Time and space complexity of reversible pebbling
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 2, pp. 137-161.

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.

DOI : https://doi.org/10.1051/ita:2004008
Classification : 68Q10,  68Q25
Mots clés : reversible computations, pebbling
@article{ITA_2004__38_2_137_0,
     author = {Kr\'alovi\v c, Richard},
     title = {Time and space complexity of reversible pebbling},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {137--161},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {2},
     year = {2004},
     doi = {10.1051/ita:2004008},
     zbl = {1082.68031},
     mrnumber = {2060774},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2004__38_2_137_0/}
}
Královič, Richard. Time and space complexity of reversible pebbling. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 2, pp. 137-161. doi : 10.1051/ita:2004008. http://www.numdam.org/item/ITA_2004__38_2_137_0/

[1] C.H. Bennett, Time-space trade-offs for reversible computation. SIAM J. Comput. 18 (1989) 766-776. | MR 1004797 | Zbl 0676.68010

[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). | MR 1863117 | Zbl 0986.68512

[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. | Zbl 0697.68043

[4] M. Li, J. Tromp and P.M.B. Vitányi, Reversible simulation of irreversible computation. Physica D 120 (1998) 168-176.

[5] M. Li and P.M.B. Vitányi, Reversibility and adiabatic computation: trading time and space for energy. Proc. Roy. Soc. Lond. Ser. A 452 (1996) 1-21. | MR 1383290 | Zbl 0869.68019

[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] M.S. Paterson and C.E. Hewitt, Comparative Schematology, in MAC Conf. on Concurrent Systems and Parallel Computation (1970) 119-127.

[8] P. Ružička, Pebbling - The Technique for Analysing Computation Efficiency. SOFSEM'89 (1989) 205-224.

[9] P. Ružička and J. Waczulík, Pebbling Dynamic Graphs in Minimal Space. RAIRO-Inf. Theor. Appl. 28 (1994) 557-565. | Numdam | MR 1305116 | Zbl 0884.68096

[10] P. Ružička and J. Waczulík, On Time-Space Trade-Offs in Dynamic Graph Pebbling. MFCS'93 711 (1993) 671-681. | Zbl 0925.68153

[11] R. Williams, Space-Efficient Reversible Simulations. DIMACS REU report (July 2000).

[12] A. Zavarský, On the Cost of Reversible Computations: Time-Space Bounds on Reversible Pebbling. Manuscript (1998).