[Paper] A Simple Distributed Deterministic Planar Separator

Source: arXiv - 2602.22916v1

Overview

A new deterministic algorithm can compute a balanced planar separator in ≈ (\tilde O(D)) communication rounds, where (D) is the graph’s diameter. This matches the best‑known (randomized) approaches while being far simpler than prior deterministic solutions, opening the door to fully deterministic distributed tools for classic planar‑graph problems.

Key Contributions

• Simple deterministic separator: Each vertex transfers its weight to a single incident face, eliminating the need for complex weight‑distribution schemes.
• Near‑optimal round complexity: The algorithm runs in (\tilde O(D)) rounds, matching the lower bound imposed by the graph’s diameter.
• Broad applicability: Directly derandomizes state‑of‑the‑art distributed algorithms for planar‑graph tasks such as SSSP, max‑flow, directed global min‑cut, and reachability.
• Cleaner proof technique: Provides an intuitive argument for why the naïve face‑weighting works, making the method easier to understand and implement.

Methodology

1. Weight assignment: Every vertex starts with unit weight (or any prescribed weight).
2. Face transfer: Each vertex picks any face it belongs to and hands its entire weight to that face. This step is completely local and deterministic.
3. Face aggregation: Faces now hold the total weight of all incident vertices. Because each vertex contributes to exactly one face, the total weight is preserved.
4. Separator extraction: Using the aggregated face weights, the algorithm identifies a simple path (or small vertex set) whose removal splits the graph into components each containing at most a constant fraction of the total weight. The identification leverages classic planar‑separator theorems (Lip‑Tarjan) but operates on the face‑weight distribution.
5. Distributed implementation: All steps are performed via standard CONGEST‑style message passing. The face‑transfer step requires only constant‑size messages, and the subsequent computation proceeds along the planar embedding, which can be explored in (\tilde O(D)) rounds.

The key insight is that any deterministic assignment of vertices to faces suffices; the algorithm does not need the sophisticated randomized balancing used in earlier work.

Results & Findings

• Correctness: The constructed separator is guaranteed to be balanced (each side ≤ (2/3) of the total weight) and of size (O(D)), i.e., a simple path whose length scales with the graph diameter.
• Complexity: The total number of communication rounds is (\tilde O(D)) (polylogarithmic factors hidden), which is asymptotically optimal because any distributed algorithm must spend at least (Ω(D)) rounds to propagate information across the graph.
• Simplicity: The algorithm’s description fits on a single page, contrasting sharply with the multi‑page, intricate constructions of the previous deterministic separator (PODC’25).

Practical Implications

• Deterministic distributed libraries: Frameworks that currently rely on randomized planar‑separator subroutines (e.g., for SSSP or max‑flow) can swap in this deterministic version, guaranteeing reproducible results—crucial for debugging, compliance, and safety‑critical systems.
• Lower latency in wide‑area networks: Since the round complexity scales with the diameter, networks with large physical spread (e.g., edge‑cloud deployments) benefit from the (\tilde O(D)) bound, avoiding the extra overhead of (\tilde O(\sqrt{n}))‑type algorithms.
• Simplified implementation: Developers need only implement a local “pick‑a‑face” rule and a standard planar‑graph traversal, making the technique accessible to engineers without deep expertise in graph theory.
• Foundation for higher‑level primitives: Many distributed algorithms for planar graphs (e.g., routing, load balancing, network design) use separators as a building block. A deterministic, easy‑to‑use separator can streamline the design of robust, provably correct protocols.

Limitations & Future Work

• Planarity requirement: The method assumes a planar embedding is known or can be computed; extending the idea to minor‑free or more general sparse graph families remains open.
• Hidden polylog factors: While (\tilde O(D)) is asymptotically optimal, the actual constants and logarithmic terms may affect performance on very large graphs; empirical evaluation is needed.
• Dynamic graphs: The current algorithm works on static graphs. Adapting it to handle edge/vertex insertions or deletions without recomputing from scratch is a promising direction.
• Weighted separators: The paper focuses on unit‑weight vertices; exploring how the simple face‑transfer idea behaves with arbitrary vertex weights (e.g., for load‑balanced partitioning) could broaden its applicability.

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Bottom line: By showing that a “pick‑any‑face” weight transfer suffices, the authors deliver a deterministic planar separator that is both theoretically optimal and practically straightforward, paving the way for reliable, high‑performance distributed algorithms on planar networks.

Authors

• Yaseen Abd-Elhaleem
• Michal Dory
• Oren Weimann

Paper Information

• arXiv ID: 2602.22916v1
• Categories: cs.DC, cs.DS
• Published: February 26, 2026
• PDF: Download PDF