Optimal Communication Complexity of Chained Index Problem

Core Concepts

The chained index problem requires Ω(n) bits of communication to solve optimally, irrespective of the number of chained instances.

Abstract

The paper studies the chained index (chain) communication problem, which is a generalization of the well-studied index problem. In the chain problem, there are k instances of the index problem, all with the same answer, shared among k+1 players in a chained fashion. The communication is one-way from each player to the next, and the last player has to output the common answer.

The key results are:

1. The authors prove an optimal lower bound of Ω(n) bits of communication for solving the chain problem, irrespective of the number of chained instances k. This settles an open conjecture posed in prior work.

2. The key technique is to use information-theoretic tools, specifically the Jensen-Shannon divergence, to analyze protocols. This allows them to obtain a stronger lower bound compared to prior approaches based on total variation distance.

3. As a corollary, the authors obtain improved streaming lower bounds for approximating maximum independent sets and submodular maximization, through reductions from the chain problem.

4. The authors also extend their lower bound to a generalized version of the problem called Augmented Chain.

The paper provides a comprehensive analysis of the communication complexity of the chained index problem and its applications in streaming algorithms.

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Key Insights Distilled From

Optimal Communication Complexity of Chained Index

by Janani Sunda... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07026.pdf

Deeper Inquiries

Can the techniques used in this paper be applied to prove tight lower bounds for other variants or generalizations of the index problem

The techniques used in the paper, such as the hybrid argument and analysis of protocols over divergence measures, can indeed be applied to prove tight lower bounds for other variants or generalizations of the index problem. Variants like the Augmented Index, where additional information is provided to one of the parties, can benefit from similar approaches. By carefully designing hybrid distributions and leveraging information-theoretic tools, one can establish optimal lower bounds for these variations of the index problem. The key lies in understanding the correlations between inputs, designing appropriate reductions, and analyzing the communication complexity using relevant measures.

Are there any applications of the chained index problem beyond the streaming lower bounds discussed in the paper

The chained index problem, as discussed in the paper, has applications beyond streaming lower bounds. One potential application could be in distributed computing scenarios where multiple parties need to collaboratively solve similar index tasks with shared information. For example, in distributed databases or decentralized systems, the chained index problem could model scenarios where each node holds a part of the data and needs to communicate with other nodes to collectively determine the final output. By understanding the communication complexity of chained index problems, insights can be gained into the efficiency and scalability of distributed algorithms and protocols.

What other information-theoretic measures, besides Jensen-Shannon divergence, could be useful for analyzing communication complexity problems

In addition to Jensen-Shannon divergence, other information-theoretic measures could be useful for analyzing communication complexity problems. One such measure is the Renyi divergence, which generalizes the KL-divergence and provides a family of divergence measures parameterized by an exponent. Renyi divergence can offer different perspectives on the distance between distributions and may reveal unique insights into the complexity of communication protocols. Additionally, measures like Bhattacharyya distance, Hellinger distance, and Wasserstein distance could also be valuable in certain contexts for characterizing the information flow and efficiency of communication in various problem settings. Each measure has its strengths and applications, and choosing the most suitable measure depends on the specific characteristics of the problem being analyzed.