UMass Lowell Computer Science 91'503 Analysis of Algorithms Prof' Karen Daniels Fall, 2001

1
UMass Lowell Computer Science 91.503 Analysis
of Algorithms Prof. Karen Daniels Fall, 2001

• Lecture 10
• Tuesday, 11/27/01
• Computational Geometry
• Chapters 35

2
Relevant Sections of Chapter 35
Youre responsible for material in this chapter
that we discuss in lecture. (Note that this
includes all sections.)
Ch35 Computational Geometry
3
Overview

• Computational Geometry Introduction
• Line Segment Intersection
• Convex Hull Algorithms
• Nearest Neighbors/Closest Points

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Computational Geometry
Computer Graphics
Visualization

Design
Analyze
Core Geometric Algorithms
Application-Based Algorithms
Apply
Telecommunications
5
Typical Problems

• maintaining line arrangements
• polygon partitioning
• nearest neighbor search
• kd-trees

• bin packing
• Voronoi diagram
• simplifying polygons
• shape similarity
• convex hull

SOURCE Steve Skienas Algorithm Design Manual
(for problem descriptions, see graphics gallery
at http//www.cs.sunysb.edu/algorith)
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Common Computational Geometry Structures
source ORourke, Computational Geometry in C
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Sample Tools of the Trade
Algorithm Design Patterns/Techniques binary
search divide-and-conquer duality randomization sw
eep-line derandomization parallelism Algorithm
Analysis Techniques asymptotic analysis,
amortized analysis Data Structures winged-edge,
quad-edge, range tree, kd-tree Theoretical
Computer Science principles NP-completeness,
hardness
MATH
Sets
Proofs
Geometry
Summations
Linear Algebra
Probability
Growth of Functions
Recurrences
Graph Theory
Combinatorics
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Computational Geometryin Context
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Line Segment Intersections(2D)

• Intersection of 2 Line Segments
• Intersection of gt 2Line Segments

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Cross-Product-Based Geometric Primitives
Some fundamental geometric questions
source 91.503 textbook Cormen et al.
(1)
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Cross-Product-Based Geometric Primitives (1)
Advantage less sensitive to accumulated
round-off error
source 91.503 textbook Cormen et al.
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Cross-Product-Based Geometric Primitives (2)
source 91.503 textbook Cormen et al.
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Intersection of 2 Line Segments
Step 1 Bounding Box Test
Step 2 Does each segment straddle the line
containing the other?
source 91.503 textbook Cormen et al.
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Segment-Segment Intersection

• Finding the actual intersection point
• Approach parametric vs. slope/intercept
• parametric generalizes to more complex
  intersections
• Parameterize each segment

Lab
c
Lcd
Cd-c
b
q(t)ctC
d
Ab-a
a
p(s)asA
Intersection values of s, t such that p(s) q(t)
asActC
source ORourke, Computational Geometry in C
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Demo

• Segment/Segment Intersection
• http/cs.smith.edu/orourke/books/CompGeom/CompGeo
  m.html

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Intersection of gt2 Line Segments
Sweep-Line Algorithmic Paradigm
source 91.503 textbook Cormen et al.
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Intersection of gt2 Line Segments
Sweep-Line Algorithmic Paradigm
source 91.503 textbook Cormen et al.
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Intersection of gt2 Line Segments
Time to detect if any 2 segments intersect
O(n lg n)
source 91.503 textbook Cormen et al.
source 91.503 textbook Cormen et al.
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Intersection of Segments

• Goal Output-size sensitive line segment
  intersection algorithm that actually computes
  intersection points
• Bentley-Ottmann plane sweep O((nk)logn) time
• k number of intersection points in output
• Intuition sweep horizontal line downwards
• just before intersection, 2 segments are adjacent
  in sweep-line intersection structure
• check for intersection only segments that
  adjacent
• event types
• top endpoint of a segment
• bottom endpoint of a segment
• intersection between 2 segments

Improved to O(nlognk) Chazelle/Edelsbrunner
source ORourke, Computational Geometry in C
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Convex Hull Algorithms

• Definitions
• Gift Wrapping
• Graham Scan
• QuickHull
• Incremental
• Divide-and-Conquer
• Lower Bound in W(nlgn)

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Convexity Convex Hulls
source ORourke, Computational Geometry in C

• A convex combination of points x1, ..., xk is a
  sum of the form a1x1... akxk where
• Convex hull of a set of points is the set of all
  convex combinations of points in the set.

source 91.503 textbook Cormen et al.
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Naive Algorithms for Extreme Points
O(n4)
source ORourke, Computational Geometry in C
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Algorithms 2D Gift Wrapping

• Use one extreme edge as an anchor for finding the
  next

source ORourke, Computational Geometry in C
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Gift Wrapping
source 91.503 textbook Cormen et al.
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Algorithms 3D Gift Wrapping
O(n2) time output sensitive O(nF) for F faces
on hull
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Algorithms 2D QuickHull

• Concentrate on points close to hull boundary
• Named for similarity to Quicksort

source ORourke, Computational Geometry in C
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Algorithms 3D QuickHull
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Algorithms Qhull (gt 2D )
http//www.geom.umn.edu/software/qhull/
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Grahams Algorithm
source ORourke, Computational Geometry in C

• Points sorted angularly provide star-shaped
  starting point
• Prevent dents as you go via convexity testing

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Graham Scan
source 91.503 textbook Cormen et al.
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Graham Scan
source 91.503 textbook Cormen et al.
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Graham Scan
source 91.503 textbook Cormen et al.
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Graham Scan
source 91.503 textbook Cormen et al.
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Graham Scan
source 91.503 textbook Cormen et al.
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Algorithms 2D Incremental
source ORourke, Computational Geometry in C

• Add points, one at a time
• update hull for each new point
• Key step becomes adding a single point to an
  existing hull.
• Idea is extended to 3D in Chapter 4.

Algorithm INCREMENTAL ALGORITHM Let H2
ConvexHullp0 , p1 , p2 for k 3 to n - 1
do Hk ConvexHull Hk-1 U pk
O(n2)
can be improved to O(nlgn)
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Algorithms 3D Incremental
O(n2) time
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Algorithms 2D Divide-and-Conquer
source ORourke, Computational Geometry in C

• Divide-and-Conquer in a geometric setting
• O(n) merge step is the challenge
• Find upper and lower tangents
• Lower tangent find rightmost pt of A leftmost
  pt of B then walk it downwards
• Idea is extended to 3D in Chapter 4.

B
A
Algorithm DIVIDE-and-CONQUER Sort points by x
coordinate Divide points into 2 sets A and B A
contains left n/2 points B contains right n/2
points Compute ConvexHull(A) and ConvexHull(B)
recursively Merge ConvexHull(A) and ConvexHull(B)
O(nlgn)
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Algorithms 3D Divide and Conquer
O(n log n) time !
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Lower Bound of O(nlgn)
source ORourke, Computational Geometry in C

• Worst-case time to find convex hull of n points
  in algebraic decision tree model is in W(nlgn)
• Proof uses sorting reduction
• Given unsorted list of n numbers (x1,x2 ,, xn)
• Form unsorted set of points (xi, xi2) for each
  xi
• Convex hull of points produces sorted list!
• Parabola every point is on convex hull
• Reduction is O(n) (which is o(nlgn))
• Finding convex hull of n points is therefore at
  least as hard as sorting n points, so worst-case
  time is in W(nlgn)

Parabola for sorting 2,1,3
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Nearest Neighbor/Closest Pair of Points
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Closest Pair
Goal Given n (2D) points in a set Q, find the
closest pair under the Euclidean metric in O(n
lgn) time.

• Divide-and-Conquer Strategy
• X points sorted by increasing x
• Y points sorted by increasing y
• Divide partition with vertical line L into PL,
  PR
• Conquer recursively find closest pair in PL, PR
• dL, dR are closest-pair distances
• d min( dL, dR )
• Combine closest-pair is either d or pair
  straddles partition line L
• Check for pair straddling partition line L
• both points must be within d of L
• create array Y Y without points in 2d
  strip
• for each point p in Y
• find (lt 7) points in Y within d of p

source 91.503 textbook Cormen et al.
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Closest Pair
Correctness
source 91.503 textbook Cormen et al.
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Closest Pair
Running Time
Key Point Presort points, then at each step form
sorted subset of sorted array in linear time
source 91.503 textbook Cormen et al.