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

TL;DR

This paper puts forth gadgetless lifting, a new framework that lifts lower bounds for a family of restricted protocols into lower bounds for general protocols, inspired by the structure-vs-pseudorandomness decomposition by G\"o\"os, Pitassi, and Watson and Yang and Zhang.

Abstract

We prove an Ω(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 Ω(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 Göös, 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.

Theorems & Definitions (18)

• Definition 1.1: Min-entropy and deficiency
• Definition 1.2: Dense distribution
• Lemma 1.3: Density-restoring partition
• Theorem 2.1: Main theorem, \ref{['thm:main:distributional']} restated
• Lemma 2.2: Loop invariant
• proof
• Lemma 2.3: Relating accuracy and avarage fixed size
• Claim 2.4
• Claim 2.5
• proof : Proof of \ref{['lem:accuracy']}