Algorithm Engineering (aka. How to Write Fast Code) CS26 S260 - - PowerPoint PPT Presentation

Algorithm Engineering

(aka. How to Write Fast Code)

An Overview of Geometry Processing

CS26 S260 – Lecture cture 12 Yan n Gu

What is geometry processing?

• Gr

Graph h studi dies es the rela latio ionsh nship ip of objects ts

• Ge

Geometry try studies dies the lo locatio ions ns of the objects ts themse mselv lves es

Lots of real-world applications

3

4

Database / Data warehouses Data mining / Data science Machine learning / Artificial intelligence

Every area requires geometry processing

Computational biology Computer graphics Geometric Information Systems (GIS)

Computational Geometry

• Started in mid 70’s
• Focused

sed on a abst strac actin ting g the ge geometri tric proble lems, ms, and design ign and analy lysis is of alg lgorit ithms hms for these e proble lems ms

• Many

y proble lems ms well ll-sol solved ved, , whil ile e many y other proble lems ms remain in open

• UC

UCR CS R CS 133 133 - Co Computa utationa tional l Ge Geometry try

• MIT 6.

6.838 - Ge Geomet etric ic Computat utation ion

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Good references for sequential geometric algorithms

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“Four Dutchmen” “The elephant book” Har-Peled’s book

Algorithm Engineering

Theory Practice

O(n log n) O(n) O(log n)

• Theory

y is is no no lo longe ger suffic fficie ient nt to g guarant antee ee go good theoretic tical al perfo forma mance nce, , becaus ause e comput uter er archite itectu cture e becomes mes sig ignific ificantl antly y more sophi histic sticate ated

• Parall

llel elism, ism, I/O ef effici ficienc ency, y, new hardw dware re such h as non-vo vola latil tile e memori

• ries,

es, and spec ecific ific applic licat ation ions

Convex Hull, Triangulation, and others

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Convex hull

• A set is convex if every line segment connecting two points in

the set is fully contained in the set

• Example applications: nearest/farthest point to a line, point

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How to construct a convex hull

• Dozens of algorithms sequentially, a few parallel ones for 2D
• Randomized incremental construction
• Only requires 𝑷(log 𝒐) rounds if adding all possible points

[BGSS SPAA 2020]

10 1 2 4 3 11 7 5 9 10 8 6

What’s triangulation for?

What’s a good triangulation?

• No points are in the circumcircle of each triangle
• Getting practical parallel algorithms for DT is notoriously hard

Delaunay triangulation (DT)

Delaunay triangulation (DT)

• Again, consider the incremental construction

• Again, consider the incremental construction

Delaunay triangulation (DT)

• Again, consider the incremental construction
• 2D parallel version is given in [BGSS SPAA 2016 / JACM 2020]
• Higher-D version is in [BGSS SPAA 2020], since 𝒍-D DT can be

subsumed by (𝒍 + 𝟐)-D convex hull

Delaunay triangulation (DT)

Point location

• In a 2D plain, decide which polygon a query point belongs to
• Example solution: trapezoidal decomposition (no parallel solution)
• Other options: Delaunay refinement

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Range Searches

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Example range searches

• Nearest neighbor search
• 𝒍-nearest neighbor (𝒍NN)

search

• Near neighbor counting
• Rectangular range search
• Rectangle queries
• Three-sided queries
• Segment (stabbing)

queries

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• Ray-scene intersection

queries

• Collision detection
• etc.

Building blocks in other algorithms / systems

Classic (sequential) data structures

• Quad/octree
• 𝒍-d tree
• Interval tree
• Segment tree
• Range tree
• Priority tree
• Other augmented trees
• R-tree, bounding volume hierarchy (BVH)

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Why trees?

21 1 2 3 A B C D E F

3 2 1

A B E F D C

Each interior tree node acts as a fast-pass check for the subtree nodes

• No need to traverse the subtree if the traversal can be pruned

22 1 2 3 C A B E F D A B C D E F

3 2 1

1 2 C 3

NNS

Each interior tree node acts as a fast-pass check for the subtree nodes

• No need to traverse the subtree if the query misses the

bounding box

23 1 2 3 C A B E F D A B C D E F

3 2 1

1 2 A B C 3

Example range searches

• Nearest neighbor search
• 𝒍-nearest neighbor (𝒍NN)

search

• Near neighbor counting
• Rectangular range search
• Rectangle queries
• Three-sided queries
• Segment (stabbing)

queries

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• Ray-scene intersection

queries

• Collision detection
• etc.

Classic (sequential) data structures

• Quad/octree
• 𝒍-d tree
• Interval tree
• Segment tree
• Range tree
• Priority tree
• Other augmented trees
• R-tree, bounding volume hierarchy (BVH)

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“Modern” Geometry Problems

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Problems that are less “classic” (from MIT 6.838)

• Clustering
• Range searches in “high” (>5) dimensions
• Low-distortion embeddings
• Geometric algorithms for modern architecture
• Geometric algorithms for streaming data

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Monday lecture (5/18)

• Efficient BVH Construction via Approximate Agglomerative

Clustering, by Lei Zhang

• Parallel Range, Segment and Rectangle Queries with

Augmented Maps, by Longze Su

• Theoretically-Efficient and Practical Parallel DBSCAN,

by Jinyang Liu

• Engineering a compact parallel Delaunay algorithm in 3D,

by Haide He

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Wednesday lecture (5/20)

Friday lecture (5/22): final project milestone

• Every of you will give a 5-minute talk about the current

progress of your final project (with slides)

• Reserve your time slot here:

https://docs.google.com/spreadsheets/d/1De2HOpzUewLK6l aMJqnxF5bahzSBu1fJUR_hSMM9qt8/edit?usp=sharing

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Paper reading is due this Friday (5/15)