A Simple Distributed Deterministic Planar Separator

TL;DR

A much simpler deterministic separator algorithm with the same (near-optimal) $\tilde O(D)$-round complexity is presented, which 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.

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.

Figures (3)

• Figure 1: The primal graph $G$ is in blue. Solid blue edges are in $T$ and dashed blue edges are not in $T$. I.e., their duals (in red) are in $T^*$. Removing the dual edge $e^*=(g,f)$ from $T^*$ breaks $T^*$ into $T^*_f$ ($f$'s subtree in $T^*$) and $T^*\setminus T^*_f$. The fundamental cycle $C(T,e)$ (highlighted in yellow) encloses all faces that correspond to nodes of $T^*_f$, and does not enclose any face that corresponds to nodes of $T^*\setminus T^*_f$.
• Figure 2: The primal graph is blue. The dual tree $T^*$ is red. The solid blue edges are edges of the primal tree $T$. Dashed edges are not in $T$ (i.e., their duals are in $T^*$). (a) $f$ is the face with vertices $v_1,v_2,\ldots,v_{10}$. The edge $e=(v_1,v_{10})$ corresponds to the dual edge $e^*=(f,\mathsf{parent}(f))$ in $T^*$. (b) Adding the artificial triangulation edges (thin dashed) to the interior of $f$ partitions $f$ into faces $f_1,f_2,\ldots, f_{8}$. In the new dual tree $T^*$, node $f_{i+1}$ is the child of $f_{i}$. The path $P(f)$ is the path from $f_1$ to $f_{8}$. (c) In this example, $f_j = f_4$. The dual edge $e^*=(f_4,f_5)$ in $T^*$ corresponds to the artificial primal edge $e=(v_5,v_{10})$ and defines the separator (highlighted in yellow).
• Figure :

Theorems & Definitions (43)

• Theorem 1
• Lemma 1: Tree and cotree, Chapter 4.5 of KM_planarity_book
• Definition 1: Fundamental cuts
• Definition 2: Fundamental cycles
• Lemma 2: Fundamental cut-cycle duality KM_planarity_book
• Definition 3: $\alpha$-proper weights
• Definition 4: Balanced separators
• Definition 5: Fundamental cycle separators
• Definition 6: Balanced node
• Definition 7: Critical node