Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage

TL;DR

The paper advances succinct representations for chordal graphs with bounded vertex leafage by deriving a nontrivial information-theoretic lower bound ${\log|\mathcal{G}_k| \ge (k-1)n\log n - kn\log k - O(\log n)}$ and constructing a matching data structure that encodes ${\mathcal G}_k$ in ${(k-1)n\log n + o(kn\log n)}$ bits. It achieves this by transforming a chordal graph with vertex leafage at most $k$ into a path-graph instance with ${kn/2}$ vertices, leveraging a proven succinct path-graph DS as a black box, and supplementing it with several auxiliary structures to support adjacency and neighbourhood queries. The resulting adjacency time is ${O(k\log n)}$, and neighbourhood queries run in ${O(k^2 d_v \log n + \log^2 n)}$ time using an additional ${2n\log n}$ bits, with the construction remaining succinct for ${k = o(n^c)}$. Overall, the work contributes to the theory of succinct representations for graph classes and points to possible generalizations of leafage-like parameters to broader graph families via tree-decomposition concepts.

Abstract

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 $v \in V$.

Figures (2)

• Figure 1: The complete rooted binary tree $T$ with $m$ nodes constructed to produce a graph in $\mathcal{G}^c_k$. $T'$ is one of the sub-trees of $T$ with exactly $k$ leaves corresponding to a dependent vertex in $U'$. $T"$ is the sub-tree corresponding to a dependent vertex in $V(H)\backslash(U \cup U')$ constructed from $\{v_1,\ldots,v_t,\ldots,v_k\}$ which are the $k$ selected nodes of $T$.
• Figure 2: (a) An example 4-vertex leafage chordal graph $G$, (b) tree representation of $G$ after pre-processing, (c) the index of the paths generated from the sub-trees along with their start node (second row), end node (third row), and index (forth row), (d) the components of the succinct data structure for $G$.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9: DBLP:journals/tcs/MunroRRR12
• Proposition 10