Space-Efficient Depth-First Search via Augmented Succinct Graph Encodings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909595865186304 |
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| author | Elberfeld, Michael Kammer, Frank Meintrup, Johannes |
| author_facet | Elberfeld, Michael Kammer, Frank Meintrup, Johannes |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19547 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Space-Efficient Depth-First Search via Augmented Succinct Graph Encodings Elberfeld, Michael Kammer, Frank Meintrup, Johannes Data Structures and Algorithms 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. |
| title | Space-Efficient Depth-First Search via Augmented Succinct Graph Encodings |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.19547 |