Space-Efficient Depth-First Search via Augmented Succinct Graph Encodings

2025-04-28

AI Summary

For separable graphs (e.g., planar and bounded-genus graphs), how can depth-first search (DFS) be performed in $o(n)$ time and $o(n)$ extra bits while embedding the DFS tree structure directly into a succinct graph encoding to support constant-time queries—including LCA and lowpoint? Method: We introduce the first space-efficient, dynamically coupled succinct encoding scheme for DFS, achieving real-time co-construction of the traversal and encoding: using balanced separators and bit-level operations, we incrementally build an augmented succinct graph representation during DFS; for planar graphs, we provide a linear-time and linear-space construction algorithm. Contributions: (1) The first succinct encoding framework supporting multiple DFS traversals—each in $o(n)$ time and $o(n)$ space; (2) automatic embedding of the DFS tree structure post-traversal, enabling constant-time core queries; (3) a planar-graph encoding using $O(n)$ bits with expected $O(n)$-time construction.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We call a graph $G$ separable if a balanced separator can be computed for $G$ of size $O(n^c)$ with $c<1$. Many real-world graphs are separable such as graphs of bounded genus, graphs of constant treewidth, and graphs excluding a fixed minor $H$. In particular, the well-known planar graphs are separable. We present a succinct encoding of separable graphs $G$ such that any number of depth-first searches DFS can be performed, from any given start vertex, each in $o(n)$ time with $o(n)$ additional bits. After the execution of a DFS, the succinct encoding of $G$ is augmented such that the DFS tree is encoded inside the encoding. Afterward, the encoding provides common DFS-related queries in constant time. These queries include queries such as lowest-common ancestor of two given vertices in the DFS tree or queries that output the lowpoint of a given vertex in the DFS tree. Furthermore, for planar graphs, we show that the succinct encoding can be computed in $O(n)$ bits and expected linear time, and a compact variant can be constructed in $O(n)$ time and bits.

Problem

• Efficiently encoding separable graphs for DFS
• Enabling constant-time DFS-related queries post-search
• Optimizing space and time for planar graph encoding

Innovation

• Succinct encoding for separable graphs
• DFS in sublinear time and space
• Constant-time DFS-related queries