Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings
Nina Hammer, Frank Kammer, Johannes Meintrup
TL;DR
The paper introduces cloud partitioning to spaces- efficiently coarsen planar graphs into clouds of size $O(\log n)$, forming a structure-maintaining minor $F$ with $|V(F)|=O(n/\log n)$ and enabling a full cloud decomposition $(F,\mathcal{P})$ without planar embeddings. This framework supports fast, embedding-free computation of balanced separators, and enables succinct encodings of planar (and $H$-minor-free) graphs, as well as planar tree decompositions with improved space and time bounds; all core constructions run in $O(n)$ time and use $O(n)$ bits. The approach generalizes to $H$-minor-free graphs via a $\phi$-critical cloud theory, preserving the same favorable time/space characteristics and facilitating separator-based encodings and decompositions in broader minor-closed classes. Empirical results on synthetic and real-world graphs corroborate practical efficiency, showing that simplified variants suffice in many cases and that the method adapts to diverse degree distributions.
Abstract
We present a novel space-efficient graph coarsening technique for $n$-vertex planar graphs $G$, called cloud partition, which partitions the vertices $V(G)$ into disjoint sets $C$ of size $O(\log n)$ such that each $C$ induces a connected subgraph of $G$. Using this partition $P$ we construct a so-called structure-maintaining minor $F$ of $G$ via specific contractions within the disjoint sets such that $F$ has $O(n/\log n)$ vertices. The combination of $(F, P)$ is referred to as a cloud decomposition. For planar graphs we show that a cloud decomposition can be constructed in $O(n)$ time and using $O(n)$ bits. Given a cloud decomposition $(F, P)$ constructed for a planar graph $G$ we are able to find a balanced separator of $G$ in $O(n/\log n)$ time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to $H$-minor-free graphs for any fixed graph $H$. This allows us to construct the succinct encoding scheme for $H$-minor-free graphs due to Blelloch and Farzan (CPM 2010) in $O(n)$ time and $O(n)$ bits improving both runtime and space by a factor of $Θ(\log n)$. As an additional application of our cloud decomposition we show that, for $H$-minor-free graphs, a tree decomposition of width $O(n^{1/2 + ε})$ for any $ε> 0$ can be constructed in $O(n)$ bits and a time linear in the size of the tree decomposition. Finally, we implemented our cloud decomposition algorithm and experimentally verified its practical effectiveness on both randomly generated graphs and real-world graphs such as road networks. The obtained data shows that a simplified version of our algorithms suffices in a practical setting, as many of the theoretical worst-case scenarios are not present in the graphs we encountered.