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Derandomization with Minimal Memory Footprint

Authors Dean Doron , Roei Tell

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Author Details

Dean Doron
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Roei Tell
  • The Institute for Advanced Study at Princeton, NJ, USA
  • DIMACS Center at Rutgers University, Piscataway, NJ, USA

Funding

The second author is supported in part by the National Science Foundation under grant number CCF-1900460.

Acknowledgements

We are grateful to Avi Wigderson for several useful conversations regarding the gap between double space blow-up and single space blow-up. We thank Lijie Chen for suggesting the idea of using catalytic space to save on complexity, early in this work, and for pointing out a gap in a previous version of the proof of Theorem 2. We also thank Ian Mertz for several useful conversations exploring the abilities of catalytic space. We are very grateful to an anonymous reviewer for a careful read of this paper and for many useful comments. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing.

Cite AsGet BibTex

Dean Doron and Roei Tell. Derandomization with Minimal Memory Footprint. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 11:1-11:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.11

Abstract

Existing proofs that deduce BPL = 𝐋 from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ⊆ DSPACE[c ⋅ S] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC⁰, that were not known before.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Error-correcting codes
Keywords
  • derandomization
  • space-bounded computation
  • catalytic space

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