{"title":"A Faster Algorithm for Finding Tarski Fixed Points","authors":"John Fearnley, Rahul Savani","doi":"10.1145/3524044","DOIUrl":null,"url":null,"abstract":"Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log k n) queries [2]. Multiple authors have conjectured that this algorithm is optimal [2, 7], and indeed this has been proven for two-dimensional instances [7]. We show that these conjectures are false in dimension three or higher by giving an O(log2 n) query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for k-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an O(log2 ⌈k/3⌉ n) query algorithm for the k-dimensional problem.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Algorithms (TALG)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3524044","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 8
Abstract
Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log k n) queries [2]. Multiple authors have conjectured that this algorithm is optimal [2, 7], and indeed this has been proven for two-dimensional instances [7]. We show that these conjectures are false in dimension three or higher by giving an O(log2 n) query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for k-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an O(log2 ⌈k/3⌉ n) query algorithm for the k-dimensional problem.
Dang等人给出了一种算法，该算法可以使用O(log k n)次查询在宽度为n的k维晶格中找到一个Tarski不动点[2]。多个作者推测该算法是最优的[2,7]，并且确实在二维实例中得到了证明[7]。我们通过给出三维Tarski问题的O(log2 n)查询算法来证明这些猜想在三维或更高维度是错误的。我们还给出了k维Tarski问题的一个新的分解定理，该定理与我们的三维问题的新算法相结合，给出了k维问题的O(log2≤k/3≤n)查询算法。
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