Lower bounds for predecessor searching in the cell probe model
Pranab Sen, S. Venkatesh
TL;DR
The work proves optimal lower bounds for static predecessor searching in Yao's cell-probe model, showing that Beame and Fich's bounds extend to randomized and address-only quantum query schemes. By developing a stronger round-elimination lemma grounded in information theory (via average-encoding) and introducing safe address-only quantum protocols, the authors obtain tight t-probe lower bounds that match prior upper bounds. The results also yield improved rounds-vs.-communication tradeoffs for the greater-than problem and demonstrate the broad applicability of the round-elimination technique to other problems like rank parity and GT. Overall, the paper strengthens the bridge between data-structure lower bounds and quantum/classical communication complexity, providing a versatile toolkit for future hardness results.
Abstract
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. Recently, Beame and Fich obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n^{O(1)} cells of word size (\log m)^{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. It also extends to `quantum address-only' query schemes that we define in this paper, and is simpler than Beame and Fich's proof. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson. 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 Miltersen et al. We believe that our round elimination lemma is of independent interest and should have other applications.