Deterministic Distributed DFS via Cycle Separators in Planar Graphs

2025-04-30

AI Summary

This paper addresses the deterministic distributed construction of depth-first search (DFS) trees for planar graphs in the CONGEST model. To overcome the limitations of existing algorithms—which rely on randomness or incur high round complexity—we propose the first deterministic distributed cycle separator algorithm for planar graphs. Leveraging a divide-and-conquer paradigm, we recursively apply this separator to decompose the graph and construct a DFS tree. Our algorithm terminates in $ ilde{O}(D)$ rounds, where $D$ denotes the graph diameter, matching the round complexity of the best-known randomized algorithms and approaching the $Omega(D)$ lower bound. This is the first deterministic distributed DFS algorithm for planar graphs achieving near-optimal round complexity. Moreover, our framework provides a scalable, deterministic divide-and-conquer methodology applicable to broader distributed graph problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

One of the most basic techniques in algorithm design consists of breaking a problem into subproblems and then proceeding recursively. In the case of graph algorithms, one way to implement this approach is through separator sets. Given a graph $G=(V,E)$, a subset of nodes $S subseteq V$ is called a separator set of $G$ if the size of each connected component of $G-S$ is at most $2/3 cdot |V|$. The most useful separator sets are those that satisfy certain restrictions of cardinality or structure. For over 40 years, various efficient algorithms have been developed for computing separators of different kinds, particularly in planar graphs. Separator sets, combined with a divide and conquer approach, have been fundamental in the design of efficient algorithms in various settings. In this work, we present the first deterministic algorithm in the distributed CONGEST model that recursively computes a cycle separator over planar graphs in $ ilde{O}(D)$ rounds. This result, as in the centralized setting, has significant implications in the area of distributed planar algorithms. In fact, from this result, we can construct a deterministic algorithm that computes a DFS tree in ${ ilde{O}}(D)$ rounds. This matches both the best-known randomized algorithm of Ghaffari and Parter (DISC, 2017) and, up to polylogarithmic factors, the trivial lower bound of $Omega(D)$ rounds.

Problem

• Develop deterministic distributed DFS via cycle separators
• Compute cycle separators in planar graphs efficiently
• Match performance of best-known randomized DFS algorithms

Innovation

• Deterministic distributed cycle separator algorithm
• Recursive computation in planar graphs
• Efficient DFS tree construction