Time and space complexity of reversible pebbling

Richard Královič

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2004)

  • Volume: 38, Issue: 2, page 137-161
  • ISSN: 0988-3754

Abstract

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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 ( n lg n ) 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.

How to cite

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Královič, Richard. "Time and space complexity of reversible pebbling." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 38.2 (2004): 137-161. <http://eudml.org/doc/245602>.

@article{Královič2004,
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 $\Omega (n\lg n)$ 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 $h+\Theta (\lg ^* h)$ 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.},
author = {Královič, Richard},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {reversible computations; pebbling; quantum computing},
language = {eng},
number = {2},
pages = {137-161},
publisher = {EDP-Sciences},
title = {Time and space complexity of reversible pebbling},
url = {http://eudml.org/doc/245602},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Královič, Richard
TI - Time and space complexity of reversible pebbling
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 2
SP - 137
EP - 161
AB - 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 $\Omega (n\lg n)$ 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 $h+\Theta (\lg ^* h)$ 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.
LA - eng
KW - reversible computations; pebbling; quantum computing
UR - http://eudml.org/doc/245602
ER -

References

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  1. [1] C.H. Bennett, Time-space trade-offs for reversible computation. SIAM J. Comput. 18 (1989) 766-776. Zbl0676.68010MR1004797
  2. [2] H. Buhrman, J. Tromp and P. Vitányi, Time and space bounds for reversible simulation, in Proc. ICALP 2001. Springer-Verlag, Lect. Notes Comput. Sci. 2076 (2001). Zbl0986.68512MR1863117
  3. [3] R.Y. Levine and A.T. Sherman, A note on Bennett’s time-space tradeoff for reversible computation. SIAM J. Comput. 19 (1990) 673-677. Zbl0697.68043
  4. [4] M. Li, J. Tromp and P.M.B. Vitányi, Reversible simulation of irreversible computation. Physica D 120 (1998) 168-176. 
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  6. [6] M. Li and P.M.B. Vitányi, Reversible simulation of irreversible computation, in Proc. 11th IEEE Conf. Computational Complexity, Philadelphia, Pennsylvania, May 24-27 (1996). 
  7. [7] M.S. Paterson and C.E. Hewitt, Comparative Schematology, in MAC Conf. on Concurrent Systems and Parallel Computation (1970) 119-127. 
  8. [8] P. Ružička, Pebbling – The Technique for Analysing Computation Efficiency. SOFSEM’89 (1989) 205-224. 
  9. [9] P. Ružička and J. Waczulík, Pebbling Dynamic Graphs in Minimal Space. RAIRO-Inf. Theor. Appl. 28 (1994) 557-565. Zbl0884.68096MR1305116
  10. [10] P. Ružička and J. Waczulík, On Time-Space Trade-Offs in Dynamic Graph Pebbling. MFCS’93 711 (1993) 671-681. Zbl0925.68153
  11. [11] R. Williams, Space-Efficient Reversible Simulations. DIMACS REU report (July 2000). 
  12. [12] A. Zavarský, On the Cost of Reversible Computations: Time-Space Bounds on Reversible Pebbling. Manuscript (1998). 

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