Time and space complexity of reversible pebbling

Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina, 842 48 Bratislava, Slovakia; rkralovi@dcs.fmph.uniba.sk.

Received: 5 November 2002
Accepted: 16 February 2004

Abstract

This paper investigates one possible model of reversible computations, an important paradigm in the context of quantum computing. Introduced by Bennett, a reversible pebble game is an abstraction of reversible computation that allows to examine the space and time complexity of various classes of problems. We present a technique for proving lower and upper bounds on time and space complexity for several types of graphs. Using this technique we show that the time needed to achieve optimal space for chain topology is Ω(nlgn) for infinitely many n and we discuss time-space trade-offs for chain. Further we show a tight optimal space bound for the binary tree of height h of the form h + Θ(lg*h) and discuss space complexity for the butterfly. These results give an evidence that reversible computations need more resources than standard computations. We also show an upper bound on time and space complexity of the reversible pebble game based on the time and space complexity of the standard pebble game, regardless of the topology of the graph.