Succinct Data Structures for Efficiently Representing and Querying Baxter Permutations and Related Families

Core Concepts

This paper introduces a novel succinct representation for Baxter permutations of size n that supports efficient queries on the permutation, including accessing the value at a given index (π(i)) and finding the index of a given value (π−1(j)), in sub-linear time.

Abstract

The key highlights and insights of this paper are:

1. The authors present a (3n+o(n))-bit representation for a Baxter permutation π of size n that can support π(i) and π−1(j) queries in O(f1(n)) and O(f2(n)) time respectively, where f1(n) and f2(n) are any increasing functions satisfying ω(log n) and ω(log2 n).

2. The representation is based on traversing the minimum Cartesian tree (MinC(π)) of the Baxter permutation using a two-stack algorithm that visits the nodes in ascending order of their labels. This allows efficient decoding of the node labels, which was a key challenge in prior representations.

3. The authors also provide a (2n+o(n))-bit representation for alternating Baxter permutations, which can support the same queries in the same time bounds.

4. As applications, the authors show how their succinct representations of Baxter and separable permutations can be used to construct efficient succinct data structures for mosaic/slicing floorplans and plane bipolar orientations that support various navigational queries.

5. The proposed representations circumvent previous lower bound results on the trade-offs between redundancy and query time for general permutations, by exploiting the structural properties of Baxter and separable permutations.

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Key Insights Distilled From

Succinct Data Structures for Baxter Permutation and Related Families

by Sankardeep C... at arxiv.org 09-26-2024

https://arxiv.org/pdf/2409.16650.pdf

Deeper Inquiries

Can the techniques used in this paper be extended to design succinct representations for other classes of permutations beyond Baxter and separable permutations?

The techniques introduced in this paper for succinct representations of Baxter and separable permutations leverage the unique structural properties of these specific classes of permutations, particularly their relationships with Cartesian trees and the absence of certain patterns. While the foundational concepts, such as the two-stack algorithm and the use of balanced parentheses for encoding, could potentially be adapted to other classes of permutations, the success of such adaptations would depend on the specific structural characteristics of those classes. For instance, classes of permutations that exhibit similar tree-like structures or have well-defined pattern restrictions may benefit from analogous succinct representations. However, for more general or less structured classes of permutations, the existing techniques may not yield efficient representations due to the increased complexity and lack of predictable patterns. Therefore, while there is potential for extension, careful consideration of the properties of the target permutation class is essential to determine the feasibility and effectiveness of such adaptations.

What are the potential applications of the efficient navigational queries supported by the succinct representations of mosaic/slicing floorplans and plane bipolar orientations?

The efficient navigational queries supported by the succinct representations of mosaic and slicing floorplans, as well as plane bipolar orientations, have significant implications in various practical applications. In the realm of computer-aided design (CAD), these representations can facilitate rapid querying of spatial relationships between rectangular objects, which is crucial for tasks such as block placement in integrated circuit design. The ability to quickly determine adjacency or retrieve neighboring objects enhances the efficiency of layout optimization processes. Additionally, in graph drawing and visualization, the efficient navigational capabilities can improve the rendering of complex structures, allowing for dynamic adjustments and interactive exploration of graph properties. Furthermore, in the context of data structures and algorithms, these succinct representations can serve as a foundation for developing more advanced data management systems that require efficient spatial queries, thereby broadening their applicability in fields such as geographic information systems (GIS) and robotics.

Are there any connections between the structural properties of Baxter and separable permutations exploited in this work and other areas of combinatorics or theoretical computer science?

Yes, the structural properties of Baxter and separable permutations have deep connections to various areas of combinatorics and theoretical computer science. For instance, Baxter permutations are closely related to Catalan structures, which appear in numerous combinatorial contexts, including binary trees, lattice paths, and polygon triangulations. The bijection established between Baxter permutations and twin binary trees highlights their relevance in tree enumeration and analysis. Similarly, separable permutations, characterized by their avoidance of specific patterns, relate to the study of pattern avoidance in permutations, a topic that has garnered significant interest in combinatorial theory. This connection extends to the analysis of sorting algorithms, where understanding the structure of permutations can lead to more efficient sorting strategies. Additionally, the techniques developed in this work, such as succinct data structures and efficient query mechanisms, contribute to the broader field of algorithm design and optimization, particularly in the context of space-efficient data management and retrieval. Thus, the insights gained from studying Baxter and separable permutations not only advance our understanding of these specific classes but also enrich the broader landscape of combinatorial mathematics and theoretical computer science.