Parallel Algorithm Design using Spectral Graph Theory Gary L Miller 1

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

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

oldest Computational Problem

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

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

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?

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

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

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

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

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

The Graph algorithms in the solver Low Stretch Spanning Trees

A better tree Recursive ‘C’ Construction

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

Thank You! 16