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A Hierarchy for Constant Communication Complexity

Andris Ambainis, Hartmut Klauck, Debbie Lim

TL;DR

A classification of communication complexity measures such that these measures are organized into equivalence classes such that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

Abstract

Similarly to the Chomsky hierarchy, we offer a classification of communication complexity measures such that these measures are organized into equivalence classes. Different from previous attempts of this endeavor, we consider two communication complexity measures as equivalent, if, when one is constant, then the other is constant as well, and vice versa. Most previous considerations of similar topics have been using polylogarithmic input length as a defining characteristic of equivalence. In this paper, two measures ${\cal C}_1, {\cal C}_2$ are constant-equivalent, if and only if for all total Boolean (families of) functions $f:\{0, 1\}^n\times\{0, 1\}^n\rightarrow \{0, 1\}$ we have ${\cal C}_1(f)=O(1)$ if and only if ${\cal C}_2(f)=O(1)$. We identify five equivalence classes according to the above equivalence relation. Interestingly, the classification is counter-intuitive in that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

A Hierarchy for Constant Communication Complexity

TL;DR

A classification of communication complexity measures such that these measures are organized into equivalence classes such that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

Abstract

Similarly to the Chomsky hierarchy, we offer a classification of communication complexity measures such that these measures are organized into equivalence classes. Different from previous attempts of this endeavor, we consider two communication complexity measures as equivalent, if, when one is constant, then the other is constant as well, and vice versa. Most previous considerations of similar topics have been using polylogarithmic input length as a defining characteristic of equivalence. In this paper, two measures are constant-equivalent, if and only if for all total Boolean (families of) functions we have if and only if . We identify five equivalence classes according to the above equivalence relation. Interestingly, the classification is counter-intuitive in that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

Paper Structure

This paper contains 19 sections, 13 theorems, 65 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main contribution.
  3. Outline.
  4. Preliminaries and notations
  5. Variants of rank and factorization norm.
  6. Information complexity.
  7. Simulation results
  8. Class 1
  9. Class 2
  10. Class 3
  11. Class 4
  12. Class 5
  13. Separation
  14. Separation between Class 1 and Class 2
  15. Separation between Class 2 and Class 3
  16. ...and 4 more sections

Key Result

Theorem 1

Class 1 includes the following complexity measures:

Figures (7)

  • Figure 1: Summary of results. We categorize communication complexity measures in to five increasingly powerful classes.
  • Figure 2: Relationship between complexity measure in Class 1.
  • Figure 3: Relationship between complexity measures in Class 2.
  • Figure 4:
  • Figure 5: Relationship between complexity measures in Class 4.
  • ...and 2 more figures

Theorems & Definitions (62)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 52 more