{"title":"漫长的路径使得模式计数变得困难,而茂密的树木使其更加困难","authors":"V'it Jel'inek, Michal Opler, Jakub Pek'arek","doi":"10.4230/LIPIcs.IPEC.2021.22","DOIUrl":null,"url":null,"abstract":"We study the counting problem known as #PPM, whose input is a pair of permutations $\\pi$ and $\\tau$ (called pattern and text, respectively), and the task is to find the number of subsequences of $\\tau$ that have the same relative order as $\\pi$. A simple brute-force approach solves #PPM for a pattern of length $k$ and a text of length $n$ in time $O(n^{k+1})$, while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time $f(k) n^{o(k/\\log k)}$ for any function $f$. In this paper, we consider the restriction of #PPM, known as $\\mathcal{C}$-Pattern #PPM, where the pattern $\\pi$ must belong to a hereditary permutation class $\\mathcal{C}$. Our goal is to identify the structural properties of $\\mathcal{C}$ that determine the complexity of $\\mathcal{C}$-Pattern #PPM. We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results: 1. If $C$ has the LPP, then $\\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(\\sqrt{k})}$ for any function $f$, and 2. if $C$ has the DTP, then $\\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(k/\\log^2 k)}$ for any function $f$. Furthermore, when $\\mathcal{C}$ is one of the so-called monotone grid classes, we show that if $\\mathcal{C}$ has the LPP but not the DTP, then $\\mathcal{C}$-Pattern #PPM can be solved in time $f(k)n^{O(\\sqrt k)}$. In particular, the lower bounds above are tight up to the polylog terms in the exponents.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Long paths make pattern-counting hard, and deep trees make it harder\",\"authors\":\"V'it Jel'inek, Michal Opler, Jakub Pek'arek\",\"doi\":\"10.4230/LIPIcs.IPEC.2021.22\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the counting problem known as #PPM, whose input is a pair of permutations $\\\\pi$ and $\\\\tau$ (called pattern and text, respectively), and the task is to find the number of subsequences of $\\\\tau$ that have the same relative order as $\\\\pi$. A simple brute-force approach solves #PPM for a pattern of length $k$ and a text of length $n$ in time $O(n^{k+1})$, while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time $f(k) n^{o(k/\\\\log k)}$ for any function $f$. In this paper, we consider the restriction of #PPM, known as $\\\\mathcal{C}$-Pattern #PPM, where the pattern $\\\\pi$ must belong to a hereditary permutation class $\\\\mathcal{C}$. Our goal is to identify the structural properties of $\\\\mathcal{C}$ that determine the complexity of $\\\\mathcal{C}$-Pattern #PPM. We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results: 1. If $C$ has the LPP, then $\\\\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(\\\\sqrt{k})}$ for any function $f$, and 2. if $C$ has the DTP, then $\\\\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(k/\\\\log^2 k)}$ for any function $f$. Furthermore, when $\\\\mathcal{C}$ is one of the so-called monotone grid classes, we show that if $\\\\mathcal{C}$ has the LPP but not the DTP, then $\\\\mathcal{C}$-Pattern #PPM can be solved in time $f(k)n^{O(\\\\sqrt k)}$. In particular, the lower bounds above are tight up to the polylog terms in the exponents.\",\"PeriodicalId\":137775,\"journal\":{\"name\":\"International Symposium on Parameterized and Exact Computation\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-11-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Symposium on Parameterized and Exact Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.IPEC.2021.22\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Parameterized and Exact Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.IPEC.2021.22","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Long paths make pattern-counting hard, and deep trees make it harder
We study the counting problem known as #PPM, whose input is a pair of permutations $\pi$ and $\tau$ (called pattern and text, respectively), and the task is to find the number of subsequences of $\tau$ that have the same relative order as $\pi$. A simple brute-force approach solves #PPM for a pattern of length $k$ and a text of length $n$ in time $O(n^{k+1})$, while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time $f(k) n^{o(k/\log k)}$ for any function $f$. In this paper, we consider the restriction of #PPM, known as $\mathcal{C}$-Pattern #PPM, where the pattern $\pi$ must belong to a hereditary permutation class $\mathcal{C}$. Our goal is to identify the structural properties of $\mathcal{C}$ that determine the complexity of $\mathcal{C}$-Pattern #PPM. We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results: 1. If $C$ has the LPP, then $\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(\sqrt{k})}$ for any function $f$, and 2. if $C$ has the DTP, then $\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(k/\log^2 k)}$ for any function $f$. Furthermore, when $\mathcal{C}$ is one of the so-called monotone grid classes, we show that if $\mathcal{C}$ has the LPP but not the DTP, then $\mathcal{C}$-Pattern #PPM can be solved in time $f(k)n^{O(\sqrt k)}$. In particular, the lower bounds above are tight up to the polylog terms in the exponents.