Near-Optimal Distance Emulator for Planar Graphs

TL;DR

This paper introduces a near-optimal method for creating small distance emulators in planar graphs, enabling efficient computation of distances among a subset of vertices with optimal size and time complexity.

Contribution

The authors present a construction of near-optimal size distance emulators for planar graphs, improving efficiency in distance computations among terminal sets.

Findings

Constructed distance emulators of size  O( ext{min}(k^2,\u007f ext{sqrt}(k mp; n))) for any terminal set.

Achieved  O(n) time complexity for building emulators, which is optimal up to logarithmic factors.

Enabled all-pairs shortest path computations among k terminals in  O(n) time for certain k values.

Abstract

Given a graph $G$ and a set of terminals $T$, a \emph{distance emulator} of $G$ is another graph $H$ (not necessarily a subgraph of $G$) containing $T$, such that all the pairwise distances in $G$ between vertices of $T$ are preserved in $H$. An important open question is to find the smallest possible distance emulator. We prove that, given any subset of $k$ terminals in an $n$-vertex undirected unweighted planar graph, we can construct in $\tilde O(n)$ time a distance emulator of size $\tilde O(\min(k^2,\sqrt{k\cdot n}))$. This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among $k$ terminals in $\tilde O(n)$ time when $k=O(n^{1/3})$.