[Paper Review] Capacity Theorems for Distributed Index Coding

Yucheng Liu, Parastoo Sadeghi+2 more|arXiv (Cornell University)|Jan 27, 2018

Cooperative Communication and Network Coding35 references3 citations

TL;DR

This paper establishes inner and outer bounds on the capacity region for distributed index coding with multiple servers, introducing a novel distributed composite coding scheme that enhances decoding flexibility and fractional rate allocation. For all 218 non-isomorphic four-message problems with equal link capacities, the bounds match in sum-rate, proving optimality in these cases.

ABSTRACT

In index coding, a server broadcasts multiple messages to their respective receivers, each with some side information that can be utilized to reduce the amount of communication from the server. Distributed index coding is an extension of index coding in which the messages are broadcast from multiple servers, each storing different subsets of the messages. In this paper, the optimal tradeoff among the message rates and the server broadcast rates, which is defined formally as the capacity region, is studied for a general distributed index coding problem. Inner and outer bounds on the capacity region are established that have matching sum-rates for all 218 non-isomorphic four-message problems with equal link capacities for all the links from servers to receivers. The proposed inner bound is built on a distributed composite coding scheme that outperforms the existing schemes by incorporating more flexible decoding configurations and enhanced fractional rate allocations into two-stage composite coding, a scheme that was originally introduced for centralized index coding. The proposed outer bound is built on the polymatroidal axioms of entropy, as well as functional dependences such as the $ m{fd}$-separation introduced by the multi-server nature of the problem. This outer bound utilizes general groupings of servers with different levels of granularity, which allows a natural tradeoff between computational complexity and tightness of the bound, and includes and improves upon all existing outer bounds for distributed index coding. Specific features of the proposed inner and outer bounds are demonstrated through concrete examples with four or five messages.

Motivation & Objective

• To characterize the optimal tradeoff between message rates and server broadcast rates in distributed index coding with multiple servers.
• To develop a unified framework for inner and outer bounds on the capacity region that subsumes and improves upon existing bounds.
• To demonstrate tightness of the proposed bounds for all non-isomorphic four-message distributed index coding problems with equal link capacities.
• To extend the centralized composite coding scheme to a distributed setting with enhanced decoding configurations and rate allocation.
• To introduce a flexible outer bound using polymatroidal axioms and functional dependencies, allowing tunable complexity-tightness tradeoffs.

Proposed method

• Proposes a distributed composite coding scheme that generalizes two-stage composite coding to multiple servers, enabling flexible decoding configurations and improved fractional rate allocation.
• Introduces a novel outer bound based on polymatroidal axioms of entropy and functional dependencies (fd-separation), incorporating general groupings of servers with variable granularity.
• Uses entropy-based inequalities and functional dependence constraints to derive a tight outer bound that captures the multi-server nature of the problem.
• Employs a hierarchical grouping strategy in the outer bound to balance computational complexity and tightness, enabling scalability.
• Validates the bounds using explicit examples with four and five messages, showing matching sum-rates for all 218 non-isomorphic four-message cases.
• Leverages the fact that the proposed inner bound subsumes previous schemes like fractional local clique covering and recursive coding, while the outer bound generalizes and improves upon existing bounds.

Experimental results

Research questions

• RQ1 What is the capacity region of the general distributed index coding problem with multiple servers and arbitrary message subsets?
• RQ2 Can a distributed composite coding scheme outperform existing schemes in terms of achievable rate regions for distributed index coding?
• RQ3 How can outer bounds be constructed to tightly characterize the capacity region in multi-server index coding, especially when link capacities are equal?
• RQ4 To what extent do the proposed inner and outer bounds match in sum-rate for all non-isomorphic four-message configurations?
• RQ5 Can the proposed outer bound framework be generalized to include and improve upon all prior outer bounds in distributed index coding?

Key findings

• The proposed inner and outer bounds achieve matching sum-rates for all 218 non-isomorphic four-message distributed index coding problems with equal link capacities, proving optimality in these cases.
• The distributed composite coding scheme outperforms existing schemes by incorporating more flexible decoding configurations and enhanced fractional rate allocations.
• The proposed outer bound is tighter than all prior outer bounds for distributed index coding and generalizes existing bounds through functional dependence and polymatroidal axioms.
• The outer bound allows a tradeoff between computational complexity and tightness via variable-granularity groupings of servers, enabling scalable analysis.
• The inner bound is computable and serves as a benchmark for evaluating linear coding schemes, with potential for simplification via elimination of unnecessary composite indices and decoding configurations.
• The results confirm that composite coding, when extended to the distributed setting, provides a powerful and tight characterization of the capacity region for small message sets.