Sorting with Asymmetric Read and Write Costs

TL;DR

This paper develops models and algorithms for sorting that minimize costly memory writes, addressing emerging memory technologies with asymmetric read/write costs, and presents efficient algorithms in various computational models.

Contribution

It introduces new algorithms and models that optimize sorting by reducing write operations in asymmetric memory and cache models.

Findings

Sorting can be performed with $O(n)$ writes and $O(n ext{log} n)$ reads in the asymmetric PRAM model.

Variants of external memory sorting algorithms significantly reduce write operations.

Parallel algorithms for sorting, FFTs, and matrix multiplication are optimized for asymmetric write costs.

Abstract

Emerging memory technologies have a significant gap between the cost, both in time and in energy, of writing to memory versus reading from memory. In this paper we present models and algorithms that account for this difference, with a focus on write-efficient sorting algorithms. First, we consider the PRAM model with asymmetric write cost, and show that sorting can be performed in $O\left(n\right)$ writes, $O\left(n \log n\right)$ reads, and logarithmic depth (parallel time). Next, we consider a variant of the External Memory (EM) model that charges $\omega > 1$ for writing a block of size $B$ to the secondary memory, and present variants of three EM sorting algorithms (multi-way mergesort, sample sort, and heapsort using buffer trees) that asymptotically reduce the number of writes over the original algorithms, and perform roughly $\omega$ block reads for every block write. Finally, we define a variant of the Ideal-Cache model with asymmetric write costs, and present write-efficient, cache-oblivious parallel algorithms for sorting, FFTs, and matrix multiplication. Adapting prior bounds for work-stealing and parallel-depth-first schedulers to the asymmetric setting, these yield parallel cache complexity bounds for machines with private caches or with a shared cache, respectively.