Unidirectional Input/Output Streaming Complexity of Reversal and Sorting

TL;DR

This paper analyzes the complexity of reversing and sorting data streams under various access restrictions, introducing a new measure called expansion and providing tight bounds and algorithms for these operations.

Contribution

It introduces the expansion complexity measure for read-write streams and establishes tight bounds for reversing and sorting streams in different models.

Findings

Reversing streams requires Θ(n/p) memory in read-only/write-only models.

With read-write streams, reversing complexity reduces to Θ(n/p^2).

The paper presents a randomized sorting algorithm with O(1) expansion, O(log n) passes, and O(log n) memory.

Abstract

We consider unidirectional data streams with restricted access, such as read-only and write-only streams. For read-write streams, we also introduce a new complexity measure called expansion, the ratio between the space used on the stream and the input size. We give tight bounds for the complexity of reversing a stream of length $n$ in several of the possible models. In the read-only and write-only model, we show that $p$-pass algorithms need memory space ${\Theta}(n/p)$. But if either the output stream or the input stream is read-write, then the complexity falls to ${\Theta}(n/p^2)$. It becomes $polylog(n)$ if $p = O(log n)$ and both streams are read-write. We also study the complexity of sorting a stream and give two algorithms with small expansion. Our main sorting algorithm is randomized and has $O(1)$ expansion, $O(log n)$ passes and $O(log n)$ memory.