# SEARCH

# ISTC-CC NEWSLETTER

# RESEARCH HIGHLIGHTS

Ling Liu's SC13 paper "Large Graph Processing Without the Overhead" featured by HPCwire.

ISTC-CC provides a listing of useful benchmarks for cloud computing.

Another list highlighting Open Source Software Releases.

Second GraphLab workshop should be even bigger than the first! GraphLab is a new programming framework for graph-style data analytics.

# ISTC-CC Abstract

## Nearly-Linear Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs

* Theory of Computer Systems, Volume 55, Issue 3, October 2014.*

**Guy E. Blelloch*, Anupam Gupta*, Ioannis Koutis^, Gary L. Miller*, Richard Peng*,
Kanat Tangwongsan**

* Carnegie Mellon University

^ University of Puerto Rico, Rio Piedras, Puerto Rico

We present the design and analysis of a nearly-linear work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input an SDD *n*-by-*n* matrix *A* with *m* nonzero entries and a vector *b*, our algorithm computes a vector ˜*x* such that ∥˜*x* − *A* + *b* ∥_{A} ≤ ε · ∥*A* + *b*∥_{A} in *O*(*m* log^{O(1)} *n* log 1/ε ) work and *
O*(

*m*

^{1/3+θ}log 1/ε ) depth for any θ > 0, where

*A*

^{+}denotes the Moore-Penrose pseudoinverse of

*A*. The algorithm relies on a parallel algorithm for generating low-stretch spanning trees or spanning subgraphs. To this end, we first develop a parallel decomposition algorithm that in

*O*(

*m*log

^{O(1)}

*n*) work and polylogarithmic depth, partitions a graph with

*n*nodes and

*m*edges into components with polylogarithmic diameter such that only a small fraction of the original edges are between the components. This can be used to generate low-stretch spanning trees with average stretch

*O*(

*n*

^{α}) in

*O*(

*m*log

^{O(1)}

*n*) work and

*O*(

*n*

^{α}) depth for any

*α*>0. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in O(mlog

^{O(1)}n) work and polylogarithmic depth. We apply this subgraph construction to derive a parallel linear solver.

By using this solver in known applications, our results imply improved parallel randomized algorithms for several problems, including single-source shortest paths, maximum flow, minimum-cost flow, and approximate maximum flow.

**FULL PAPER: pdf**