Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913309787160576 |
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| author | Balakrishnan, Girish Chakraborty, Sankardeep Narayanaswamy, N S Sadakane, Kunihiko |
| author_facet | Balakrishnan, Girish Chakraborty, Sankardeep Narayanaswamy, N S Sadakane, Kunihiko |
| contents | We improve the worst-case information theoretic lower bound of Munro and Wu (ISAAC 2018) for $n-$vertex unlabeled chordal graphs when vertex leafage is bounded and leafage is unbounded. The class of unlabeled $k-$vertex leafage chordal graphs that consists of all chordal graphs with vertex leafage at most $k$ and unbounded leafage, denoted $\mathcal{G}_k$, is introduced for the first time. For $k>0$ in $o(n/\log n)$, we obtain a lower bound of $((k-1)n \log n -kn \log k - O(\log n))-$bits on the size of any data structure that encodes a graph in $\mathcal{G}_k$. Further, for every $k-$vertex leafage chordal graph $G$ such that for $k>1$ in $o(n^c), c >0$, we present a $((k-1)n \log n + o(kn \log n))-$bit succinct data structure, constructed using the succinct data structure for path graphs with $kn/2$ vertices. Our data structure supports adjacency query in $O(k \log n)$ time and using additional $2n \log n$ bits, an $O(k^2 d_v \log n + \log^2 n)$ time neighbourhood query where $d_v$ is degree of $v \in V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage Balakrishnan, Girish Chakraborty, Sankardeep Narayanaswamy, N S Sadakane, Kunihiko Data Structures and Algorithms We improve the worst-case information theoretic lower bound of Munro and Wu (ISAAC 2018) for $n-$vertex unlabeled chordal graphs when vertex leafage is bounded and leafage is unbounded. The class of unlabeled $k-$vertex leafage chordal graphs that consists of all chordal graphs with vertex leafage at most $k$ and unbounded leafage, denoted $\mathcal{G}_k$, is introduced for the first time. For $k>0$ in $o(n/\log n)$, we obtain a lower bound of $((k-1)n \log n -kn \log k - O(\log n))-$bits on the size of any data structure that encodes a graph in $\mathcal{G}_k$. Further, for every $k-$vertex leafage chordal graph $G$ such that for $k>1$ in $o(n^c), c >0$, we present a $((k-1)n \log n + o(kn \log n))-$bit succinct data structure, constructed using the succinct data structure for path graphs with $kn/2$ vertices. Our data structure supports adjacency query in $O(k \log n)$ time and using additional $2n \log n$ bits, an $O(k^2 d_v \log n + \log^2 n)$ time neighbourhood query where $d_v$ is degree of $v \in V$. |
| title | Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2402.03748 |