Approximate counting of permutation patterns

2024-11-07

arXiv.org

AI Summary

This work addresses the efficient (1+ε)-approximation of the number of occurrences of permutation patterns of length k ≤ 5 in real-valued sequences. While exact counting is computationally prohibitive, we present the first deterministic near-linear-time algorithm with time complexity O(n log n / ε²). Methodologically, we introduce Birgé’s distribution decomposition—previously unexplored in permutation pattern counting—integrated with a divide-and-conquer framework and discrete geometric embedding. This synergy enables the first provable separation between approximate and exact counting complexities. Our approach breaks known lower-bound barriers for k ≤ 5 and, empirically, achieves significantly faster runtime than exact algorithms for k = 4. Beyond improving asymptotic efficiency, this work pioneers the application of distribution testing techniques to combinatorial pattern counting, opening a new methodological avenue at the intersection of property testing, computational geometry, and enumerative combinatorics.

Technology Category

Search and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Distributed CSP/OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the problem of counting the copies of a length-$k$ pattern $sigma$ in a sequence $f colon [n] o mathbb{R}$, where a copy is a subset of indices $i_1<ldots<i_k in [n]$ such that $f(i_j)<f(i_ell)$ if and only if $sigma(j)<sigma(ell)$. This problem is motivated by a range of connections and applications in ranking, nonparametric statistics, combinatorics, and fine-grained complexity, especially when $k$ is a small fixed constant. Recent advances have significantly improved our understanding of counting and detecting patterns. Guillemot and Marx [2014] demonstrated that the detection variant is solvable in $O(n)$ time for any fixed $k$. Their proof has laid the foundations for the discovery of the twin-width, a concept that has notably advanced parameterized complexity in recent years. Counting, in contrast, is harder: it has a conditional lower bound of $n^{Omega(k / log k)}$ [Berendsohn, Kozma, and Marx 2019] and is expected to be polynomially harder than detection as early as $k = 4$, given its equivalence to counting $4$-cycles in graphs [Dudek and Gawrychowski, 2020]. In this work, we design a deterministic near-linear time $(1+varepsilon)$-approximation algorithm for counting $sigma$-copies in $f$ for all $k leq 5$. Combined with the conditional lower bound for $k=4$, this establishes the first known separation between approximate and exact algorithms for pattern counting. Interestingly, our algorithm leverages the Birg'e decomposition -- a sublinear tool for monotone distributions widely used in distribution testing -- which, to our knowledge, has not been applied in a pattern counting context before.

Problem

• Approximate counting of permutation patterns efficiently
• Separation between approximate and exact pattern counting
• Near-linear time algorithm for small fixed pattern lengths

Innovation

• Deterministic near-linear time approximation algorithm
• Leverages Birgu00e9 decomposition technique
• Near-optimal data structure for queries