Deterministic Expander Routing: Faster and More Versatile

TL;DR

The paper addresses deterministic expander routing in CONGEST, achieving a bound that matches the randomized GKS17 result while enabling preprocessing/query tradeoffs. It introduces a one-shot hierarchical decomposition, a randomized-meeting-in-the-middle approach, and a derandomization via pre-embedded shufflers, augmented by coarse-grained shufflers and deterministic expander sorting. The core contributions include a deterministic expander routing algorithm with single-instance rounds $2^{O( ext{sqrt}( ext{log } n ext{ log } ext{log } n))}$, plus preprocessing and per-query tradeoffs that yield near-optimal deterministic performance for problems like $k$-clique enumeration, approaching the current randomized bounds. This work advances the practicality and scope of deterministic distributed routing on expanders, enabling efficient MST-like tasks and broader subgraph enumeration in deterministic CONGEST settings, with potential cross-application to other EXPANDER-based primitives and decompositions.

Abstract

We consider the expander routing problem formulated by Ghaffari, Kuhn, and Su (PODC 2017), where the goal is to route all the tokens to their destinations given that each vertex is the source and the destination of at most $°(v)$ tokens. They developed $\textit{randomized algorithms}$ that solve this problem in $\text{poly}(φ^{-1}) \cdot 2^{O(\sqrt{\log n \log \log n})}$ rounds in the $\textsf{CONGEST}$ model, where $φ$ is the conductance of the graph. Later, Ghaffari and Li (DISC 2018) gave an improved algorithm. However, both algorithms are randomized, which means that all the resulting applications are also randomized. Recently, Chang and Saranurak (FOCS 2020) gave a deterministic algorithm that solves an expander routing instance in $2^{O(\log^{2/3} n \cdot \log^{1/3} \log n)}$ rounds. The deterministic algorithm is less efficient and does not allow preprocessing/query tradeoffs, which precludes the de-randomization of algorithms that require this feature, such as the $k$-clique enumeration algorithm in general graphs. The main contribution of our work is a new deterministic expander routing algorithm that not only matches the randomized bound of [GKS 2017] but also allows preprocessing/query tradeoffs. Our algorithm solves a single instance of routing query in $2^{{O}(\sqrt{\log n \cdot \log \log n})}$ rounds. Our algorithm achieves the following preprocessing and query tradeoffs: For $0 < ε< 1$, we can answer every routing query in $\log^{O(1/ε)} n$ rounds at the cost of a $(n^{O(ε)} + \log^{O(1/ε)} n)$-round preprocessing procedure. Combining this with the approach of Censor-Hillel, Leitersdorf, and Vulakh (PODC 2022), we obtain a near-optimal $\tilde{O}(n^{1-2/k})$-round deterministic algorithm for $k$-clique enumeration in general graphs, improving the previous state-of-the-art $n^{1-2/k+o(1)}$.

Figures (1)

• Figure 1: An illustration of the hierarchical decomposition. The gray dotted edges denote the expander embedding as described in \ref{['prop:hierarchical']}(\ref{['itm:hierarchical:embedding']}). For example, the gray dotted edges inside $X_1$ is the virtual graph $H_{X_{1}}$. The base graph of the child node with vertex set $X_1$ is now $H_{X_1}$. The black dotted edges between $X_i$ and $X'_i$ form a matching embedding described in \ref{['prop:hierarchical']}(\ref{['itm:hierarchical:matching']}).

Theorems & Definitions (80)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• proof
• Corollary 1.4
• proof
• Lemma 2.3
• Theorem 3.1: ChangS20
• Definition 3.2
• Corollary 3.3