Algorithms for Distance Problems in Continuous Graphs

TL;DR

This paper introduces subquadratic algorithms for computing the diameter and mean distance in continuous graphs, focusing on two sparse graph classes: graphs of bounded treewidth and planar graphs. The authors formulate a subgraph-versus-ambient-graph framework and leverage orthogonal range searching, portal-based separations, and dynamic planar data structures to manage cross-interactions across separations. For constant treewidth $k$, they achieve $O(n\log^{4k-2}n)$ time (and $O(n^{1+\varepsilon}2^{O(k)})$ for general $k$) to compute both diameter and mean; for planar graphs, they obtain $O(nF\log n)$ time, where $F$ is the number of faces. Planarity-specific techniques rely on dynamic forests and top-tree based handling of edge-embedding constraints to compute eccentricities and means from face boundaries, enabling $O(n\log n)$ per face and thus $O(nF\log n)$ overall. The work advances subquadratic strategies for distance-based graph parameters in continuous settings and outlines directions toward further generalization to planar graphs and additional distance statistics.

Abstract

We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous graphs with $m$ edges these values can be computed in roughly $O(m^2)$ time. In this paper, we use geometric techniques to obtain subquadratic time algorithms to compute the diameter and the mean distance of a continuous graph for two well-established classes of sparse graphs. We show that the diameter and the mean distance of a continuous graph of treewidth at most $k$ can be computed in $O(n\log^{O(k)} n)$ time, where $n$ is the number of vertices in the graph. We also show that computing the diameter and mean distance of a continuous planar graph with $n$ vertices and $F$ faces takes $O(n F \log n)$ time.

Figures (9)

• Figure 1: Types of diametral pairs of a continuous graph.
• Figure 2: Visualization of the setting in the proof of \ref{['lem:closed_walks']}. Left: Combining a shortest $ab$-path, a shortest $a'b'$-path and the edges $aa'$ and $bb'$ is a shortest closed walk passing through all the interior points of $aa'$ and $bb'$. Right: To take maximum in \ref{['lem:closed_walks']} is necessary (the distance between two points is not given, in general, by half the length of a closed walk.)
• Figure 3: Visualization of the divide-and-conquer approach to compute $\mathop{\mathrm{diam}}\nolimits(\mathcal{G}\xspace)$ (see \ref{['for:dc']}).
• Figure 4: Concrete example of the minimization diagram of $d_\mathcal{K}\xspace(p(\lambda),q(\mu))$. In the center, some values of $d_\mathcal{K}\xspace(p(\lambda),q(\mu))$ are shown in red; on the right, 3D visualization of the roofs.
• Figure 7: Left: operation $\textsc{Link}(u,v,e_u,e_v,\lambda_{uv})$ for edge-weighted embedded forest. Right: with the operation $\textsc{AddLeftPath}(\Delta,u,v)$, the edges in the shadow subtrees increase their value by $\Delta$.

Theorems & Definitions (24)

• Theorem 1: Theorems \ref{['thm:diameter-treewidth-1']} and \ref{['thm:mean-treewidth-1']}
• Theorem 2: Theorems \ref{['thm:diameter-treewidth-2']} and \ref{['thm:mean-treewidth-2']}
• Theorem 3
• Lemma 4: Cabello and Knauer CABELLO2009815
• Lemma 5: Bringmann, Husfeldt, and Magnusson BHM20
• Theorem 6: Cabello Cabello22
• Lemma 6
• Corollary 7
• Lemma 7
• Lemma 7