1. Parallel Algorithm Design using Spectral Graph Theory
Gary L Miller 1

2. Spectral Graph Theory Use linear algebra to solve graph problems
Use graph theory to solve linear algebra problems
We start with 1.

3. Talking Point
We need to develop fast parallel primitives that have many applications
An important primitive: linear system solvers.

4. oldest Computational Problem

5. Image Denoising Image Segmentation
Given image + noise, recover image.
Space X Moon Launch

6. Convex Programs as a higher Primitive
Linear programs
Maximum Flow in a graph.
Minimum cost maximum flow
Single source shortest path
General Convex Programs
Image denoising
Machine Learning

7. Interior point methods for Convex Programs
Karmarker, Renagar, Ye, Nesterov, Nemirovsky, Boyd, Vanderburghe, INPUT: Convex program with m constraints RUNTIME: O(√ m) linear system solves.
One of the major breakthroughs of the 20th century.
How fast is the solver?

8. Special Linear Systems
A is Symmetric Diagonally Dominant (SDD)
Symmetric.
Diagonal entry at least sum of absolute values of off diagonals.

9. Problems whose interior point pivot is a SDD system
Linear programs
Maximum Flow in a graph.
Minimum cost maximum flow
Single source shortest path
General Convex Programs
Image denoising
Several Machine Learning problems

10. Fundamental Problem: Solving Linear Systems
n-by-n
m non-zero entries
Given matrix A, vector b
Find vector x such that Ax=b
Maybe draw matrix
Ok with almost exact solutions

11. Fast SDD solvers Input: n by n SDD matrix A with m non-zeros vector b
Output: Approximate solution x to Ax=b.
Runtime: O(m log n log(1/ε))
Parallel solver, O(m1/3) depth and nearly-linear work

12. Theoretical Applications of SDD Solvers: Multipule Iterations
Learning on graphical models. Planar graph embeddings. Finite Element PDEs Generating random spanning Maximum flow

13. The Graph algorithms in the solver
Low Stretch Spanning Trees

14. A better tree
Recursive ‘C’ Construction

15. Future directions Going from theory to practice.
Fundamental algorithm design missing
More applications
Work-efficient parallelizations?
Fast solvers for Symmetric Positive Definite systems.

16. Thank You! 16