Pub Date : 1998-05-01DOI: 10.1002/(SICI)1098-2418(199805)12:3%3C271::AID-RSA4%3E3.0.CO;2-S
L. Devroye, Amar Goudjil
We study binary search trees constructed from Weyl sequences fnng; n 1, where is an irrational and f:g denotes mod 1". We explore various properties of the structure of these trees, and relate them to the continued fraction expansion of. If H n is the height of the tree with n nodes when is chosen at random and uniformly on 0; 1], then we show that in probability, H n (12== 2) log n log log n.
研究了基于Weyl序列的二叉搜索树;N 1，其中是无理数，f:g表示mod 1 '。我们探索了这些树结构的各种性质，并将它们与的连分式展开联系起来。设H n为n节点树的高度，在0上随机均匀选择;1]，然后我们证明在概率中，H n (12== 2) log n log n。
{"title":"A study of random Weyl trees","authors":"L. Devroye, Amar Goudjil","doi":"10.1002/(SICI)1098-2418(199805)12:3%3C271::AID-RSA4%3E3.0.CO;2-S","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199805)12:3%3C271::AID-RSA4%3E3.0.CO;2-S","url":null,"abstract":"We study binary search trees constructed from Weyl sequences fnng; n 1, where is an irrational and f:g denotes mod 1\". We explore various properties of the structure of these trees, and relate them to the continued fraction expansion of. If H n is the height of the tree with n nodes when is chosen at random and uniformly on 0; 1], then we show that in probability, H n (12== 2) log n log log n.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124275485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-05-01DOI: 10.1002/(SICI)1098-2418(199805)12:3%3C253::AID-RSA3%3E3.0.CO;2-U
L. Kirousis, E. Kranakis, D. Krizanc, Y. Stamatiou
Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by Kamath et al. to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253
设f是一个随机布尔公式，它是3-SAT的一个实例。我们考虑计算最小实数k的问题，如果子句数与f的变量数之比严格超过k，则f几乎肯定是不满足的。通过一个众所周知的或多或少直截了当的论证，可以证明kF5.191。Kamath等人首先为占用问题提供了新的改进边界，将该上界改进为4.758。有强有力的实验证据表明，k的值在4.2左右。在这项工作中，我们根据随机公式f定义了一个递减的随机变量序列，如果其中任何一个的期望值收敛于零，那么f几乎肯定是不满足的。通过让序列第一项的期望值收敛于零，通过简单的初等计算，我们得到k的上界等于4.667。从序列第二项的期望值，我们得到值4.601q。这项工作是在第一作者访问卡尔顿Ž大学计算机科学学院期间进行的，并得到了NSERC自然科学与工程研究委员会的部分支持。由加拿大帕特雷大学(University of Patras)授予的公休假。第二和第三位Ž作者得到了NSERC自然科学与工程研究的部分资助。加拿大理事会。在这项研究的最后阶段，第一和最后的作者也部分Ž。欧盟ESPRIT长期研究项目ALCOM-IT项目(20244)资助。†本文的扩展摘要发表在第四届欧洲年会论文集Ž算法研讨会，ESA ' 96, September 25]27, 1996，巴塞罗那，西班牙。[27]该扩展摘要是由本文的前三位作者共同撰写的。致:l.m. Kirousis Q 1998 John Wiley & Sons, Inc。CCC 1042-9832r98r030253-17
{"title":"Approximating the unsatisfiability threshold of random formulas","authors":"L. Kirousis, E. Kranakis, D. Krizanc, Y. Stamatiou","doi":"10.1002/(SICI)1098-2418(199805)12:3%3C253::AID-RSA3%3E3.0.CO;2-U","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199805)12:3%3C253::AID-RSA3%3E3.0.CO;2-U","url":null,"abstract":"Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by Kamath et al. to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116749554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-03-01DOI: 10.1002/(SICI)1098-2418(199807)12:4%3C405::AID-RSA6%3E3.0.CO;2-X
A. Meir, J. Moon
We determine the asymptotic behavior of the expected value and the variance of the log-product of the subtree-sizes of trees T belonging to simply generated families of n rooted trees. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 197]212, 1998
{"title":"Addendum to On the log-product of the subtree-sizes of random trees","authors":"A. Meir, J. Moon","doi":"10.1002/(SICI)1098-2418(199807)12:4%3C405::AID-RSA6%3E3.0.CO;2-X","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199807)12:4%3C405::AID-RSA6%3E3.0.CO;2-X","url":null,"abstract":"We determine the asymptotic behavior of the expected value and the variance of the log-product of the subtree-sizes of trees T belonging to simply generated families of n rooted trees. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 197]212, 1998","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133541146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1997-12-01DOI: 10.1002/(SICI)1098-2418(199712)11:4%3C369::AID-RSA5%3E3.0.CO;2-X
M. Henzinger, M. Thorup
In dynamic graph algorithms the following provide-or-bound problem has to be solved quickly: Given a set S containing a subset R and a way of generating random elements from S testing for membership in R, either (i) provide an element of R, or (ii) give a (small) upper bound on the size of R that holds with high probability. We give an optimal algorithm for this problem. This algorithm improves the time per operation for various dynamic graph algorithms by a factor of O(log n). For example, it improves the time per update for fully dynamic connectivity from O(log3n) to O(log2n).
{"title":"Sampling to provide or to bound: With applications to fully dynamic graph algorithms","authors":"M. Henzinger, M. Thorup","doi":"10.1002/(SICI)1098-2418(199712)11:4%3C369::AID-RSA5%3E3.0.CO;2-X","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199712)11:4%3C369::AID-RSA5%3E3.0.CO;2-X","url":null,"abstract":"In dynamic graph algorithms the following provide-or-bound problem has to be solved quickly: Given a set S containing a subset R and a way of generating random elements from S testing for membership in R, either (i) provide an element of R, or (ii) give a (small) upper bound on the size of R that holds with high probability. We give an optimal algorithm for this problem. This algorithm improves the time per operation for various dynamic graph algorithms by a factor of O(log n). For example, it improves the time per update for fully dynamic connectivity from O(log3n) to O(log2n).","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114751556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1997-12-01DOI: 10.1002/(SICI)1098-2418(199712)11:4%3C395::AID-RSA7%3E3.0.CO;2-U
Lars Elistrup Rasmussen
We present a very simple, randomized approximation algorithm for determining the number of cliques in a random graph. Supported in part by NSF Grant CCR-9505448 and a UC Berkeley Faculty Research Grant
{"title":"Approximately counting cliques","authors":"Lars Elistrup Rasmussen","doi":"10.1002/(SICI)1098-2418(199712)11:4%3C395::AID-RSA7%3E3.0.CO;2-U","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199712)11:4%3C395::AID-RSA7%3E3.0.CO;2-U","url":null,"abstract":"We present a very simple, randomized approximation algorithm for determining the number of cliques in a random graph. Supported in part by NSF Grant CCR-9505448 and a UC Berkeley Faculty Research Grant","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130756160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1997-10-01DOI: 10.1002/(SICI)1098-2418(199710)11:3%3C245::AID-RSA3%3E3.0.CO;2-0
Y. Kohayakawa, Bernd Kreuter
We consider the binomial random graphGn,p and determine a sharp threshold function for the edge-Ramsey property Gn,p → (C1 , . . . , Cr ) for all `1, . . . , `r, where C denotes the cycle of length `. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. AMS subject classification code (1991): 05C55, 05C80, 05C38
{"title":"Threshold functions for asymmetric Ramsey properties involving cycles","authors":"Y. Kohayakawa, Bernd Kreuter","doi":"10.1002/(SICI)1098-2418(199710)11:3%3C245::AID-RSA3%3E3.0.CO;2-0","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199710)11:3%3C245::AID-RSA3%3E3.0.CO;2-0","url":null,"abstract":"We consider the binomial random graphGn,p and determine a sharp threshold function for the edge-Ramsey property Gn,p → (C1 , . . . , Cr ) for all `1, . . . , `r, where C denotes the cycle of length `. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. AMS subject classification code (1991): 05C55, 05C80, 05C38","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129496654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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