6.850: Geometric Computing
Spring 2007
Problem Set 1
Due: March 1st, 2007
Note: You should solve all the problems in the Mandatory Part, and one of the two problems
in the Optional Part.
Mandatory Part
Problem 1. Consider a function T : N2 N, satisfying
n

6.850: Geometric Computing
Spring 2007
Problem Set 2
Due: March 20, 2007
Note: You should solve all the problems in the Mandatory Part, and one of the two problems
in the Optional Part.
Mandatory Part
Problem 1. Intersection check Given a set S of n verti

Algorithms for Streaming Data
Piotr Indyk
April 12, 2007
Lecture 17: Algorithms for
Streaming Data
Streaming Data
Problems defined over elements
P=cfw_p1,pn
The algorithm sees p1, then p2, then p3,
Key fact: it has limited storage
Can store only s<n p

Motion Planning
Piotr Indyk
March 10, 2005
Lecture 11: Motion Planning
Piano Movers Problem
Given:
A set of obstacles
The initial position of a robot
The final position of a robot
Goal: find a path that
Moves the robot from the initial to final
posi

Geometric Optimization
Piotr Indyk
March 20, 2007
Geometric Optimization
Geometric Optimization
Minimize/maximize something subject to
some constraints
Have seen:
Linear Programming
Minimum Enclosing Ball
Diameter/NN (?)
All had easy polynomial time

Algorithmic Applications of
Low-distortion Geometric
Embeddings
Piotr Indyk
MIT
Low-distortion geometric embeddings
Formally: a mapping f : PA PB :
PA: points from metric space with distance D(, )
d
PB : points from some normed space, e.g., l2
For an

6.850: Geometric Computing
Spring 2007
Problem Set 4
Due: May 15, 2007
Note: You should solve all the problems in the Mandatory Part, and one of the two problems
in the Optional Part.
Mandatory Part
Problem 1. External queue. Show how to eciently implemen

6.850: Geometric Computing
Spring 2007
Problem Set 3
Due: April 26, 2007
Note: You should solve all the problems in the Mandatory Part, and one of the two problems
in the Optional Part.
Mandatory Part
Problem 1. Fast Approximate Near Neighbor in 1 . Const

Combinatorial Geometry
Piotr Indyk
May 3, 2007
Combinatorial Geometry
1
Previous Lecture
Algorithm for matching A in B:
Take any pair a,aA, let r=|a-a|
Find all pairs b,bB such that |b-b|=r
For all such pairs
Compute t that transforms (a,a) into (b,b

The Visibility Problem
and
Binary Space Partition
Piotr Indyk
(and Nati Srebro)
March 6, 2007
The Visibility Problem
b
a
c
d
March 6, 2007
e
Algorithms
Painters algorithm:
draw objects in order, from back to front
Z-buffer:
Draw objects in arbitrary o

Voronoi Diagrams
(Slides mostly by Allen Miu)
March 1, 2005
Lecture 8: Voronoi Diagrams
Post Office: What is the area of service?
pi : site points
q : free point
e : Voronoi edge
v : Voronoi vertex
q
March 1, 2005
v
pi
Lecture 8: Voronoi Diagrams
e
Defini

External Memory Algorithms
for Geometric Problems
Piotr Indyk
(slides partially by Lars Arge and
Jeff Vitter)
Compared to Previous Lectures
Another way to tackle large data sets
Exact solutions (no more embeddings)
April 14, 2005
Lecture 17: External Me

Arrangements and Duality
Motivation: Ray-Tracing
Slides mostly by Darius Jazayeri
March 1, 2007
Lecture 7: Arrangements and Duality
Ray-Tracing
Render a scene by shooting a ray from the
viewer through each pixel in the scene, and
determining what object

Delaunay Triangulations
(slides mostly by Glenn Eguchi)
March 3, 2005
Lecture 9: Delaunay triangulations
Motivation: Terrains
Set of data points A R2
Height (p) defined at each point p in A
How can we most naturally approximate
height of points not in

Geometric Computation:
Introduction
Piotr Indyk
February 6, 2007
Lecture 1: Introduction to Geometric Computation
Welcome to 6.850 !
Overview and goals
Course Information
Closest pair
Signup sheet
February 6, 2007
Lecture 1: Introduction to Geometric Comp

Polygon Triangulation
Piotr Indyk
(and Daniel Vlasic )
February 22, 2007
Triangulation: Definition
Triangulation of a simple polygon P:
decomposition of P into triangles by a
maximal set of non-intersecting
diagonals
Diagonal: an open line segment that

Orthogonal Range Queries
Piotr Indyk
February 15, 2007
Lecture 5: Orthogonal Range Queries
Range Searching in 2D
Given a set of n points,
build a data structure
that for any query
rectangle R, reports all
points in R
February 15, 2007
Lecture 5: Orthogon

Segment Intersection
Piotr Indyk
2003 by Piotr Indyk
6.850 Geometric Computation
February 8, 2007
L2.1
Problem
Segment intersection problem:
Given: a set of n distinct
segments s1sn, represented
by coordinates of endpoints
Detection: detect if there i

Point Location
Piotr Indyk
(and Sergi Elizalde and David
Pritchard)
February 27, 2007
Lecture 6: Point Location
Point Location
February 27, 2007
Lecture 6: Point Location
Definition
Given: a planar subdivision S
Goal: build a data structure that, given