common intersection of allobjects

COUNT: How many pairs intersect? QUERY: Preprocess Sto support fast queries of

the form Does object Qintersect any member ofS? (or Report all intersections with Q)

3May also want to insert/delete in S, orallow objects of S to move.


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Warm Up: 1D Problem

Given n segments (intervals) on a line

DETECT: O(n log n), based on sorting

REPORT: O(k+n log n), based on sorting,then marching through, left to right

Lower bound to DETECT: : (n log n) ,

from ELEMENT UNIQUENESS

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Element UniquenessInput: {x1,x2,,xn}Are they distinct?(yes/no)

(n log n)


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Intersection Search: Segments

Segment intersection: Given a set Sofnline segments in the plane, determine: Does some pair intersect? (DETECT)

Compute all points of intersection(REPORT)

Nave: O(n2)CG: O(n log n) DETECT, O(k+n log n) REPORT

Lower Bound to DETECT: (n log n) , from ELEMENT UNIQUENESS

Element UniquenessInput: {x1,x2,,xn}Are they distinct?(yes/no)

(n log n)


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Primitive Computation

Does segment abintersect segment cd? Types of intersect

Test using Left tests (sign of a crossproduct (determinant), which determinesthe orientation of 3 points)

Time O(1) (constant)

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a

bc

Left( a, b, c ) = TRUE ab ac > 0c is left of ab


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Bentley-Ottmann Sweep

Main idea: Process eventsin order ofdiscovery as a vertical line, L, sweepsover the scene, from left to right

Two data structures: SLS: Sweep Line Status: bottom-to-top

ordering of the segments intersecting L

EQ: Event Queue

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L


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Two data structures:

SLS: Sweep Line Status: bottom-to-topordering of the segments intersecting L

EQ: Event Queue

Initialize: SLS =

EQ = sorted list of segment endpoints

How to store: SLS: balanced binary search tree

(dictionary, support O(log n) insert, delete, search)

EQ: priority queue (e.g., heap)8

O(n log n)

L


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Hit left endpt of e

Hit right endpt of e

Hit crossing point e f (only needed in REPORT)

Event Handling

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O(log n)L

a

b eFind segs a and b above/below ein SLS. Inserte into SLS.Test(a,e), Test(b,e) and insertcrossings (if any) in EQ

a

b

e

L

Find segs a and b above/below ein SLS. Deletee from SLS.Test(a,b), and insert crossing(if any) in EQ

a

b

L

e

f

O(log n)

O(log n)

Exchangee, f in SLS.Test(a,f), Test(b,e) and insertcrossings (if any) in EQ


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Summary

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[Lecture Notes of David Mount, Lecture 6]


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Algorithm Analysis Invariantsof algorithm:

SLS is correctly ordered. Test(a,b) for intersection is done whenever segments a and b

become adjacent in the SLS order

Discovered crossings are inserted into EQ

(for REPORT; for DETECT, stop at first detected crossing) Claim:All crossings are discovered

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[Lecture Notes of David Mount, Lecture 6]


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Algorithm Analysis

Time: O(n log n) to DETECT (O(n) events @ O(log n) )

Time: O((n+k)log n) to REPORT (O(n+k) events @ O(log n) )

Example Applet

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k = # crossings = output size
http://www.lems.brown.edu/~wq/projects/cs252.htmlhttp://www.lems.brown.edu/~wq/projects/cs252.htmlhttp://www.lems.brown.edu/~wq/projects/cs252.htmlhttp://www.lems.brown.edu/~wq/projects/cs252.htmlhttp://www.lems.brown.edu/~wq/projects/cs252.htmlhttp://www.lems.brown.edu/~wq/projects/cs252.html


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Space: Working Memory

Basic (original) version of B-O sweep:Naively, the EQ has size at most O(n+k),since we store the SLS (O(n)) and theEQ (size 2n+k )

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Better bounds on size of EQ: O(n log2n)[Pach and Sharir]


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Space: Working Memory

Modified B-O Sweep: Any time 2 segs STOP being adjacent in

SLS, remove from EQ the corresponding

crossing point (if any); we will re-insert thecrossing point again (at least once) beforethe actual crossing event.

Note: Now the EQ has at most n-1 crossingevents (and at most 2n endpoint events)

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Optimal REPORT algorithms exist: O(k + n log n), O(k+n) space (working memory)

[Chazelle-Edelsbrunner] O(k + n log n), O(n) space [Balaban]

Special Case: REPORT for horiz/vert segs Bentley-Ottmann sweep: O(k + n log n) (optimal!)

Special case: Simplicity testing O(n), from Chazelle triangulation

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(Complex)

LAll crossings happen when L hits a verticalsegment, si ; just locate (O( log n)) the endpointand walk along vertical segment in SLS to reportall kihorizontal segments crossed along si


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Advertisement

Today, 4:00pm, in Math P-131

Talk: Watchman Route Problem

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Sweeping applies also to Unions

Intersections

Arrangements

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Map Overlay

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Data Structures for Planar Subdivisions

Winged Edge Data

Structure. Quad Edge Data

Structure.

Our focus on Doubly

Connected Edge List.

[DCEL: doubly connected edge list]

f0

f1e

e4,0

v1

v0

-represent edge as 2 halves

-lists: vertex, face, edge/twin-facilitates face traversal

es

twin

v2

v7

v6

v5

v3

v4

e0,1

Every edge e is struct: e.org, e.dest: pointer to origin

and dest vertices.

e.face is face on left of edge e.twin is the same edge,

oriented the other direction.

e.next is next edge along theface

Ack: R. Pless


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DCEL

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Practical Methods, 3D

Uniform grid

Quadtrees

Bounding volume hierarchies26

With each pixel, store alist of objectsintersecting it.Do brute force on

pixel-by-pixel basis.

See Samet books, SANDwebsite
http://www.cs.umd.edu/~brabec/sandjava/index.htmlhttp://www.cs.umd.edu/~brabec/sandjava/index.html


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QuickCD: Collision Detection


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The 2-Box Cover Problem

Find smallest (tightest fitting) pairof bounding boxes

Motivation: Best outer approximation

Bounding volume hierarchies


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Bounding Volume Hierarchy

BV-tree:

Level 0 k-dops


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BV-tree: Level 1

26-dops

14-dops6-dops

18-dops


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BV-tree: Level 2


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BV-tree: Level 5


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BV-tree: Level 8

lecture intersection search(1)

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