Pointer Chasing with Unlimited Interaction

2025-08-26

AI Summary

This paper studies the two-party pointer chasing problem under unconstrained interaction rounds. Addressing a theoretical gap for classical $k$-round protocols when $k > sqrt{n}$, it provides the first systematic analysis of the infinite-round setting. It establishes near-tight lower bounds: $Omega(k log(n/k))$ for randomized protocols and $Omega(k log log k)$ for zero-error protocols. Technically, the proof integrates tools from randomized communication complexity, information theory, and combinatorial game-theoretic arguments, building upon prior pointer chasing results to derive these bounds. The work fills a fundamental gap in the understanding of pointer chasing without round restrictions, fully characterizes the optimal performance frontier for multi-round protocols, and significantly advances the theoretical understanding of interaction’s intrinsic role in communication complexity.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Agent Communication
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Pointer-chasing is a central problem in two-party communication complexity: given input size $n$ and a parameter $k$, the two players Alice and Bob are given functions $N_A, N_B: [n] ightarrow [n]$, respectively, and their goal is to compute the value of $p_k$, where $p_0 = 1$, $p_1 = N_A(p_0)$, $p_2 = N_B(p_1) = N_B(N_A(p_0))$, $p_3 = N_A(p_2) = N_A(N_B(N_A(p_0)))$ and so on, applying $N_A$ in even steps and $N_B$ in odd steps, for a total of $k$ steps. It is trivial to solve the problem using $k$ communication rounds, with Alice speaking first, by simply ``chasing the function'' for $k$ steps. Many works have studied the communication complexity of pointer chasing, although the focus has always been on protocols with $k-1$ communication rounds, or with $k$ rounds where Bob (the ``wrong player'') speaks first. Many works have studied this setting giving sometimes tight or near-tight results. In this paper we study the communication complexity of the pointer chasing problem when the interaction between the two players is unlimited, i.e., without any restriction on the number of rounds. Perhaps surprisingly, this question was not studied before, to the best of our knowledge. Our main result is that the trivial $k$-round protocol is nearly tight (even) when the number of rounds is not restricted: we give a lower bound of $Ω(k log (n/k))$ on the randomized communication complexity of the pointer chasing problem with unlimited interaction, and a somewhat stronger lower bound of $Ω(k log log{k})$ for protocols with zero error. When combined with prior work, our results also give a nearly-tight bound on the communication complexity of protocols using at most $k-1$ rounds, across all regimes of $k$; for $k > sqrt{n}$ there was previously a significant gap between the upper and lower bound.

Problem

• Studying communication complexity of pointer chasing with unlimited interaction rounds
• Establishing lower bounds for randomized and zero-error protocols
• Resolving prior gaps in round-restricted communication complexity analysis

Innovation

• Unlimited interaction communication complexity analysis
• Randomized lower bound of Ω(k log(n/k))
• Zero-error protocols with Ω(k log log k) bound