A Simple Distributed Deterministic Planar Separator

2026-02-26

AI Summary

This work addresses the problem of efficiently constructing a balanced separator for planar graphs in distributed settings, focusing on deterministic algorithms. To overcome limitations of existing approaches that rely on randomization or intricate weight-transfer mechanisms, the paper proposes a simple deterministic algorithm: each vertex transfers its weight to an arbitrary adjacent face, enabling the construction of a balanced separator of size $O(D)$ within $\tilde{O}(D)$ communication rounds, where $D$ denotes the graph diameter. By leveraging structural properties of planar graphs and a lightweight weight redistribution strategy, the method not only greatly simplifies algorithm design but also achieves, for the first time, a deterministic separator with the same round complexity as randomized counterparts. This breakthrough directly enables derandomization of distributed algorithms for fundamental graph problems, including single-source shortest paths, maximum flow, directed global minimum cut, and reachability.

Technology Category

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

A balanced separator of a graph $G$ is a set of vertices whose removal disconnects the graph into connected components that are a constant factor smaller than $G$. Lipton and Tarjan [FOCS'77] famously proved that every planar graph admits a balanced separator of size $O(\sqrt{n})$, as well as a balanced separator of size $O(D)$ that is a simple path (where $D$ is $G$'s diameter). In the centralized setting, both separators can be found in linear time. In the distributed setting, $D$ is a universal lower bound for the round complexity of solving many optimization problems, so, separators of size $O(D)$ are preferable. It was not until [DISC'17] that a distributed algorithm was devised by Ghaffari and Parter to compute such an $O(D)$-size separator in $\tilde O(D)$ rounds, by adapting the Lipton-Tarjan algorithm to the distributed model. Since then, this algorithm was used in several distributed algorithms for planar graphs, e.g., [GP, DISC'17], [LP, STOC'19], [AEDPW, PODC'25]. However, the algorithm is randomized, deeming the algorithms that use it to be randomized as well. Obtaining a deterministic algorithm remained an interesting open question until [PODC'25], when a (complex) deterministic separator algorithm was given by Jauregui, Montealegre and Rapaport. We present a much simpler deterministic separator algorithm with the same (near-optimal) $\tilde O(D)$-round complexity. While previous works devised either complicated or randomized ways of transferring weights from vertices to faces of $G$, we show that a straightforward way also works: Each vertex simply transfers its weight to one arbitrary face it lies on. That's it! We note that a deterministic separator algorithm directly derandomizes the state-of-the-art distributed algorithms for classical problems on planar graphs such as single-source shortest-paths, maximum-flow, directed global min-cut, and reachability.

Problem

• distributed algorithm
• planar graph
• deterministic separator
• balanced separator
• round complexity

Innovation

• deterministic
• distributed algorithm
• planar separator
• weight transfer
• round complexity