Implicit Decomposition for Write-Efficient Connectivity Algorithms

TL;DR

This paper introduces new algorithms for graph connectivity that significantly reduce write operations, leveraging an implicit graph decomposition to improve efficiency in memory systems with high write costs.

Contribution

It presents the first algorithms with sublinear write complexity for graph connectivity, utilizing an implicit decomposition approach for both sequential and parallel processing.

Findings

Achieves sublinear write complexity in graph algorithms.

Provides parallel algorithms with o(m) writes for general graphs.

Demonstrates practical efficiency in systems with read-write asymmetry.

Abstract

The future of main memory appears to lie in the direction of new technologies that provide strong capacity-to-performance ratios, but have write operations that are much more expensive than reads in terms of latency, bandwidth, and energy. Motivated by this trend, we propose sequential and parallel algorithms to solve graph connectivity problems using significantly fewer writes than conventional algorithms. Our primary algorithmic tool is the construction of an $o(n)$-sized "implicit decomposition" of a bounded-degree graph $G$ on $n$ nodes, which combined with read-only access to $G$ enables fast answers to connectivity and biconnectivity queries on $G$. The construction breaks the linear-write "barrier", resulting in costs that are asymptotically lower than conventional algorithms while adding only a modest cost to querying time. For general non-sparse graphs on $m$ edges, we also provide the first $o(m)$ writes and $O(m)$ operations parallel algorithms for connectivity and biconnectivity. These algorithms provide insight into how applications can efficiently process computations on large graphs in systems with read-write asymmetry.