Faster Construction of a Planar Distance Oracle with Õ(1) Query Time

TL;DR

The paper tackles exact distance querying on planar graphs by achieving a near-linear preprocessing target of $ ilde{O}(n^{4/3})$ and constant-query time $ ilde{O}(1)$, improving the previous bound of $ ilde{O}(n^{3/2})$. It introduces a near-optimal method for constructing additively weighted Voronoi diagrams in undirected planar graphs, centered on a relaxed-partition framework and an enhanced MSSP data structure to enable fast partitioning and robust path analysis. The resulting static distance oracle uses $ ilde{O}(n^{4/3})$ space and preprocessing with $ ilde{O}(1)$ query time, and yields a dynamic variant with $ ilde{O}(1)$-time queries and $ ilde{O}(n^{2/3})$ updates, matching the state of the art in many respects while improving preprocessing time. Overall, the work advances exact planar distance oracles by closing the gap toward optimal preprocessing, leveraging sophisticated Voronoi-structure techniques and shortest-path separators.

Abstract

We show how to preprocess a weighted undirected $n$-vertex planar graph in $\tilde O(n^{4/3})$ time, such that the distance between any pair of vertices can then be reported in $\tilde O(1)$ time. This improves the previous $\tilde O(n^{3/2})$ preprocessing time [JACM'23]. Our main technical contribution is a near optimal construction of \emph{additively weighted Voronoi diagrams} in undirected planar graphs. Namely, given a planar graph $G$ and a face $f$, we show that one can preprocess $G$ in $\tilde O(n)$ time such that given any weight assignment to the vertices of $f$ one can construct the additively weighted Voronoi diagram of $f$ in near optimal $\tilde O(|f|)$ time. This improves the $\tilde O(\sqrt{n |f|})$ construction time of [JACM'23].

Figures (16)

• Figure 1: A graph (piece) $X$. The vertices of $X$'s infinite face $f$ 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 $f^*$) and 5 internal nodes (corresponding to the 5 trichromatic faces of $\textsf{VD}$). The edges of the tree correspond to contracted subpaths in $X^*$.
• Figure 2: The partition of $P$ into parts corresponding to the Voronoi cells of $\textsf{VD}(s_1,s_2,s_3)$ is very fragmented.
• Figure 3: The path $P$ with endpoints $x$ and $y$. We think of $P$ as being oriented from $y$ to $x$. The corresponding sites $s_x$ and $s_y$ with $R_x=R_{s_x,x}$ and $R_y=R_{s_y,y}$. The site $s$ is on $F_{\mathsf{left}}$, the path $R_{s,v}$ is a non-swirly path, and the path $R_{s,u}$ is a $y$-swirly path.
• Figure 4: Left: The (orange) cycle $C$ in the proof of \ref{['clm:winner_takes_something']}. The three options for $R_{s_2,u}$ are in red. Dashed red contradicts $v$ being closer to $s_1$ than to $s_2$, dotted red contradicts $x$ being closer to $s_x$ than to $s_2$, and wavy red contradicts $R_{s_2,u}$ entering $P$ from the left. Right: The (orange) cycle $C$ in the proof of \ref{['clm:no-criss-cross']}. Since $s_1$ is (strictly) on one side of $C$ and $v_2$ is on the other side of $C$, the dashed $R_{s_1,v_2}$ path must cross $C$ in either $P[v_2,v_1)$ or $R_{s_2,v_2}$. Both these crosses are impossible.
• Figure 5: The (orange) cycles $C$ (left image) and $C_2$ (right image) in the proof of \ref{['clm:criss-cross']}. The situation here is that $s_1$ reaches $v_1$ from the left and loses to $s_2$ that reaches $v_1$ from the right, and the symmetric issue for $v_2$. Thus, $v_1$ and $v_2$ both $(\{s_1,s_2\},\mathsf{left})$-likes both $s_1$ and $s_2$. To recognize a winner we detect which site is closer to $z$.

Theorems & Definitions (57)

• Definition 1: $(S,\mathsf{left})$-like
• Definition 2: Relaxed Partition
• Lemma 1: Enhanced MSSP data structure
• Lemma 2
• Lemma 3
• Remark 1
• Lemma 4
• Claim 1
• proof
• Claim 2