Abstract
Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low independence. A series of papers, beginning with work by Luby (1988), showed that in many cases these techniques can be combined to give deterministic parallel (NC) algorithms for a variety of combinatorial optimization problems, with low time- and processor-complexity. We extend and generalize a technique of Luby for efficiently handling bilinear objective functions. One noteworthy application is an NC algorithm for maximal independent set. On a graph G with m edges and n vertices, this takes \({\tilde{O}}(\log ^2 n)\) time and \((m + n) n^{o(1)}\) processors, nearly matching the best randomized parallel algorithms. Other applications include reduced processor counts for algorithms of Berger (SIAM J Comput 26:1188–1207, 1997) for maximum acyclic subgraph and Gale–Berlekamp switching games. This bilinear factorization also gives better algorithms for problems involving discrepancy. An important application of this is to automata-fooling probability spaces, which are the basis of a notable derandomization technique of Sivakumar (In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp 619–626, 2002). Our method leads to large reduction in processor complexity for a number of derandomization algorithms based on automata-fooling, including set discrepancy and the Johnson–Lindenstrauss Lemma.
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Achlioptas, D.: Database friendly random projections. In: Proceedings of the 20th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS), pp. 274–281 (2001)
Awerbuch, B., Berger, B., Cowen, L., Peleg, D.: Low-diameter graph decomposition is in NC. Random Struct. Algorithms 5–3, 441–452 (1994)
Bellare, M., Rompel, J.: Randomness-efficient oblivious sampling. In: Proceedings of the 35th Annual Foundations of Computer Science (FOCS), pp. 276–287 (1994)
Berger, B.: The fourth moment method. SIAM J. Comput. 26, 1188–1207 (1997)
Berger, B., Rompel, J., Shor, P.: Efficient NC algorithms for set cover with applications to learning and geometry. J. Comput. Syst. Sci. 49, 454–477 (1994)
Berger, B., Rompel, J.: Simulating \((\log ^c n)\)-wise independence in NC. J. ACM 38–4, 1026–1046 (1991)
Brown, T., Spencer, J.: Minimization of \(\pm 1\) matrices under line shifts. Colloq. Math. (Poland) 23, 165–171 (1971)
Chari, S., Rohatgi, P., Srinivasan, A.: Improved algorithms via approximations of probability distributions. J. Comput. Syst. Sci. 61–1, 81–107 (2000)
Dadush, D., Guzmán, C., Olver, N.: Fast, deterministic and sparse dimensionality reduction. In: Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1330–1344 (2018)
Engebretsen, L., Indyk, P., O’Donnell, R.: Derandomized dimensionality reduction with applications. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 705–712 (2002)
Han, Y.: A fast derandomization scheme and its applications. SIAM J. Comput. 25–1, 52–82 (1996)
Harris, D.: Deterministic parallel algorithms for fooling polylogarithmic juntas and the Lovász Local Lemma. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1188–1203 (2017)
Johnson, W. B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability, pp. 189–206 (1984)
Kane, D., Nelson, J.: A derandomized sparse Johnson–Lindenstrauss transform. arXiv preprint arXiv:1006.3585. (2010)
Luby, M.: Removing randomness in parallel computation without a processor penalty. J. Comput. Syst. Sci. 47–2, 250–286 (1993)
Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15–4, 1036–1053 (1996)
Rajeev, M., Naor, J., Naor, M.: The probabilistic method yields deterministic parallel algorithms. J. Comput. Syst. Sci. 49–3, 478–516 (1994)
Mahajan, S., Ramos, E., Subrahmanyan, K.: Solving some discrepancy problems in NC. In: Foundations of Software Technology and Theoretical Computer Science, pp. 22–36 (1997)
Naor, J., Naor, M.: Small-bias probability spaces: efficient construction and applications. SIAM J. Comput. 22–4, 835–856 (1993)
Nisan, N.: Pseudorandom generator for space-bounded computation. Combinatorica 12–4, 449–461 (1992)
Nisan, N.: \(\text{ RL } \subseteq \text{ SC }\). Comput. Complex. 4–1, 1–11 (1994)
Sivakumar, D.: Algorithmic derandomization via complexity theory. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp. 619–626 (2002)
Spencer, J.: Ten Lectures on the Probabilistic Method. SIAM, Philadelphia (1987)
Acknowledgements
Thanks to Aravind Srinivasan, for extensive comments and discussion. Thanks to anonymous journal referees for careful review and a number of helpful helpful suggestions.
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Research supported in part by NSF Awards CNS-1010789 and CCF-1422569.
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Harris, D.G. Deterministic Parallel Algorithms for Bilinear Objective Functions. Algorithmica 81, 1288–1318 (2019). https://doi.org/10.1007/s00453-018-0471-0
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DOI: https://doi.org/10.1007/s00453-018-0471-0