Compact representations of pattern-avoiding permutations

2025-10-23

AI Summary

This paper designs a compact data structure for π-avoiding permutations—permutations of size $n$ that avoid a fixed pattern $pi$. Methodologically, it leverages combinatorial structural properties of such permutations, integrating divide-and-conquer encoding, geometric point-set mapping, and efficient indexing to achieve linear-time construction. The main contributions are: (i) the first data structure achieving the optimal space bound of $O(n log s_pi)$ bits, where $s_pi$ denotes the number of $n$-element $pi$-avoiding permutations—breaking the information-theoretic lower bound for general permutations; (ii) constant-time support for forward and inverse position queries; (iii) $O(log log n)$-time query performance for geometric range counting (e.g., axis-aligned rectangle counting); and (iv) extensibility to broader permutation classes with bounded treewidth, enabling structural compression at the data structure level.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Pattern-avoiding permutations are a central object of study in both combinatorics and theoretical computer science. In this paper we design a data structure that can store any size-$n$ permutation $τ$ that avoids an arbitrary (and unknown) fixed pattern $π$ in the asymptotically optimal $O(n lg{s_π})$ bits, where $s_π$ is the Stanley-Wilf limit of $π$. Our data structure supports $τ(i)$ and $τ^{-1}(i)$ queries in $O(1)$ time, sidestepping the lower bound of Golynski (SODA 2009) that holds for general permutations. Comparable results were previously known only in more restricted cases, e.g., when $τ$ is separable, which means avoiding the patterns 2413 and 3142. We also extend our data structure to support more complex geometric queries on pattern-avoiding permutations (or planar point sets) such as rectangle range counting in $O(lglg{n})$ time. This result circumvents the lower bound of $Ω{(lg{n}/lglg{n})}$ by Pătraşcu (STOC 2007) that holds in the general case. For bounded treewidth permutation classes (which include the above-mentioned separable class), we further reduce the space overhead to a lower order additive term, making our data structure succinct. This extends and improves results of Chakraborty et al. (ISAAC 2024) that were obtained for separable permutations via different techniques. All our data structures can be constructed in linear time.

Problem

• Designing compact data structures for pattern-avoiding permutations
• Achieving optimal space usage with constant-time query support
• Enabling efficient geometric queries while bypassing general lower bounds

Innovation

• Compact data structure for pattern-avoiding permutations
• Supports O(1) queries for permutation functions
• Enables geometric queries like rectangle counting