{"title":"用exp(O(n1/3))个样本进行痕量重建","authors":"F. Nazarov, Y. Peres","doi":"10.1145/3055399.3055494","DOIUrl":null,"url":null,"abstract":"In the trace reconstruction problem, an unknown bit string x ∈ {0,1}n is observed through the deletion channel, which deletes each bit of x with some constant probability q, yielding a contracted string x. How many independent copies of x are needed to reconstruct x with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was exp(O(n1/2)). We improve this bound to exp(O(n1/3)) using statistics of individual bits in the output and show that this bound is sharp in the restricted model where this is the only information used. Our method, that uses elementary complex analysis, can also handle insertions. Similar results were obtained independently and simultaneously by Anindya De, Ryan O'Donnell and Rocco Servedio.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2016-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"58","resultStr":"{\"title\":\"Trace reconstruction with exp(O(n1/3)) samples\",\"authors\":\"F. Nazarov, Y. Peres\",\"doi\":\"10.1145/3055399.3055494\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the trace reconstruction problem, an unknown bit string x ∈ {0,1}n is observed through the deletion channel, which deletes each bit of x with some constant probability q, yielding a contracted string x. How many independent copies of x are needed to reconstruct x with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was exp(O(n1/2)). We improve this bound to exp(O(n1/3)) using statistics of individual bits in the output and show that this bound is sharp in the restricted model where this is the only information used. Our method, that uses elementary complex analysis, can also handle insertions. Similar results were obtained independently and simultaneously by Anindya De, Ryan O'Donnell and Rocco Servedio.\",\"PeriodicalId\":20615,\"journal\":{\"name\":\"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-12-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"58\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3055399.3055494\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3055399.3055494","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In the trace reconstruction problem, an unknown bit string x ∈ {0,1}n is observed through the deletion channel, which deletes each bit of x with some constant probability q, yielding a contracted string x. How many independent copies of x are needed to reconstruct x with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was exp(O(n1/2)). We improve this bound to exp(O(n1/3)) using statistics of individual bits in the output and show that this bound is sharp in the restricted model where this is the only information used. Our method, that uses elementary complex analysis, can also handle insertions. Similar results were obtained independently and simultaneously by Anindya De, Ryan O'Donnell and Rocco Servedio.