🤖 AI Summary
This work addresses the challenge of efficiently performing breadth-first search (BFS) and related graph algorithms directly on highly compressed representations of planar graphs while strictly limiting auxiliary space usage. The authors propose the first succinct encoding of planar graphs that can be constructed in linear time, supports native BFS traversal, and requires only o(n) additional bits of space. This representation enables constant-time queries on both the BFS tree and the original graph structure. Furthermore, it extends to several advanced applications, including computing balanced separators of size O(√n), diameter-based tree decompositions, triangulation, and bipartiteness testing. By integrating techniques from succinct data structures, planar graph embedding theory, and dual-graph traversal, the method achieves sublinear space complexity without sacrificing linear-time efficiency.
📝 Abstract
We present a succinct encoding of planar graphs that supports executing a breadth-first search directly on the encoding. The succinct encoding can be constructed in expected $O(n)$ time using $O(n)$ bits during construction; a compact variant can be constructed in deterministic $O(n)$ time using $O(n)$ bits. Once the encoding is constructed, a BFS from any start vertex can be computed in $O(n)$ time using $o(n)$ additional bits, including the space needed to represent the BFS tree. The resulting BFS tree $T$ remains available for standard tree operations, such as traversal, parent and child queries, layer queries, and lowest common ancestor queries, in constant time per query or output element. The encoding also supports standard graph queries. For plane graphs $G=(V, E)$, we provide traversal of the interdigitating tree $\hat T$, i.e., the spanning tree of the dual graph whose edges correspond to $E \setminus E(T)$.
As our main application, we implement the well-known planar separator theorem in a space-efficient way. For biconnected plane graphs, our encoding allows us to compute a balanced separator of size $O(\sqrt n)$ in $O(n)$ time using $o(n)$ additional bits. Along the way, we show that biconnected plane graphs encoded by our representation can be triangulated in expected $O(n)$ time and $o(n)$ bits in the succinct variant, or in deterministic $O(n)$ time using $O(n)$ bits in the compact variant. Further applications include computation of a tree decomposition of width $O(d)$ where $d$ is the diameter of the plane graph at hand and testing for bipartiteness. Finally, all results that do not rely on a plane embedding generalize to separable graph classes.