Approximate counting of permutation patterns
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909868054544384 |
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| author | Ben-Eliezer, Omri Mitrović, Slobodan Srivastava, Pranjal |
| author_facet | Ben-Eliezer, Omri Mitrović, Slobodan Srivastava, Pranjal |
| contents | We consider the problem of counting the copies of a length-$k$ pattern $σ$ in a sequence $f \colon [n] \to \mathbb{R}$, 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 $σ(j) < σ(\ell)$. 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] obtained an $O(n)$ time algorithm for the detection variant 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 $n^{Ω(k / \log k)}$ [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 $σ$-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 pattern counting. Interestingly, while neither the sequence $f$ nor the pattern $σ$ are monotone, our algorithm makes extensive use of coresets for monotone functions [Har-Peled, 2006]. Along the way, we develop a near-optimal data structure for $(1+\varepsilon)$-approximate increasing pair range queries in the plane, which exhibits a conditional separation from the exact case and may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximate counting of permutation patterns Ben-Eliezer, Omri Mitrović, Slobodan Srivastava, Pranjal Data Structures and Algorithms We consider the problem of counting the copies of a length-$k$ pattern $σ$ in a sequence $f \colon [n] \to \mathbb{R}$, 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 $σ(j) < σ(\ell)$. 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] obtained an $O(n)$ time algorithm for the detection variant 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 $n^{Ω(k / \log k)}$ [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 $σ$-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 pattern counting. Interestingly, while neither the sequence $f$ nor the pattern $σ$ are monotone, our algorithm makes extensive use of coresets for monotone functions [Har-Peled, 2006]. Along the way, we develop a near-optimal data structure for $(1+\varepsilon)$-approximate increasing pair range queries in the plane, which exhibits a conditional separation from the exact case and may be of independent interest. |
| title | Approximate counting of permutation patterns |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2411.04718 |