Higher Cell Probe Lower Bounds for Evaluating Polynomials

Abstract

In this paper, we study the cell probe complexity of evaluating an n-degree polynomial P over a finite field F of size at least n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+Ω(1)</sup> . More specifically, we show that any static data structure for evaluating P(x), where x ∈ F, must use Ω(lg |F|/ lg(Sw/n lg |F|)) cell probes to answer a query, where S denotes the space of the data structure in number of cells and w the cell size in bits. This bound holds in expectation for randomized data structures with any constant error probability δ <; 1/2. Our lower bound not only improves over the Ω(lg |F|/ lg S) lower bound of Miltersen [TCS'95], but is in fact the highest static cell probe lower bound to date: For linear space (i.e. S = O(n lg |F|/w)), our query time lower bound simplifies to Ω(lg |F|), whereas the highest previous lower bound for any static data structure problem having d different queries is Ω(lg d/ lg lg d), which was first achieved by Pátrascu and Thorup [SICOMP'10]. We also use the recent technique of Larsen [STOC'12] to show a lower bound of tq = Ω(lg |F| lg n/lg(wtu/ lg |F|) lg(wt <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> )) for dynamic data structures for polynomial evaluation over a finite field F of size Ω(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Here t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> denotes the expected query time and tu the worst case update time. This lower bound holds for randomized data structures with any constant error probability δ <; 1/2. This is only the second time a lower bound beyond max{t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> , t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> } = Ω(max{lg n, lg d/ lg lg d}) has been achieved for dynamic data structures, where d denotes the number of different queries and updates to the problem. Furthermore, it is the first such lower bound that holds for randomized data structures with a constant probability of error.