[Paper Review] Faster and simpler algorithms for finding large patterns in permutations

László Kozma|arXiv (Cornell University)|Feb 23, 2019

Algorithms and Data Compression29 references4 citations

TL;DR

This paper presents two new algorithms for the permutation pattern matching (PPM) problem, achieving improved time bounds: an $ n^{0.44k + o(k)} $-time algorithm and a simpler $ O(1.6181^n) $-time algorithm with polynomial space. The key innovation lies in a novel dynamic programming strategy that matches even-valued pattern entries first, then greedily matches odd-valued entries left-to-right, significantly simplifying prior approaches while improving asymptotic performance for large patterns.

ABSTRACT

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length $n$ contains a given pattern of length $k$. In this work we give two new algorithms for this well-studied problem, one whose running time is $n^{0.44k+o(k)}$, and one whose running time is the better of $O(1.6181^n)$ and $n^{k/2+o(k)}$. These results improve the earlier best bounds of Ahal and Rabinovich (2000), and Bruner and Lackner (2012), and are the fastest algorithms for the problem when $k = Ω(\log n)$. When $k = o(\log n)$, the parameterized algorithm of Guillemot and Marx (2013) dominates. Our second algorithm uses polynomial space and is significantly simpler than all previous approaches with comparable running times, including an $n^{k/2+o(k)}$ algorithm proposed by Guillemot and Marx. Our approach can be summarized as follows: "for every matching of the even-valued entries of the pattern, try to match all odd-valued entries left-to-right". For the special case of patterns that are Jordan-permutations, we show an improved, subexponential running time.

Motivation & Objective

• To improve the time complexity of the permutation pattern matching (PPM) problem for large patterns, where the pattern length $ k $ is $ heta(\log n) $ or larger.
• To develop a deterministic, simpler algorithm with comparable or better running time than prior methods, particularly avoiding complex probabilistic analyses.
• To explore structural properties of permutation incidence graphs, especially pathwidth, to derive tighter bounds on algorithmic complexity.
• To provide a new, transparent analysis framework that simplifies prior approaches relying on random walk arguments or intricate dynamic programming state management.

Proposed method

• Introduce a new dynamic programming strategy that processes even-valued entries of the pattern before odd-valued ones, exploiting a partitioning of the pattern into even and odd components.
• For each valid matching of even entries, attempt to extend it by matching odd entries in left-to-right order, ensuring order-consistency with the pattern.
• Use a global optimization strategy to select the order of pattern entry matching, minimizing the size of the state boundary that must be tracked during dynamic programming.
• Leverage structural properties of permutation incidence graphs, particularly pathwidth, to bound the number of states in the dynamic programming table.
• Construct a family of permutations $ \pi(a,b) $ that avoid the pattern $ (4,3,1,2) $ but have incidence graphs with large grid minors, demonstrating that pathwidth cannot be bounded by minor-closed parameters alone.
• Analyze the algorithm's performance using a refined state space reduction technique, showing that only a subexponential number of partial matchings need to be considered.

Experimental results

Research questions

• RQ1 Can the running time of permutation pattern matching be improved beyond the $ n^{0.47k + o(k)} $ bound of Ahal and Rabinovich for large patterns?
• RQ2 Is it possible to design a simpler, deterministic algorithm for PPM with comparable or better asymptotic performance than existing methods?
• RQ3 What is the relationship between the pathwidth of a permutation's incidence graph and the complexity of PPM, especially in relation to forbidden minors?
• RQ4 Can the pathwidth of incidence graphs of permutations be bounded by structural parameters such as minor containment or grid size?

Key findings

• The paper presents a new algorithm, Algorithm M, that solves PPM in $ n^{0.44k + o(k)} $ time, improving upon the previous best bound of $ n^{0.47k + o(k)} $.
• A second algorithm achieves $ O(1.6181^n) $ time complexity with polynomial space, significantly simplifying prior approaches while maintaining competitive running time.
• The new approach avoids complex probabilistic analyses by using a deterministic, global matching strategy based on even-odd partitioning, leading to a more transparent and verifiable proof.
• The paper constructs a family of permutations that avoid a fixed pattern but have incidence graphs with $ \Theta(a) \times \Theta(b) $ grid minors, showing that pathwidth cannot be bounded by minor-closed parameters alone.
• For Jordan-permutations, the paper achieves a subexponential running time, indicating potential for further improvements in special cases.
• The analysis demonstrates that the pathwidth of the incidence graph is not tightly correlated with minor containment, challenging assumptions about structural parameterization in PPM.