Succinct Data Structures for Baxter Permutation and Related Families

TL;DR

The first succinct representation of the separable permutation $\rho: [n] \rightarrow [n]$ of size $n$ that supports both $\rho(i)$ and $\rho^{-1}(j)$ queries in $O(1)$ time is provided.

Abstract

A permutation $π: [n] \rightarrow [n]$ is a Baxter permutation if and only if it does not contain either of the patterns $2-41-3$ and $3-14-2$. Baxter permutations are one of the most widely studied subclasses of general permutation due to their connections with various combinatorial objects such as plane bipolar orientations and mosaic floorplans, etc. In this paper, we introduce a novel succinct representation (i.e., using $o(n)$ additional bits from their information-theoretical lower bounds) for Baxter permutations of size $n$ that supports $π(i)$ and $π^{-1}(j)$ queries for any $i \in [n]$ in $O(f_1(n))$ and $O(f_2(n))$ time, respectively. Here, $f_1(n)$ and $f_2(n)$ are arbitrary increasing functions that satisfy the conditions $ω(\log n)$ and $ω(\log^2 n)$, respectively. This stands out as the first succinct representation with sub-linear worst-case query times for Baxter permutations. Additionally, we consider a subclass of Baxter permutations called \textit{separable permutations}, which do not contain either of the patterns $2-4-1-3$ and $3-1-4-2$. In this paper, we provide the first succinct representation of the separable permutation $ρ: [n] \rightarrow [n]$ of size $n$ that supports both $ρ(i)$ and $ρ^{-1}(j)$ queries in $O(1)$ time. In particular, this result circumvents Golynski's [SODA 2009] lower bound result for trade-offs between redundancy and $ρ(i)$ and $ρ^{-1}(j)$ queries. Moreover, as applications of these permutations with the queries, we also introduce the first succinct representations for mosaic/slicing floorplans, and plane bipolar orientations, which can further support specific navigational queries on them efficiently.

Figures (7)

• Figure 1: (a) the case when $\phi(i)$ is in the left subtree of $\phi(k)$, and (b) the case when $\phi(i)$ is in the right subtree of $\phi(k)$.
• Figure 2: An example of the representation of the Baxter permutation $\pi = (9, 8, 10, 1, 7, 4, 5, 6, 2, 3, 11)$. Note that the data structure maintains only $E$ and $lr$ along with $o(n)$-bit auxiliary structures.
• Figure 3: An example of the representation of the separable permutation $\rho = (2, 1, 9, 10, 11, 12, 8, 4, 6, 5, 7, 3)$. Each tree within the red area represents a mini-tree of $T_{\rho}$ with $\ell_1 = 3$.
• Figure 4: (a) and (b) are equivalent mosaic floorplans that are not slicing floorplans. (c) is a slicing floorplan.
• Figure 5: A mosaic floorplan corresponds to $\pi=(2, 5, 6, 3, 1, 4, 8, 7)$. The first number of each block is a bottom-left deletion order, while the second number is a top-left deletion order.

Theorems & Definitions (32)

• Theorem 1: Farzan2014
• Lemma 1
• proof
• Lemma 2
• Theorem 2
• proof
• Example 1
• Corollary 1
• Lemma 3
• proof