Higher Cell Probe Lower Bounds for Evaluating Polynomials

Abstract

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 1+Ω(1) . 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 Pǎtraşcu 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(wtu)) for dynamic data structures for polynomial evaluation over a finite field F of size Ω(n 2 ). Here tq 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{tu, tq} = 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.

Near-Optimal Range Reporting Structures for Categorical Data

“…The main motivating factor for studying the pointer machine is that we can prove polynomially high and often tight lower bounds for range reporting problems using the framework of Chazelle [11]. This stands in sharp contrast to the highest query time lower bound proved for any static data structure problem in the word-RAM, a mere Ω(lg N ), even for linear space data structures [18]. Note that since any memory cell stores a constant number of pointers, one cannot generally hope for a query time below Θ(lg N ).…”

Time Lower Bounds for Nonadaptive Turnstile Streaming Algorithms

“…More specifically, we use ideas from the technique now formally known as cell sampling [18,38,32]. This technique derives lower bounds based on one key observation: if a data structure/streaming algorithm probes t memory words on an update, then there is a set C of t memory words such that at least m/S t updates probe only memory words in C, where m is the number of distinct updates in the problem (for data structure lower bound proofs, we typically consider queries rather than updates, and we obtain tighter lower bounds by forcing C to have near-linear size).…”

From Independence to Expansion and Back Again

“…The problem with this construction is that the Vandermonde matrix is dense, resulting in an evaluation time of Ω(k) if we simply store the coefficients of the polynomial. The lower bounds by Siegel [22], and later Larsen [17], as presented in Table 1, show that a data structure for evaluating a polynomial of degree k − 1 using time t < k must use space at least k(u/k) 1/t . The data structure of [15] presents a step in this direction, but is still far from the lower bound for k-independent functions.…”