E0 244: Computational Geometry and Topology

Lecture 0: OverviewJanuary 3, 2018

What?

» Design and analysis of algorithms with focus on geometric and topological problems

Sathish and Vijay E0 244 : Computational Geometry and Topology 1

Collections of points/disks

Voronoi / Power Diagram

Delaunay Triangulation

Why?

» Delaunay Triangulation

Sathish and Vijay E0 244 : Computational Geometry and Topology 2

www.ics.uci.edu

www.inria.fr

eecs.berkeley.edu

» Meshing (CAD, CAM, CFD)» Graphics» GIS» Molecular modeling

Why?

» Computer graphics, CAD, CAM

» Computer vision» GIS» Robotics» Scientific visualization» Structural biology» TDA

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math.upenn.edu

VGL, IISc

cvg.

ethz

.ch

Blend of Geometry and Topology Topics

» Classical Computational Geometry§ Delaunay triangulation§ Voronoi diagram

» Computational Topology§ Complexes§ Homology§ Persistence

» Contemporary geometry§ Geometric approximation algorithms§ Hitting sets, packing, covering

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Blend of Mathematical and Algorithmic Topics

» Epsilon nets» Intersection graphs» Algebraic topology» Geometry in 2D, 3D

» Incremental algorithms» Linear programming» Geometric optimization» Local search» Algorithms for

topological invariants» Matrix reduction

algorithms

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Administrivia» Instructors

§ Sathish Govindarajan and Vijay Natarajan§ E-mail: {gsat,vijayn}@iisc.ac.in§ Subject: “[E0 244] : …”

» Time and place§ MW 02:00 – 03:30, CSA 252

» Web page§ http://www.csa.iisc.ac.in/~gsat/Course/CGT§ https://canvas.instructure.com

» Online registration§ http://acadserver.admin.iisc.ac.in/course/§ Canvas invitation after your register (credit / audit)

Sathish and Vijay 6E0 244 : Computational Geometry and Topology

Prerequisites

» Fundamental data structures and algorithms» E0 225 : Design and Analysis of Algorithms

Sathish and Vijay 7E0 244 : Computational Geometry and Topology

Course material» Recommended books

§ Computational Geometry: Algorithms and Applications, Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars, Third Edition, Springer (SIE), 2011.

§ Computational Topology : An Introduction, Herbert Edelsbrunner and John L. Harer, American Mathematical Society, Indian Edition, 2010.

§ Recent literature

» Additional Material§ Research articles§ Notes§ Lecture slides

Sathish and Vijay 8E0 244 : Computational Geometry and Topology

Lecture Plan : I

0. Convex Hull (Today)1. Convex Hull, Voronoi Diagram, and

Delaunay Triangulation2. Delaunay Triangulation - Properties3. Geometric Data Structures4. Delaunay Triangulation - Algorithm

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Lecture Plan : II

5. Geometric covering, hitting, packing problems

6. TDA : Topological Data Analysis

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Lecture Plan : III7. Complexes : Simplicial, Cell, Vietoris-Rips, Cech8. Complexes : Applications9. Homology : Groups, k-cycles10. Homology : Betti numbers, algorithm11. Homology : Matrix reduction, applications12. Persistent Homology : Filtration, birth-death13. Persistent Homology : Algorithm14. Persistent Homology : Application15. Reeb graph : Definition, contour tree, algorithm16. Reeb graph : ApplicationsSathish and Vijay E0 244 : Computational Geometry and Topology 11

Lecture Plan : IV

Geometric covering, hitting, packing[17-20] Linear programming based techniques[21-23] Local search based techniques

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Evaluation» Assignments (35%)

§ 3 assignments§ Electronic submission (LaTeX)

» Midterm exam (35%)§ Week of March 5

» Final project or Final exam (30%)§ Decide before Feb 15

» Collaboration§ Learn and understand concepts§ Have lively discussions§ BUT … § All work you hand in must be your own work§ Penalty: ZERO in the corresponding component and

grade reduction

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