Distances in Planar Graphs are Almost for Free!

Abstract

We prove that, up to subpolynomial or polylogarithmic factors, there is no tradeoff between preprocessing time, query time, and size of exact distance oracles for planar graphs. Namely, we show how given an $n$-vertex weighted directed planar graph $G$, one can compute in $n^{1+o(1)}$ time and space a representation of $G$ from which one can extract the exact distance between any two vertices of $G$ in $\log^{2+o(1)}(n)$ time. Previously, it was only known how to construct oracles with these space and query time in $n^{3/2+o(1)}$ time [STOC 2019, SODA 2021, JACM 2023]. We show how to construct these oracles in $n^{1+o(1)}$ time.

Figures (12)

• Figure 1: A graph (region) $H$. The vertices of $H$'s infinite face $h$ are the sites of the Voronoi diagram $\textsf{VD}$. Each site is represented by a unique color, which is also used to shade its Voronoi cell. The dual representation $\textsf{VD}^*$ is illustrated as the blue tree. The tree has 7 leaves (corresponding to 7 copies of $h^*$) and 5 internal nodes (corresponding to the 5 trichromatic faces of $\textsf{VD}$). The edges of the tree correspond to contracted subpaths in $H^*$.
• Figure 2: A schematic diagram for Lemma \ref{['lem:VD0-Tx']}. The shortest path tree $T_x$ is shown in blue, and its dual co-tree $T^*_x$ is shown in red. The distinguished face $h$ is depicted as a black cycle, with five designated sites on $h$ indicated by solid blue nodes. The subgraph $\textsf{VD}^*_0$ is highlighted in thick purple. Edges of $\textsf{VD}^*_0$ that do not belong to $T^*_x$ are additionally highlighted in yellow. The duals of these edges are exactly the edges of $T_x$ that enter a blue site.
• Figure 4: A schematic diagram of the Proof of Lemma \ref{['lem:TwoColorMonotonicity']}. A green edge $e \in T_g$ is indicated by the bold green arrow. One of the paths comprising the fundamental cycle of $e^*$ w.r.t. $T^*_g$, say, $P_1^*$, is indicated by a dashed black line (the remainder of its rootward path in $T^*_g$ is indicated by a dashed red line). A green edge $a^*$ on $P^*_1$ is marked in a thick black line. The fundamental cycle $C_a$ of $a$ w.r.t. $T_g$ is shaded green. It contains only green vertices and encloses the prefix of $P^*_1$ ending at $a^*$.
• Figure 5: Possible cases in the Proof of Lemma \ref{['lem:SimpleTrichromaticDecisionCorrectness']}. The bold green arrow represents edge $e$ and the dashed black line represents the fundamental cycle of $e^*$ w.r.t. $T^*_g$. The bisectors are indicated by a thin dashed purple line and the trichromatic face $\tilde{f}$ is the bold purple triangle. In cases (a)-(c) the green vertex of $\tilde{f}$ belongs to $T^e$, whereas in cases (d)-(f) the green vertex of $\tilde{f}$ does not belong to $T^e$.
• Figure 6: The partition described in the Proof of Lemma \ref{['lem:SimpleTrichromaticDecisionCorrectness']}. The shaded areas indicate the parts. The $GR$-curve and $GB$-curve are highlighted in yellow. The bold green arrow represents edge $e$ and the dashed black lines represent the the green prefixes of paths $P^*_1, P^*_2$ of the fundamental cycle of $e^*$ w.r.t. $T^*_g$. The bisectors are indicated by thin dashed purple lines, and the trichromatic face $\tilde{f}$ is the bold purple triangle. The highlighted green node is the green vertex of $\tilde{f}$. The figures correspond to case (1) where both critical edges exist.

Theorems & Definitions (31)

• Theorem 1.1
• Definition 2.1: edge centroid decomposition
• Definition 2.2
• Lemma 2.3: Lemma 2.1 of GawrychowskiKMS18
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 3.1
• Lemma 3.2