Abstract We give a deterministic algorithm for computing the diameter of an n-point set in three dimensions with O(n logcn) running time, where c is a constant.
摘要给出了一种计算三维n点集直径的确定性算法，其运行时间为O(n logcn)，其中c为常数。
{"title":"A deterministic algorithm for the three-dimensional diameter problem","authors":"J. Matoušek, O. Cheong","doi":"10.1145/167088.167217","DOIUrl":"https://doi.org/10.1145/167088.167217","url":null,"abstract":"Abstract We give a deterministic algorithm for computing the diameter of an n-point set in three dimensions with O(n logcn) running time, where c is a constant.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114388368","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}
Checking Approximate Computations over the Reals S. Ar” M. Blumt B. Codenotti$ P. Gemmell~ This paper provides the first systematic investigation of checking approximate numerical computations over subsets of the reals. In most cases, approximate checking is more challenging than exact checking. Problem conditioning, i.e., the measure of sensitivity of the output to slight changes in the input, and the presence of approximate ion parameters foil the direct transformation of many exact checkers to the approximate setting. Furthermore, approximate checking over the reals is complicated by the lack of nice finite field properties such as the existence of a samplable distribution which is invariant under addition or multiplication by a scalar. We overcome the above problems by using such techniques as testing and checking over similar but distinct distributions, using functions’ random and downward self-reducibility properties, and taking advantage of the small variance of the sum of independent identically distributed random variables. We provide approximate checkers for a variety of computations, including matrix multiplication, linear system solution, matrix inversion, and computation of the determinant. We also present an approximate version of Beigel’s trick and extend the approximate linear self tester/corrector of [8] and the trigonometric selftester/corrector of [5] to more general computations. *Department of Computer Science, Princeton University, Princeton, NJ 08544-2087. Supported by NSF PYI grant CCR9057486 and a grant from MITL. t Computer Science Division, UC Berkeley, Berkeley, CA 94720, and International Computer Science Institute, Berkeley CA 94704. Supported by NSF grant CCR88-13632. t International Computer Science Institute, Berkeley, CA 94704, and IEI-CNR, Piss (Italy). Partially supported by the “ Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo”. Subproject 2. e-mail: codenotti@iei.pi .cnr.it ~Computer Science Division, UC Berkeley, Berkeley, CA 94720. Supported by GTE, Schlumberger Fellowships, and by NSF grant CCR88-13632.
S. Ar " M. Blumt B. Codenotti$ P. Gemmell~本文首次系统地研究了实数子集上的近似数值计算的校核。在大多数情况下，近似检查比精确检查更具挑战性。问题调节，即输出对输入微小变化的灵敏度测量，以及近似离子参数的存在，阻碍了许多精确检查器直接转换为近似设置。此外，由于缺乏良好的有限域性质，例如在标量的加法或乘法下不变的可抽样分布的存在，对实数的近似检查变得复杂。利用函数的随机性和向下自约性，利用独立同分布随机变量和的方差小等技术，克服了上述问题。我们为各种计算提供近似检查器，包括矩阵乘法，线性系统解，矩阵反演和行列式计算。我们还提出了Beigel技巧的近似版本，并将[8]的近似线性自测试仪/校正器和[5]的三角自测试仪/校正器扩展到更一般的计算中。*普林斯顿大学计算机科学系，普林斯顿，新泽西08544-2087。由NSF PYI拨款CCR9057486和MITL资助。t加州大学伯克利分校计算机科学部，加州伯克利，加州94720;国际计算机科学研究所，加州伯克利，加州94704。国家科学基金CCR88-13632资助。1国际计算机科学研究所，伯克利，CA 94704, IEI-CNR, Piss(意大利)。部分由“Progetto Finalizzato Sistemi Informatici和Calcolo Parallelo”支持。子项目2。电邮:codenotti@iei.pi .cnr.it ~加州大学伯克利分校计算机科学部，加州伯克利94720。由GTE、斯伦贝谢奖学金和NSF基金CCR88-13632资助。
{"title":"Checking approximate computations over the reals","authors":"S. Ar, M. Blum, B. Codenotti, P. Gemmell","doi":"10.1145/167088.167288","DOIUrl":"https://doi.org/10.1145/167088.167288","url":null,"abstract":"Checking Approximate Computations over the Reals S. Ar” M. Blumt B. Codenotti$ P. Gemmell~ This paper provides the first systematic investigation of checking approximate numerical computations over subsets of the reals. In most cases, approximate checking is more challenging than exact checking. Problem conditioning, i.e., the measure of sensitivity of the output to slight changes in the input, and the presence of approximate ion parameters foil the direct transformation of many exact checkers to the approximate setting. Furthermore, approximate checking over the reals is complicated by the lack of nice finite field properties such as the existence of a samplable distribution which is invariant under addition or multiplication by a scalar. We overcome the above problems by using such techniques as testing and checking over similar but distinct distributions, using functions’ random and downward self-reducibility properties, and taking advantage of the small variance of the sum of independent identically distributed random variables. We provide approximate checkers for a variety of computations, including matrix multiplication, linear system solution, matrix inversion, and computation of the determinant. We also present an approximate version of Beigel’s trick and extend the approximate linear self tester/corrector of [8] and the trigonometric selftester/corrector of [5] to more general computations. *Department of Computer Science, Princeton University, Princeton, NJ 08544-2087. Supported by NSF PYI grant CCR9057486 and a grant from MITL. t Computer Science Division, UC Berkeley, Berkeley, CA 94720, and International Computer Science Institute, Berkeley CA 94704. Supported by NSF grant CCR88-13632. t International Computer Science Institute, Berkeley, CA 94704, and IEI-CNR, Piss (Italy). Partially supported by the “ Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo”. Subproject 2. e-mail: codenotti@iei.pi .cnr.it ~Computer Science Division, UC Berkeley, Berkeley, CA 94720. Supported by GTE, Schlumberger Fellowships, and by NSF grant CCR88-13632.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114968879","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}
This paper considers the problem of scheduling dynamic parallel computations to achieve linear speedup without using significantly more space per processor than that required for a single-processor execution. Utilizing a new graph-theoretic model of multithreaded computation, execution efficiency is quantified by three important measures: T1 is the time required for executing the computation on a 1 processor, $T_infty$ is the time required by an infinite number of processors, and S1 is the space required to execute the computation on a 1 processor. A computation executed on P processors is time-efficient if the time is $O(T_1/P + T_infty)$, that is, it achieves linear speedup when $P=O(T_1/T_infty)$, and it is space-efficient if it uses O(S1P) total space, that is, the space per processor is within a constant factor of that required for a 1-processor execution. The first result derived from this model shows that there exist multithreaded computations such that no execution schedule can simultaneously achieve efficient time and efficient space. But by restricting attention to "strict" computations---those in which all arguments to a procedure must be available before the procedure can be invoked---much more positive results are obtainable. Specifically, for any strict multithreaded computation, a simple online algorithm can compute a schedule that is both time-efficient and space-efficient. Unfortunately, because the algorithm uses a global queue, the overhead of computing the schedule can be substantial. This problem is overcome by a decentralized algorithm that can compute and execute a P-processor schedule online in expected time $O(T_1/P + T_inftylg P)$ and worst-case space $O(S_1Plg P)$, including overhead costs.
{"title":"Space-efficient scheduling of multithreaded computations","authors":"R. Blumofe, C. Leiserson","doi":"10.1145/167088.167196","DOIUrl":"https://doi.org/10.1145/167088.167196","url":null,"abstract":"This paper considers the problem of scheduling dynamic parallel computations to achieve linear speedup without using significantly more space per processor than that required for a single-processor execution. Utilizing a new graph-theoretic model of multithreaded computation, execution efficiency is quantified by three important measures: T1 is the time required for executing the computation on a 1 processor, $T_infty$ is the time required by an infinite number of processors, and S1 is the space required to execute the computation on a 1 processor. A computation executed on P processors is time-efficient if the time is $O(T_1/P + T_infty)$, that is, it achieves linear speedup when $P=O(T_1/T_infty)$, and it is space-efficient if it uses O(S1P) total space, that is, the space per processor is within a constant factor of that required for a 1-processor execution. \u0000The first result derived from this model shows that there exist multithreaded computations such that no execution schedule can simultaneously achieve efficient time and efficient space. But by restricting attention to \"strict\" computations---those in which all arguments to a procedure must be available before the procedure can be invoked---much more positive results are obtainable. Specifically, for any strict multithreaded computation, a simple online algorithm can compute a schedule that is both time-efficient and space-efficient. Unfortunately, because the algorithm uses a global queue, the overhead of computing the schedule can be substantial. This problem is overcome by a decentralized algorithm that can compute and execute a P-processor schedule online in expected time $O(T_1/P + T_inftylg P)$ and worst-case space $O(S_1Plg P)$, including overhead costs.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"138 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115691658","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}
In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.
{"title":"A linear time algorithm for finding tree-decompositions of small treewidth","authors":"H. Bodlaender","doi":"10.1145/167088.167161","DOIUrl":"https://doi.org/10.1145/167088.167161","url":null,"abstract":"In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122170489","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}
E. Kushilevitz, Y. Mansour, M. Rabin, David Zuckerman
We establish, for the rst time, lower bounds for randomized mutual exclusion algorithms (with a read-modify-write operation). Our main result is that a constant-size shared variable cannot guarantee strong fairness, even if randomization is allowed. In fact, we prove a lower bound of ›(log logn) bits on the size of the shared variable, which is also tight. We investigate weaker fairness conditions and derive tight (upper and lower) bounds for them as well. Surprisingly, it turns out that slightly weakening the fairness condition results in an exponential reduction in the size of the required shared variable. Our lower bounds rely on an analysis of Markov chains that may be of interest on its own and may have applications elsewhere.
{"title":"Lower bounds for randomized mutual exclusion","authors":"E. Kushilevitz, Y. Mansour, M. Rabin, David Zuckerman","doi":"10.1145/167088.167139","DOIUrl":"https://doi.org/10.1145/167088.167139","url":null,"abstract":"We establish, for the rst time, lower bounds for randomized mutual exclusion algorithms (with a read-modify-write operation). Our main result is that a constant-size shared variable cannot guarantee strong fairness, even if randomization is allowed. In fact, we prove a lower bound of ›(log logn) bits on the size of the shared variable, which is also tight. We investigate weaker fairness conditions and derive tight (upper and lower) bounds for them as well. Surprisingly, it turns out that slightly weakening the fairness condition results in an exponential reduction in the size of the required shared variable. Our lower bounds rely on an analysis of Markov chains that may be of interest on its own and may have applications elsewhere.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"162 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133117804","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}
We show that for every family of arithmetic circuits of polynomial size and degree over the algebra (Z*, max, concat ), there is an equivalent family of arithmetic circuits of depth log2 n. (The depth can be reduced to log n if unbounded fan-in is allowed.) This is the first depth-reduction result for arithmetic circuits Olrer a nonco~utative semiring, and it complements the lower bounds of [Ni91, K090] showing that depth reduction cannot be done in the general noncommutative setting. The (max,concat) semiring is of interest, because it characterizes certain classes of optimization problems [AJ92, Vi91]. In particular, our results show that OptSACi is contained in AC1. We also prove other results relating Boolean and arithmetic circuit complexity. We show that ACl has no more power than arithmetic circuits of polynomial size and degree n“(log log’) (improving the trivial bound of nOIIOg‘)). Connections are drawn between TCl and arithmetic circuits of polynomial size and degree.
{"title":"Depth reduction for noncommutative arithmetic circuits","authors":"E. Allender, Jia Jiao","doi":"10.1145/167088.167226","DOIUrl":"https://doi.org/10.1145/167088.167226","url":null,"abstract":"We show that for every family of arithmetic circuits of polynomial size and degree over the algebra (Z*, max, concat ), there is an equivalent family of arithmetic circuits of depth log2 n. (The depth can be reduced to log n if unbounded fan-in is allowed.) This is the first depth-reduction result for arithmetic circuits Olrer a nonco~utative semiring, and it complements the lower bounds of [Ni91, K090] showing that depth reduction cannot be done in the general noncommutative setting. The (max,concat) semiring is of interest, because it characterizes certain classes of optimization problems [AJ92, Vi91]. In particular, our results show that OptSACi is contained in AC1. We also prove other results relating Boolean and arithmetic circuit complexity. We show that ACl has no more power than arithmetic circuits of polynomial size and degree n“(log log’) (improving the trivial bound of nOIIOg‘)). Connections are drawn between TCl and arithmetic circuits of polynomial size and degree.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133042927","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}
We present an O(n) time algorithm for computing row-wise maxima or minima of an implicit, totally monotone nn matrix whose entries represent shortest-path distances between pairs of vertices in a simple polygon. We apply this result to derive improved algorithms for several well- known problems in computational geometry. Most prominently, we obtain linear-time algorithms for computing the geodesic diameter, all farthest neighbors, and external farthest neighbors of a simple polygon, improving the previous best result by a factor of O(logn) in each case.
{"title":"Matrix searching with the shortest path metric","authors":"J. Hershberger, S. Suri","doi":"10.1145/167088.167220","DOIUrl":"https://doi.org/10.1145/167088.167220","url":null,"abstract":"We present an O(n) time algorithm for computing row-wise maxima or minima of an implicit, totally monotone nn matrix whose entries represent shortest-path distances between pairs of vertices in a simple polygon. We apply this result to derive improved algorithms for several well- known problems in computational geometry. Most prominently, we obtain linear-time algorithms for computing the geodesic diameter, all farthest neighbors, and external farthest neighbors of a simple polygon, improving the previous best result by a factor of O(logn) in each case.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123659012","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}
We investigate cryptographic lower bounds on the learnability of Boolean formulas and constant depth circuits on the {niform distribution and other specifi; distributions. We first show that weakly learning Boolean formulas and constant depth threshold circuits with membership queries on the uniform distribution in polynomial time is as hard as factoring Blum integers (or inverting RSA, or deciding 1? quadratic residuosity . We formalize the notion of a trivially learnable distri ution and extend these hardness results to all non-trivial distributions. Moreover, we show that under appropriate assumptions on the hardness of factoring, the learnability of Boolean formulas and constant depth threshold circuits on any distribution is characterized by the distribution’s Renyi entropy. Furthermore, we show that a sub-exponential lower bound for factoring implies a Q(2’Og@ ‘‘) lower bound (for some constant ~) for learning Boolean circuits of depth d on the uniform distribution (with membership queries), which matches the upper bound of Linial, M ansour, and Nisan [19]. From this we conclude that, assuming such a lower bou-nd for factoring, there is no O(npOLy 10gn ) algorithm to learn all of ACO on the uniform distribution. We observe that, under cryptographic assumptions, all our bounds can be used to establish trade~trs between the running time and the number of samples necessary to learn.
研究了布尔公式和等深度电路在均匀分布和其他特殊情况下的可学习性的密码下界;分布。我们首先证明，弱学习布尔公式和具有多项式时间内均匀分布上的成员查询的等深度阈值电路与分解Blum整数(或逆RSA，或决定1?二次残差。我们形式化了平凡可学分布的概念，并将这些结果推广到所有的非平凡分布。此外，我们还证明了在适当的分解难度假设下，布尔公式和等深度阈值电路在任何分布上的可学习性都是由分布的Renyi熵表征的。此外，我们证明了分解的次指数下界意味着在均匀分布上学习深度为d的布尔电路的Q(2'Og@”)下界(对于某些常数~)，它与Linial, M ansour和Nisan的上界相匹配[19]。由此我们得出结论，假设因式分解的下限是这样的，不存在O(npOLy 10gn)算法来学习均匀分布上的所有ACO。我们观察到，在密码学假设下，我们所有的边界都可以用来建立运行时间和学习所需样本数量之间的交易区间。
{"title":"Cryptographic hardness of distribution-specific learning","authors":"M. Kharitonov","doi":"10.1145/167088.167197","DOIUrl":"https://doi.org/10.1145/167088.167197","url":null,"abstract":"We investigate cryptographic lower bounds on the learnability of Boolean formulas and constant depth circuits on the {niform distribution and other specifi; distributions. We first show that weakly learning Boolean formulas and constant depth threshold circuits with membership queries on the uniform distribution in polynomial time is as hard as factoring Blum integers (or inverting RSA, or deciding 1? quadratic residuosity . We formalize the notion of a trivially learnable distri ution and extend these hardness results to all non-trivial distributions. Moreover, we show that under appropriate assumptions on the hardness of factoring, the learnability of Boolean formulas and constant depth threshold circuits on any distribution is characterized by the distribution’s Renyi entropy. Furthermore, we show that a sub-exponential lower bound for factoring implies a Q(2’Og@ ‘‘) lower bound (for some constant ~) for learning Boolean circuits of depth d on the uniform distribution (with membership queries), which matches the upper bound of Linial, M ansour, and Nisan [19]. From this we conclude that, assuming such a lower bou-nd for factoring, there is no O(npOLy 10gn ) algorithm to learn all of ACO on the uniform distribution. We observe that, under cryptographic assumptions, all our bounds can be used to establish trade~trs between the running time and the number of samples necessary to learn.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115053486","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}
In this paper, we study the problem of learning in the presence of classification noise in the probabilistic learning model of Valiant and its variants. In order to identify the class of “robust” learning algorithms in the most general way, we formalize a new but related model of learning from statistical queries. Intuitively, in this model, a learning algorithm is forbidden to examine individual examples of the unknown target function, but is given access to an oracle providing estimates of probabilities over the sample space of random examples. One of our main results shows that any class of functions learnable from statistical queries is in fact learnable with classification noise in Valiant’s model, with a noise rate approaching the informationtheoretic barrier of 1/2. We then demonstrate the generality of the statistical query model, showing that practically every class learnable in Valiant’s model and its variants can also be learned in the new model (and thus can be learned in the presence of noise). A notable exception to this statement is the class of parity functions, which we prove is not learnable from statistical queries, and for which no noise-tolerant algorithm is known.
{"title":"Efficient noise-tolerant learning from statistical queries","authors":"M. Kearns","doi":"10.1145/167088.167200","DOIUrl":"https://doi.org/10.1145/167088.167200","url":null,"abstract":"In this paper, we study the problem of learning in the presence of classification noise in the probabilistic learning model of Valiant and its variants. In order to identify the class of “robust” learning algorithms in the most general way, we formalize a new but related model of learning from statistical queries. Intuitively, in this model, a learning algorithm is forbidden to examine individual examples of the unknown target function, but is given access to an oracle providing estimates of probabilities over the sample space of random examples. One of our main results shows that any class of functions learnable from statistical queries is in fact learnable with classification noise in Valiant’s model, with a noise rate approaching the informationtheoretic barrier of 1/2. We then demonstrate the generality of the statistical query model, showing that practically every class learnable in Valiant’s model and its variants can also be learned in the new model (and thus can be learned in the presence of noise). A notable exception to this statement is the class of parity functions, which we prove is not learnable from statistical queries, and for which no noise-tolerant algorithm is known.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125425398","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}
In this paper, we precisely characterize the randomness complexity of the unique element isolation problem, a crucial step in the RNC algorithm for perfect matching due to Mulmuley, Vazirani & Vazirani[21] and in several other applications. Given a set $S$ and an unknown family $cal F subseteq$ $2^{S}$ with $|cal F| leq$ $Z$, we present a scheme to assign polynomially bounded weights to the elements of $S$, using only $O(log Z + log |S|)$ ransom bits, such that the minimum weight set in $cal F$ is unique with high probability. This generalizes and improves the results of Mulmuley, Vazirani & Vazirani who give a scheme which uses $O(S log S)$ random bits independent of $Z$. We also prove a matching lower bound for the randomness complexity of this problem. This new weight assignment scheme yields a randomness-efficient $RNC^{2}$ algorithm for perfect matching which uses $O(log Z + log n)$ random bits where $Z$ is any given upper bound on the number of perfect matchings in the input graph. This generalizes the result of Grigoriev & Karpinski[11] who present an $NC^{3}$ algorithm when $Z$ is polynomially bounded and also gives an improvement on the running time in this case. The worst-case randomness complexity of our algorithm is $O(n log (m/n))$ random bits, as opposed to the previous bound of $O(m log n)$ bits. Our technique also gives randomness-efficient solutions for several problems in which the unique element isolation tool is used, such as $RNC$ algorithms for variants of matching and basic problems on linear matroids such as matroid intersection and matroid matching. We also obtain a randomness-efficient alternative to the random reduction from $SAT$ to $USAT$, the language of uniquely satisfiable formulas, due to Valiant and Vazirani[32]. This reduction can be derandomized in the case of languages in $F ew P$ to yield new proofs of the results $F ew P subseteq oplus P$ and $F ew P subseteq C_{=} P$.
由于Mulmuley, Vazirani ＆ Vazirani[21]以及其他一些应用，我们精确地描述了唯一元素隔离问题的随机复杂性，这是RNC算法中完美匹配的关键一步。给定一个集合$S$和一个未知族$cal F subseteq$$2^{S}$和一个未知族$|cal F| leq$$Z$，我们提出了一种方案，为$S$的元素分配多项式有界权值，仅使用$O(log Z + log |S|)$勒索位，使得$cal F$中的最小权值集具有高概率唯一性。这推广并改进了Mulmuley, Vazirani ＆ Vazirani的结果，他们给出了一个使用$O(S log S)$独立于$Z$的随机比特的方案。我们还证明了该问题随机复杂度的匹配下界。这个新的权重分配方案产生了一个随机高效的$RNC^{2}$完美匹配算法，它使用$O(log Z + log n)$随机位，其中$Z$是输入图中完美匹配数量的任意给定上界。这推广了Grigoriev ＆ Karpinski[11]的结果，他们在$Z$为多项式有界时提出了$NC^{3}$算法，并改进了这种情况下的运行时间。我们算法的最坏情况随机复杂度是$O(n log (m/n))$随机位，而不是之前的$O(m log n)$位。我们的技术还为使用唯一单元隔离工具的几个问题提供了随机高效的解决方案，例如用于匹配变体的$RNC$算法和线性拟阵的基本问题，如拟阵相交和拟阵匹配。由于Valiant和Vazirani[32]，我们还获得了从$SAT$到$USAT$的随机约简的随机高效替代方案，这是唯一可满足公式的语言。对于$F ew P$中的语言，可以对这种约简进行非随机化，以产生对结果$F ew P subseteq oplus P$和$F ew P subseteq C_{=} P$的新证明。
{"title":"Randomness-optimal unique element isolation, with applications to perfect matching and related problems","authors":"Suresh Chari, P. Rohatgi, A. Srinivasan","doi":"10.1145/167088.167213","DOIUrl":"https://doi.org/10.1145/167088.167213","url":null,"abstract":"In this paper, we precisely characterize the randomness complexity of the unique element isolation problem, a crucial step in the RNC algorithm for perfect matching due to Mulmuley, Vazirani & Vazirani[21] and in several other applications. Given a set $S$ and an unknown family $cal F subseteq$ $2^{S}$ with $|cal F| leq$ $Z$, we present a scheme to assign polynomially bounded weights to the elements of $S$, using only $O(log Z + log |S|)$ ransom bits, such that the minimum weight set in $cal F$ is unique with high probability. This generalizes and improves the results of Mulmuley, Vazirani & Vazirani who give a scheme which uses $O(S log S)$ random bits independent of $Z$. We also prove a matching lower bound for the randomness complexity of this problem. This new weight assignment scheme yields a randomness-efficient $RNC^{2}$ algorithm for perfect matching which uses $O(log Z + log n)$ random bits where $Z$ is any given upper bound on the number of perfect matchings in the input graph. This generalizes the result of Grigoriev & Karpinski[11] who present an $NC^{3}$ algorithm when $Z$ is polynomially bounded and also gives an improvement on the running time in this case. The worst-case randomness complexity of our algorithm is $O(n log (m/n))$ random bits, as opposed to the previous bound of $O(m log n)$ bits. Our technique also gives randomness-efficient solutions for several problems in which the unique element isolation tool is used, such as $RNC$ algorithms for variants of matching and basic problems on linear matroids such as matroid intersection and matroid matching. We also obtain a randomness-efficient alternative to the random reduction from $SAT$ to $USAT$, the language of uniquely satisfiable formulas, due to Valiant and Vazirani[32]. This reduction can be derandomized in the case of languages in $F ew P$ to yield new proofs of the results $F ew P subseteq oplus P$ and $F ew P subseteq C_{=} P$.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"131 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130778428","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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