Distributed Compression for Computation and Bounds on the Optimal Rate

TL;DR

This work addresses distributed functional compression for two correlated sources by encoding source characteristic graphs and their $n$-fold OR products to enable asymptotically lossless computation of arbitrary functions. It introduces a graph-entropy framework where rates are bounded by chromatic entropy and exact chromatic numbers are derived for cycle families, with recursive/polynomial coloring schemes for odd cycles and spectral bounds via adjacency eigenvalues. Extending to general graphs, the paper develops Gershgorin-circle-based bounds and a decomposition method to bound eigenvalues of $n$-fold OR products, tying expansion properties to rate tradeoffs. The results offer a principled, spectra-informed coding approach that connects graph structure, entropy, and distributed computation performance, with implications for efficient information sharing in multicast and networked computation settings.

Abstract

We address the problem of distributed computation of arbitrary functions of two correlated sources $X_1$ and $X_2$, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple instances using the $n$-fold OR product of this graph with itself). For regular graphs and general graphs, we establish bounds on the optimal rate -- characterized by the chromatic entropy for the $n$-fold graph products -- that allows a receiver for asymptotically lossless computation of arbitrary functions over finite fields. For the special class of cycle graphs (i.e., $2$-regular graphs), we establish an exact characterization of chromatic numbers and derive bounds on the required rates. Next, focusing on the more general class of $d$-regular graphs, we establish connections between $d$-regular graphs and expansion rates for $n$-fold graph powers using graph spectra. Finally, for general graphs, we leverage the Gershgorin Circle Theorem (GCT) to provide a characterization of the spectra, which allows us to build new bounds on the optimal rate. Our codes leverage the spectra of the computation and provide a graph expansion-based characterization to efficiently/succinctly capture the computation structure, providing new insights into the problem of distributed computation of arbitrary functions.

Figures (6)

• Figure 1: Distributed functional compression with two sources and a receiver, where $G_{X_1}$ is cyclic.
• Figure 2: A valid coloring of $C_{4}^{3}$ with 8 colors.
• Figure 3: The $2$-fold OR product of $C_5$, i.e., $C_5^2$, and its valid coloring.
• Figure 4: (Left) $(\chi({C_{2k+1}}))^n$ (dashed (blue) curve), and $\chi({C_{2k+1}^n})$ (solid (orange) curve) for any $k\geq 2$. (Right) The gain, i.e., $\eta_n$, of the coloring approach in Proposition \ref{['prop-chromatic_number_cycles-odd']} compared to the Greedy algorithm.
• Figure 5: A $3$-regular graph, $G_{3,6}$, is distinguished by a dashed square and $\chi({G_{3,6}^2})=9$.

Theorems & Definitions (63)

• Definition 1: Degree of a vertex havel1955remark
• Definition 2: Independent set, and maximal independent set beigel1999finding
• Definition 3: Source characteristic graphs korner1973coding
• Definition 4: Entropy of a characteristic graph korner1973coding
• Definition 5: Path and Hamiltonian path bermond1979hamiltonian
• Definition 6: $d$-regular graphs chen1997graphnagle1966ordering
• Definition 7: Expansion rate bang2008digraphs
• Example 1
• Definition 8: $n$-fold OR product graph nagle1966orderingnilli1991secondalon2002graph
• Definition 9: Sub-graphs of $G_{{X}}^n$