Deterministic Distributed DFS via Cycle Separators in Planar Graphs

TL;DR

This work achieves a deterministic CONGEST model algorithm for cycle separators in planar graphs, running in tilde-O(D) rounds and enabling parallel computation across components. By introducing deterministic face weights and augmentations, the authors avoid randomized geometric simulations and triangulations, yet still locate good separator edges. They then leverage these cycle separators in a recursive framework to construct a DFS tree in tilde-O(D) rounds, matching the best-known randomized results up to polylog factors and approaching fundamental lower bounds. The approach rests on deterministic subroutines for DFS orders, LCA, path marking, and face-aware augmentations, supported by low-congestion shortcuts in planar graphs. The results have practical impact for distributed planar algorithms, offering near-optimal deterministic performance for DFS and related separator-based procedures, and open avenues for further deterministic separator methods in broader graph classes.

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 $\tilde{O}(D)$ 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 ${\tilde{O}}(D)$ 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 $Ω(D)$ rounds.

Figures (11)

• Figure 1: Given embedding $\mathcal{E}$, black solid edges are $T$-edges, dotted black edges are real $T$-fundamental edges. Nodes $z_1$ and $z_2$ are $\mathcal{E}$-compatible, but there are multiple ways to insert the edge $e=\{z_1,z_2\}$. The subgraph induced by red and green nodes corresponds to $F_e$. Red nodes are the border $C_e$, green nodes are the node inside $F_e$ (i.e., in $\mathring{F}_e$) and white nodes are the nodes outside $F_e$. The dotted red edges represent three ways to insert the virtual fundamental edge in $t_{z_1}$ and $t_{z_1}$, each insertion generates different faces. The nodes in each face are marked with red.
• Figure 2: In (a) a left-dfs-order of $T$ and in (b) a right-dfs-order of $T$. The solid edges correspond to the edges of $T$, while the non-tree edges are dotted. The clockwise order $t_v$ of the embedding is consistent with this figure.
• Figure 3: How we count the number of nodes in $F_e$ when $e=uv$, $u$ is not an ancestor of $v$. Nodes in the light green area are the nodes counted in $\omega(F_e)$
• Figure 4: How we count the number of nodes in $F_e$ when $e=uv$ when node $u$ is an ancestor of $v$ and $t_u(v)>t_u(z)$. Nodes in the light green area are the nodes counted in $\omega(F_e)$
• Figure 5: An example of the $(T,F_e)$-compatible nodes with the node $u$. Black edges are $T$-edges, black dashed edges represent real fundamental edges, red dashed edges are the virtual fundamental edges added in a $(T,F_e)$-compatible way and purple nodes are not $(T,F_e)$-compatible with node $u$.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Claim 1