Lower bounds for predecessor searching in the cell probe model ✩

We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form “What is the predecessor of x in S? ” can be answered efficiently. We study this problem in the cell probe model introduced by Yao [A.C. -C. Yao, Should tables be sorted, J. Assoc. Comput. Mach. 28 (3) (1981) 615–628]. Recently, Beame and Fich [P. Beame, F. Fich, Optimal bounds for the predecessor problem and related problems, J. Comput. System Sci. 65 (1) (2002) 38–72] obtained optimal bounds as functions of either m or n only on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only nO(1) cells of word size (logm)O(1) bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich’s proof works for deterministic query schemes only. In addition, it is simpler than Beame and Fich’s proof. In fact, our lower bound for predecessor searching extends to the ‘quantum address-only’ query schemes that we define in this paper. In these query schemes, quantum parallelism is allowed only over the ‘address lines’ of the queries. These query schemes subsume classical randomised query schemes, and include many quantum query algorithms like Grover’s algorithm [L. Grover, A fast quantum mechanical algorithm for database search, in: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 212–219]. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson [P. Bro Miltersen, Noam Nisan, S. Safra, A. Wigderson, On data structures and asymmetric communication complexity, J. Comput. System Sci. 57 (1) (1998) 37–49]. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. Our strong round elimination lemma also extends to quantum communication complexity. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the ‘greater-than’ problem, improving on the tradeoff in [P. Bro Miltersen, Noam Nisan, S. Safra, A. Wigderson, On data ✩ This paper is the journal version of the conference papers [P. Sen, S. Venkatesh, Lower bounds in the quantum cell probe model, in: Proceedings of the 28th International Colloquium on Automata, Languages and Programming, in: Lecture Notes in Comput. Sci., vol. 2076, Springer-Verlag, 2001, pp. 358–369] and [P. Sen, Lower bounds for predecessor searching in the cell probe model, in: Proceedings of the 18th Annual IEEE Conference on Computational Complexity, 2003, pp. 73–83]. * Corresponding author. E-mail addresses: pgdsen@tcs.tifr. res.in (P. Sen), venkat@cs.uvic. ca (S. Venkatesh). 1 Work done while the author was a graduate student at TIFR, Mumbai, India visiting UC Berkeley and DIMACS under a Sarojini Damodaran International Fellowship grant, and a postdoctoral researcher at LRI, Orsay, France supported by the EU 5th framework program QAIP IST-199911234 and by CNRS/STIC 01N80/0502 and 01N80/0607 grants. 2 Present address: School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India. 3 Work done while the author was a postdoctoral researcher at IAS, Princeton, USA supported by NSF grant CCR-9987845 and a joint IASDIMACS post doctoral fellowship. 0022-0000/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcss. 2007.06. 016 P. Sen, S. Venkatesh / Journal of Computer and System Sciences 74 (2008) 364–385 365 structures and asymmetric communication complexity, J. Comput. System Sci. 57 (1) (1998) 37–49]. We believe that our round elimination lemma is of independent interest and should have other applications. © 2007 Elsevier Inc. All rights reserved.