{"title":"随机递归树定向切割中根删除的淬火最坏情况假设","authors":"Laura Eslava, Sergio I. López, Marco L. Ortiz","doi":"10.1017/jpr.2024.40","DOIUrl":null,"url":null,"abstract":"We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size <jats:italic>n</jats:italic> according to the decreasing order of their degrees; namely, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline1.png\"/> <jats:tex-math>$(v^{(i)})_{i=1}^{n}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> be such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline2.png\"/> <jats:tex-math>$\\deg(v^{(1)}) \\geq \\cdots \\geq \\deg (v^{(n)})$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The targeted vertex-cutting process is performed by sequentially removing vertices <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline3.png\"/> <jats:tex-math>$v^{(1)}, v^{(2)}, \\ldots, v^{(n)}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline4.png\"/> <jats:tex-math>$K_n$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is upper bounded by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline5.png\"/> <jats:tex-math>$Z_{\\geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline6.png\"/> <jats:tex-math>$\\ln Z_{\\geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> grows as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline7.png\"/> <jats:tex-math>$\\ln n$</jats:tex-math> </jats:alternatives> </jats:inline-formula> asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the <jats:italic>k</jats:italic>th moment of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline8.png\"/> <jats:tex-math>$\\ln Z_{\\geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is proportional to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline9.png\"/> <jats:tex-math>$(\\!\\ln n)^k$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As a consequence, we obtain that the first-order growth of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline10.png\"/> <jats:tex-math>$K_n$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is upper bounded by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000408_inline11.png\"/> <jats:tex-math>$n^{1-\\ln 2}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quenched worst-case scenario for root deletion in targeted cutting of random recursive trees\",\"authors\":\"Laura Eslava, Sergio I. López, Marco L. Ortiz\",\"doi\":\"10.1017/jpr.2024.40\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size <jats:italic>n</jats:italic> according to the decreasing order of their degrees; namely, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline1.png\\\"/> <jats:tex-math>$(v^{(i)})_{i=1}^{n}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> be such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline2.png\\\"/> <jats:tex-math>$\\\\deg(v^{(1)}) \\\\geq \\\\cdots \\\\geq \\\\deg (v^{(n)})$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The targeted vertex-cutting process is performed by sequentially removing vertices <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline3.png\\\"/> <jats:tex-math>$v^{(1)}, v^{(2)}, \\\\ldots, v^{(n)}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline4.png\\\"/> <jats:tex-math>$K_n$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is upper bounded by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline5.png\\\"/> <jats:tex-math>$Z_{\\\\geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline6.png\\\"/> <jats:tex-math>$\\\\ln Z_{\\\\geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> grows as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline7.png\\\"/> <jats:tex-math>$\\\\ln n$</jats:tex-math> </jats:alternatives> </jats:inline-formula> asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the <jats:italic>k</jats:italic>th moment of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline8.png\\\"/> <jats:tex-math>$\\\\ln Z_{\\\\geq D}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is proportional to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline9.png\\\"/> <jats:tex-math>$(\\\\!\\\\ln n)^k$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As a consequence, we obtain that the first-order growth of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline10.png\\\"/> <jats:tex-math>$K_n$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is upper bounded by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000408_inline11.png\\\"/> <jats:tex-math>$n^{1-\\\\ln 2}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random.\",\"PeriodicalId\":50256,\"journal\":{\"name\":\"Journal of Applied Probability\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/jpr.2024.40\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/jpr.2024.40","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Quenched worst-case scenario for root deletion in targeted cutting of random recursive trees
We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size n according to the decreasing order of their degrees; namely, let $(v^{(i)})_{i=1}^{n}$ be such that $\deg(v^{(1)}) \geq \cdots \geq \deg (v^{(n)})$. The targeted vertex-cutting process is performed by sequentially removing vertices $v^{(1)}, v^{(2)}, \ldots, v^{(n)}$ and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, $K_n$, is upper bounded by $Z_{\geq D}$, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that $\ln Z_{\geq D}$ grows as $\ln n$ asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the kth moment of $\ln Z_{\geq D}$ is proportional to $(\!\ln n)^k$. As a consequence, we obtain that the first-order growth of $K_n$ is upper bounded by $n^{1-\ln 2}$, which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random.
期刊介绍:
Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used.
A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.