Sublinear Geometric Algorithms Sublinear Geometric Algorithms B. - - PowerPoint PPT Presentation

Sublinear Geometric Algorithms Sublinear Geometric Algorithms

• B. Chazelle, D. Liu, A. Magen

Princeton University / University of Toronto STOC 2003

• B. Chazelle, D. Liu, A. Magen

Princeton University / University of Toronto STOC 2003

CS 468 Geometric Algorithms Seminar Winter 2005/2006

2 CS 468 – Geometric Algorithms Seminar – Winter 2005/2006

Overview Overview

What is this paper about? Sublinear geometric algorithms, for convex polygons (2D) and convex polyhedra (3D):

• Intersection
• Ray shooting
• Volume approximation
• Shortest path approximation

What is this paper about? Sublinear geometric algorithms, for convex polygons (2D) and convex polyhedra (3D):

• Intersection
• Ray shooting
• Volume approximation
• Shortest path approximation

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Sublinear Algorithms Sublinear Algorithms

What means sublinear?

• Not reading the complete input
• Hence: No preprocessing
• Assuming “standard” input formats
• Randomized Las-Vegas algorithms
• No wrong answers, but run time may vary
• Expected runtimes are sublinear

What means sublinear?

• Not reading the complete input
• Hence: No preprocessing
• Assuming “standard” input formats
• Randomized Las-Vegas algorithms
• No wrong answers, but run time may vary
• Expected runtimes are sublinear

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Overview Overview

Overview:

• Basic technique
• Sublinear search algorithm
• Optimality
• Polygons / polyhedra intersection
• Ray shooting & applications
• Volume approximation
• Shortest path approximation

Overview:

• Basic technique
• Sublinear search algorithm
• Optimality
• Polygons / polyhedra intersection
• Ray shooting & applications
• Volume approximation
• Shortest path approximation

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Overview Overview

Overview:

• Basic technique
• Sublinear search algorithm
• Optimality
• Polygons / polyhedra intersection
• Ray shooting & applications
• Volume approximation
• Shortest path approximation

Overview:

• Basic technique
• Sublinear search algorithm
• Optimality
• Polygons / polyhedra intersection
• Ray shooting & applications
• Volume approximation
• Shortest path approximation

each new technique builds upon previous techniques each new technique builds upon previous techniques

6

Basic Technique Basic Technique

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Search Problem Search Problem

Successor Search:

• Given a sorted doubly-linked list of numbers
• And a search value k
• Find first element greater than k
• List stored in continuous memory block

(allowing sampling of list elements) Successor Search:

• Given a sorted doubly-linked list of numbers
• And a search value k
• Find first element greater than k
• List stored in continuous memory block

(allowing sampling of list elements)

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Data Structure Data Structure

Looks like this: Memory Layout: Looks like this: Memory Layout:

1 3 7 8 9 1 3 9 7 8

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The Search Algorithm The Search Algorithm

Search Algorithm:

• Pick √n random list elements
• Determine closest predecessor and successor
• f k (within the random sample)
• Perform plain, linear search towards k

(starting from these two elements) Search Algorithm:

• Pick √n random list elements
• Determine closest predecessor and successor
• f k (within the random sample)
• Perform plain, linear search towards k

(starting from these two elements)

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The Search Algorithm (2) The Search Algorithm (2)

Looks like this: Looks like this:

• list -
• random sample -

succ(k)

• linear search -

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Analysis Analysis

The Algorithm:

• Takes expected O(√n ) time.
• This is optimal.

The Algorithm:

• Takes expected O(√n ) time.
• This is optimal.

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Why? Why?

Runtime: Runtime:

Sampling: Good case: Bad case:

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Runtime Runtime

Runtime Analysis Runtime depends on “empty” block around succ(k) Probability of no hit within Factor c√n of target is O(exp(-c)) Hence: Expected distance O(√n ) after √n trials. Runtime Analysis Runtime depends on “empty” block around succ(k) Probability of no hit within Factor c√n of target is O(exp(-c)) Hence: Expected distance O(√n ) after √n trials.

factor c away

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Optimality Optimality

Why can’t we do O(any better)? The proof employs Yao’s minimax principle:

The expected running time of an optimal deterministic algorithm for any chosen input distribution is a lower bound

• n the expected running time of the optimal Las Vegas

randomized algorithm.

Why can’t we do O(any better)? The proof employs Yao’s minimax principle:

The expected running time of an optimal deterministic algorithm for any chosen input distribution is a lower bound

• n the expected running time of the optimal Las Vegas

randomized algorithm.

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Constructing the Bound Constructing the Bound

A Lower Bound for an Opt. Determ. Algorithm Choose the following scenario:

• Input: Random permutation
• Search for last element

What can the algorithm do?

• Op. “A”: Visit neighbor of known node
• Op. “B”: Go to unknown node in memory

A Lower Bound for an Opt. Determ. Algorithm Choose the following scenario:

• Input: Random permutation
• Search for last element

What can the algorithm do?

• Op. “A”: Visit neighbor of known node
• Op. “B”: Go to unknown node in memory

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( )

b

n b a n n ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − ≥ 1 " elem. last not saw " Pr

Constructing the Bound (2) Constructing the Bound (2)

Random input:

• Consider the question: How likely is it, to

discover one of the last √n elements?

• With every newly visited node, the chance of

discovery increases

• After a A-Op’s and b B-Op’s:

Random input:

• Consider the question: How likely is it, to

discover one of the last √n elements?

• With every newly visited node, the chance of

discovery increases

• After a A-Op’s and b B-Op’s:

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Constructing the Bound (3) Constructing the Bound (3)

Looks like this: Looks like this:

( )

b

n b a n n ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − ≥ 1 " elem. last not saw " Pr

A A B B A A A B A B unknown target: last √n

n a b

b

Pr(“no discovery”) new random choice no-go

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Constructing the Bound (3) Constructing the Bound (3)

Number of B-Ops:

• Expected value of A and B-op’s until visiting
• ne of the last √n is O(√n).
• If last √n elements have not been visited

⇒ √n more A-Ops necessary Number of B-Ops:

• Expected value of A and B-op’s until visiting
• ne of the last √n is O(√n).
• If last √n elements have not been visited

⇒ √n more A-Ops necessary

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What’s next? What’s next?

Now: Construct sublinear geometric algorithms

• Similar basic idea
• All with O(√n ) time complexity
• Some problems are “more general” than

successor-searching Thus: run-time also optimal Now: Construct sublinear geometric algorithms

• Similar basic idea
• All with O(√n ) time complexity
• Some problems are “more general” than

successor-searching Thus: run-time also optimal

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Polygon I ntersection Polygon I ntersection

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Problem Statement Problem Statement

Polygon Intersection:

• Given two polygons A, B
• Encoded as densely stored

linked lists of vertices (same as before)

• Do they intersect?
• If so, report one intersection point.

Polygon Intersection:

• Given two polygons A, B
• Encoded as densely stored

linked lists of vertices (same as before)

• Do they intersect?
• If so, report one intersection point.

A B

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I dea of the Algorithm I dea of the Algorithm

Idea: Employ the same sampling approach Idea: Employ the same sampling approach

two input polygons simplified by random subsampling

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I dea of the Algorithm I dea of the Algorithm

Two step Approach: Two step Approach:

(I) test for simplified polys intersection (II) test potentially overlapping remaining region

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First Step First Step

First Step:

• Random subsampling
• Choose O(√n ) sample points
• Compute intersection in linear time

First Step:

• Random subsampling
• Choose O(√n ) sample points
• Compute intersection in linear time

A B

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First Step (2) First Step (2)

First Step:

• If no intersection found:
• Compute separating plane
• Go to step 2

First Step:

• If no intersection found:
• Compute separating plane
• Go to step 2

A B

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Second Step Second Step

Second Step:

• Compute expected O(√n ) vertices of A

exceeding separating plane

• Test intersection with simplified B

Second Step:

• Compute expected O(√n ) vertices of A

exceeding separating plane

• Test intersection with simplified B

A B

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Second Step (2) Second Step (2)

Second Step:

• If no intersection is found:
• Compute new separating plane
• Test against corresponding part of B

Second Step:

• If no intersection is found:
• Compute new separating plane
• Test against corresponding part of B

A B

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Second Step (3) Second Step (3)

Second Step:

• If no intersection is found:
• Compute new separating plane
• Test against corresponding part of B

Second Step:

• If no intersection is found:
• Compute new separating plane
• Test against corresponding part of B

A B

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Second Step (4) Second Step (4)

Second Step:

• If no intersection is found:
• Compute new separating plane
• Test against corresponding part of B

Second Step:

• If no intersection is found:
• Compute new separating plane
• Test against corresponding part of B

A B

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Second Step (5) Second Step (5)

Second Step:

• Report intersection if found
• Otherwise repeat second step

with A and B exchanged for final answer Second Step:

• Report intersection if found
• Otherwise repeat second step

with A and B exchanged for final answer

A B

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Polyhedra I ntersection Polyhedra I ntersection

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Polyhedral I ntersection Polyhedral I ntersection

Intersection of Two Convex Polyhedra

• Same approach as for polygons
• Some additional technical problems...

Intersection of Two Convex Polyhedra

• Same approach as for polygons
• Some additional technical problems...

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First Step First Step

First Step:

• Simplify Polyhedra
• Take O(√n ) sample vertices

First Step:

• Simplify Polyhedra
• Take O(√n ) sample vertices

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First Step First Step

First Step:

• Test for intersection (lin. programming)
• If no intersection: Compute separating plane

First Step:

• Test for intersection (lin. programming)
• If no intersection: Compute separating plane

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Second Step Second Step

Second Step:

• Compute full resolution part of B

which exceeds separating plane

• Test with simplified A

Second Step:

• Compute full resolution part of B

which exceeds separating plane

• Test with simplified A

A B

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Second Step Second Step

Second Step:

• If no intersection is found:

Compute new supporting plane... Second Step:

• If no intersection is found:

Compute new supporting plane...

A B

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Second Step Second Step

Second Step:

• Test for intersection with full resolution piece of A
• n B-side of plane
• If no intersection found: repeat second step with

A and B swapped (as for polygons) Second Step:

• Test for intersection with full resolution piece of A
• n B-side of plane
• If no intersection found: repeat second step with

A and B swapped (as for polygons)

A B

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Additional Problem Additional Problem

Additional Problem for 3D-Polyhedra

• How compute full resolution

cut-off pieces?

• Start at a low-res point
• Adjacency search
• Problem: Starting vertex

with large out-degree Additional Problem for 3D-Polyhedra

• How compute full resolution

cut-off pieces?

• Start at a low-res point
• Adjacency search
• Problem: Starting vertex

with large out-degree

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Solution Solution

Handle Vertices with Large Out-Degree

• First, sample O(√n ) edges randomly from

polyhedron (only global sampling possible!)

• If out-degree larger than O(√n ), then one can

expect to find some of those edges

• Hence: Start linear search at edges closest

to plane (c.f. successor search)

• Otherwise (small out-degree): All edges can be

searched linearly Handle Vertices with Large Out-Degree

• First, sample O(√n ) edges randomly from

polyhedron (only global sampling possible!)

• If out-degree larger than O(√n ), then one can

expect to find some of those edges

• Hence: Start linear search at edges closest

to plane (c.f. successor search)

• Otherwise (small out-degree): All edges can be

searched linearly

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Complexity Complexity

Complexity

• The cut-off caps are O(√n )
• Intuition: Similar to successor searching
• Formal proof: See paper
• All other steps are also O(√n )
• Yields O(√n ) overall complexity
• The problem is a generalization of successor
• searching. Hence: Optimal complexity.

Complexity

• The cut-off caps are O(√n )
• Intuition: Similar to successor searching
• Formal proof: See paper
• All other steps are also O(√n )
• Yields O(√n ) overall complexity
• The problem is a generalization of successor
• searching. Hence: Optimal complexity.

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Ray Shooting & Applications Ray Shooting & Applications

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Ray Shooting Ray Shooting

Same Technique Can Handle Ray Shooting:

• Rays are very skinny polyhedra
• Can use the same algorithm
• Computes intersections in O(√n ) time
• This is also optimal

Same Technique Can Handle Ray Shooting:

• Rays are very skinny polyhedra
• Can use the same algorithm
• Computes intersections in O(√n ) time
• This is also optimal

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Applications Applications

Point Location

• Point location in Delaunay triangulations
• Shoot vertical ray towards

convex hull of (x, y, x2 + y2)

• Similar construction also works for Voronoi

Diagrams

• Yields point location algorithms with optimal

O(√n ) complexity Point Location

• Point location in Delaunay triangulations
• Shoot vertical ray towards

convex hull of (x, y, x2 + y2)

• Similar construction also works for Voronoi

Diagrams

• Yields point location algorithms with optimal

O(√n ) complexity

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Other Applications Other Applications

Nearest Neighbor Search:

• Find nearest neighbor in polyhedron to a

given point in space in O(√n ) time

• Uses (slightly) modified polyhedron

intersection algorithm Nearest Neighbor Search:

• Find nearest neighbor in polyhedron to a

given point in space in O(√n ) time

• Uses (slightly) modified polyhedron

intersection algorithm

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Other Applications (2) Other Applications (2)

Direction Search

• Find point with maximum

projection on a line

• Find point with maximum

projection on a line restricting input polyhedron to a plane

• Both in O(√n )

Direction Search

• Find point with maximum

projection on a line

• Find point with maximum

projection on a line restricting input polyhedron to a plane

• Both in O(√n )

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Volume Approximation Volume Approximation

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Volume Approximation Volume Approximation

The Task:

• Determine volume of polyhedron
• With accuracy up to factor ε

Main Idea:

• Dudley approximation
• Non-uniform rescaling to be able

to guarantee ε-precision The Task:

• Determine volume of polyhedron
• With accuracy up to factor ε

Main Idea:

• Dudley approximation
• Non-uniform rescaling to be able

to guarantee ε-precision

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Dudley Approximation Dudley Approximation

Dudley Approximation of Convex Polyhedra

• Given a convex

polygon /polyhedron enclosed in a unit sphere

• An approximation of

complexity O(1/ε (d-1)/2) with max. Hausdorff-distance ε can be constructed Dudley Approximation of Convex Polyhedra

• Given a convex

polygon /polyhedron enclosed in a unit sphere

• An approximation of

complexity O(1/ε (d-1)/2) with max. Hausdorff-distance ε can be constructed

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Dudley Approximation (2) Dudley Approximation (2)

Dudley Approximation of Convex Polyhedra Dudley Approximation of Convex Polyhedra O(1/ε (d-1)/2) uniformely distributed sample points O(1/ε (d-1)/2) uniformely distributed sample points

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Dudley Approximation (3) Dudley Approximation (3)

Dudley Approximation of Convex Polyhedra Dudley Approximation of Convex Polyhedra find nearest neighbor of each point find nearest neighbor of each point

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Dudley Approximation (4) Dudley Approximation (4)

Dudley Approximation of Convex Polyhedra Dudley Approximation of Convex Polyhedra convex hull of “supporting” planes convex hull of “supporting” planes

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Dudley Approximation (5) Dudley Approximation (5)

Dudley Approximation of Convex Polyhedra Dudley Approximation of Convex Polyhedra convex hull of “supporting” planes convex hull of “supporting” planes

maximum error: ε (with respect to unit sphere)

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Volume Approximation Volume Approximation

Volume Approximation using Dudley Construction

• Observation: All steps in Dudley Construction

can be peformed in O(√n ) time

• Remaining problem: Poor approximation

for skinny polyhedra Volume Approximation using Dudley Construction

• Observation: All steps in Dudley Construction

can be peformed in O(√n ) time

• Remaining problem: Poor approximation

for skinny polyhedra

bounded absolute error larger relative volume error

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Rescaling Rescaling

Solution: Non-Uniform Rescaling Solution: Non-Uniform Rescaling

1 2 3 4

• riginal polyhedron

main axis transform helper polyhedron

vol.

≥

c· original vol.

enclosed ellipsoid

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Result Result

Rescaling:

• Enclosed ellipsoid has

volume ≥ c·original volume

• Volume can be measured with arbitrary,

constant accuracy by rescaling Overall:

• Sublinear operations only
• Overall: O(ε -1√n ) running time for

ε -approximation of volume Rescaling:

• Enclosed ellipsoid has

volume ≥ c·original volume

• Volume can be measured with arbitrary,

constant accuracy by rescaling Overall:

• Sublinear operations only
• Overall: O(ε -1√n ) running time for

ε -approximation of volume

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Shortest-Path Approximation Shortest-Path Approximation

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• Approx. Shortest Path
• Approx. Shortest Path

The Task:

• Compute shortest path between

two surface points on convex polyhedron

• Approximate up to factor ε
• Allow path to leave surface of polyhedron

(otherwise, any path might be

• f complexity Ω(n))

The Task:

• Compute shortest path between

two surface points on convex polyhedron

• Approximate up to factor ε
• Allow path to leave surface of polyhedron

(otherwise, any path might be

• f complexity Ω(n))

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How does it work? How does it work?

Main Idea:

• Same basic idea: Dudley approximation
• Construct simplified polyhedron
• Constant number of faces

⇒ Can then use standard algorithm to compute shortest path

• Problem: Guaranteeing accuracy (again)
• Solution: Clipping to O(path-length)-Box

Main Idea:

• Same basic idea: Dudley approximation
• Construct simplified polyhedron
• Constant number of faces

⇒ Can then use standard algorithm to compute shortest path

• Problem: Guaranteeing accuracy (again)
• Solution: Clipping to O(path-length)-Box

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Using the Dudley Approx. Using the Dudley Approx.

Dudley Approximation:

• Simplify polyhedron (Dudley)
• Build path along simplified w. std. method

Dudley Approximation:

• Simplify polyhedron (Dudley)
• Build path along simplified w. std. method
• riginal

simplified approximate path

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The Problem The Problem

Remaining Problem: Relative Accuracy

• ε-approx for long paths
• Larger relative error for short paths

Remaining Problem: Relative Accuracy

• ε-approx for long paths
• Larger relative error for short paths

long path – smaller relative error short path – large relative error

ε ε

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Solution Solution

Clipping before Dudleying

• Estimate length of shortest

path up to factor 8

• Place box of side length

16·est around start point

• Clip simplified polyhedron to box (additional

constraints during computation)

• By scaling ε correspondingly, this guarantees

ε-approximation of shortest path Clipping before Dudleying

• Estimate length of shortest

path up to factor 8

• Place box of side length

16·est around start point

• Clip simplified polyhedron to box (additional

constraints during computation)

• By scaling ε correspondingly, this guarantees

ε-approximation of shortest path

new ε

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Result Result

Overall result:

• Can approximate shortest path

up to factor (1+ε)

• Using time O(ε -5/4 √n ) + f(ε -5/4)

where f is time for exact solution

• Known: f(k) ∈ O(k log2 k)
• Authors use modified Dudley construction to

achieve given bounds; see paper for details. Overall result:

• Can approximate shortest path

up to factor (1+ε)

• Using time O(ε -5/4 √n ) + f(ε -5/4)

where f is time for exact solution

• Known: f(k) ∈ O(k log2 k)
• Authors use modified Dudley construction to

achieve given bounds; see paper for details.

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Conclusions Conclusions

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Conclusions Conclusions

Sublinear Algorithms for Geometric Problems

• Convex polygons / polyhedra intersection
• Ray shooting (convex P’s, and similar problems)
• Volume approximation (convex P’s)
• Shortest path approximation (convex P’s)

Properties

• All in O(√n )
• Techniques are generalizations of

the successor searching algorithm Sublinear Algorithms for Geometric Problems

• Convex polygons / polyhedra intersection
• Ray shooting (convex P’s, and similar problems)
• Volume approximation (convex P’s)
• Shortest path approximation (convex P’s)

Properties

• All in O(√n )
• Techniques are generalizations of

the successor searching algorithm