Gadgetless Lifting Beats Round Elimination: Improved Lower Bounds for Pointer Chasing

TL;DR

This paper establishes a nearly tight lower bound for the k-step pointer chasing problem's communication complexity, introducing a novel gadgetless lifting framework inspired by structure-vs-pseudorandomness decomposition.

Contribution

The paper presents a new gadgetless lifting method to derive lower bounds for general protocols, improving upon previous techniques like round elimination and information complexity.

Findings

Proved an (n/k + k) lower bound on communication complexity.

Introduced gadgetless lifting framework for lower bounds.

Bridged gap between lower and upper bounds for pointer chasing.

Abstract

We prove an \Omega(n/k+k) communication lower bound on (k-1)-round distributional complexity of the k-step pointer chasing problem under uniform input distribution, improving the \Omega(n/k - k log n) lower bound due to Yehudayoff (Combinatorics Probability and Computing, 2020). Our lower bound almost matches the upper bound of O(n/k + k) communication by Nisan and Wigderson (STOC 91). As part of our approach, we put forth gadgetless lifting, a new framework that lifts lower bounds for a family of restricted protocols into lower bounds for general protocols. A key step in gadgetless lifting is choosing the appropriate definition of restricted protocols. In this paper, our definition of restricted protocols is inspired by the structure-vs-pseudorandomness decomposition by G\"o\"os, Pitassi, and Watson (FOCS 17) and Yang and Zhang (STOC 24). Previously, round-communication trade-offs were mainly obtained by round elimination and information complexity. Both methods have some barriers in some situations, and we believe gadgetless lifting could potentially address these barriers.