Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

TL;DR

This paper introduces a space-efficient graph coarsening method called cloud partition for planar and minor-free graphs, enabling faster algorithms and succinct encodings without relying on graph embeddings.

Contribution

The authors develop a novel cloud decomposition technique that improves space and time efficiency for graph algorithms on planar and minor-free graphs, avoiding the need for embeddings.

Findings

Constructed in linear time and space for planar graphs.

Enabled efficient balanced separator and tree decomposition computations.

Improved succinct encoding schemes for minor-free graphs.

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 $\Theta(\log n)$. As an additional application of our cloud decomposition we show that, for $H$-minor-free graphs, a tree decomposition of width $O(n^{1/2 + \epsilon})$ for any $\epsilon > 0$ 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.