Unifying the Landscape of Super-Logarithmic Dynamic Cell-Probe Lower Bounds
Young Kun Ko
TL;DR
This work presents a unified framework to translate one-way communication lower bounds under product distributions into super-logarithmic dynamic cell-probe lower bounds for a broad class of generalized Multiphase Problems. Central to the approach is a two-stage strategy: (i) a one-way simulation that converts a dynamic data-structure algorithm into a short, high-advantage communication protocol, and (ii) an information-theoretic impossibility argument that rules out such short protocols for hard functions f, using average min-entropy and discrepancy tools. By harnessing chronogram/epoch techniques alongside min-entropy analyses, the authors derive the first super-logarithmic lower bounds for the Multiphase Problem with Inner Product, and they provide a versatile lifting theorem and several hard-function constructions (including 0-XOR variants) that broaden the applicability of the method. The results consolidate and extend prior dynamic lower-bound techniques, offering a principled path to future super-logarithmic bounds in dynamic data structures with a broad spectrum of inner gadgets. Overall, the paper advances the theoretical understanding of dynamic lower bounds and presents a flexible toolkit for establishing harder-than-logarithmic barriers across dynamic data-structure problems.
Abstract
We prove a general translation theorem for converting one-way communication lower bounds over a product distribution to dynamic cell-probe lower bounds. Specifically, we consider a class of problems considered in [Pat10] where: 1. $S_1, \ldots, S_m \in \{0, 1\}^n$ are given and publicly known. 2. $T \in \{0, 1\}^n$ is a sequence of updates, each taking $t_u$ time. 3. For a given $Q \in [m]$, we must output $f(S_Q, T)$ in $t_q$ time. Our main result shows that for a "hard" function $f$, for which it is difficult to obtain a non-trivial advantage over random guessing with one-way communication under some product distribution over $S_Q$ and $T$ (for example, a uniform distribution), then the above explicit dynamic cell-probe problem must have $\max \{ t_u, t_q \} \geq \tildeΩ(\log^{3/2}(n))$ if $m = Ω(n^{0.99})$. This result extends and unifies the super-logarithmic dynamic data structure lower bounds from [LWY20] and [LY25] into a more general framework. From a technical perspective, our approach merges the cell-sampling and chronogram techniques developed in [LWY20] and [LY25] with the new static data structure lower bound methods from [KW20] and [Ko25], thereby merging all known state-of-the-art cell-probe lower-bound techniques into one. As a direct consequence of our method, we establish a super-logarithmic lower bound against the Multiphase Problem [Pat10] for the case where the data structure outputs the Inner Product (mod 2) of $S_Q$ and $T$. We suspect further applications of this general method towards showing super-logarithmic dynamic cell-probe lower bounds. We list some example applications of our general method, including a novel technique for a one-way communication lower bound against small-advantage protocols for a product distribution using average min-entropy, which could be of independent interest.