A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.<>
{"title":"On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines","authors":"L. Blum, M. Shub, S. Smale","doi":"10.1109/SFCS.1988.21955","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21955","url":null,"abstract":"A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121706220","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}
G.A. Dirac's classical theorem (1952) asserts that if every vertex of a graph G on n vertices has degree at least n/2, the G has a Hamiltonian cycle. A fast parallel algorithm on a concurrent-read-exclusive-write parallel random-access machine (CREW PRAM) is given to find a Hamiltonian cycle in such graphs. The algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. It works in O(log/sup 4/n) parallel time and uses linear number of processors on a CREW PRAM. It is also proved that a perfect matching in dense graphs can be found in NC/sup 2/. The cost of improved time is a quadratic number of processors. It is also proved that finding an NC algorithm for perfect matching in slightly less dense graphs is as hard as the same problem for all graphs, and the problem of finding a Hamiltonian cycle becomes NP-complete.<>
{"title":"Optimal parallel algorithm for the Hamiltonian cycle problem on dense graphs","authors":"E. Dahlhaus, P. Hajnal, Marek Karpinski","doi":"10.1109/SFCS.1988.21936","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21936","url":null,"abstract":"G.A. Dirac's classical theorem (1952) asserts that if every vertex of a graph G on n vertices has degree at least n/2, the G has a Hamiltonian cycle. A fast parallel algorithm on a concurrent-read-exclusive-write parallel random-access machine (CREW PRAM) is given to find a Hamiltonian cycle in such graphs. The algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. It works in O(log/sup 4/n) parallel time and uses linear number of processors on a CREW PRAM. It is also proved that a perfect matching in dense graphs can be found in NC/sup 2/. The cost of improved time is a quadratic number of processors. It is also proved that finding an NC algorithm for perfect matching in slightly less dense graphs is as hard as the same problem for all graphs, and the problem of finding a Hamiltonian cycle becomes NP-complete.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115444643","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}
Novel algorithms for computation in permutation groups are presented. They provide an order-of-magnitude improvement in the worst-case analysis of the basic permutation-group problems, including membership testing and computing the order of the group. For deeper questions about the group, including finding composition factors, an improvement of up to four orders of magnitude is realized. These and other essential investigations are all accomplished in O(n/sup 4/log/sup c/n) time. The approach is distinguished by its recognition and use of the intrinsic structure of the group at hand.<>
{"title":"Fast management of permutation groups","authors":"László Babai, E. Luks, Á. Seress","doi":"10.1109/SFCS.1988.21943","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21943","url":null,"abstract":"Novel algorithms for computation in permutation groups are presented. They provide an order-of-magnitude improvement in the worst-case analysis of the basic permutation-group problems, including membership testing and computing the order of the group. For deeper questions about the group, including finding composition factors, an improvement of up to four orders of magnitude is realized. These and other essential investigations are all accomplished in O(n/sup 4/log/sup c/n) time. The approach is distinguished by its recognition and use of the intrinsic structure of the group at hand.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"270 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123269942","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}
Randomized algorithms for approximating the number of perfect matchings in a graph are considered. An algorithm that is a natural simplification of one suggested and analyzed previously is introduced and analyzed. One of the key ideas is to view the analysis from a geometric perspective: it is proved that for any graph G the k-slice of the well-known Edmonds matching polytope has magnification 1. For a bipartite graph G=(U, V, E), mod U mod = mod V mod =n, with d edge-disjoint perfect matchings, it is proved that the ratio of the number of almost perfect matchings to the number of perfect matchings is at most n/sup 3n/d/. For any constant alpha >0 this yields a a fully polynomial randomized algorithm for approximating the number of perfect matchings in bipartite graphs with d>or= alpha n. Moreover, for some constant c>0 it is the fastest known approximation algorithm for bipartite graphs with d>or= clog n.<>
{"title":"Polytopes, permanents and graphs with large factors","authors":"P. Dagum, M. Luby, M. Mihail, U. Vazirani","doi":"10.1109/SFCS.1988.21957","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21957","url":null,"abstract":"Randomized algorithms for approximating the number of perfect matchings in a graph are considered. An algorithm that is a natural simplification of one suggested and analyzed previously is introduced and analyzed. One of the key ideas is to view the analysis from a geometric perspective: it is proved that for any graph G the k-slice of the well-known Edmonds matching polytope has magnification 1. For a bipartite graph G=(U, V, E), mod U mod = mod V mod =n, with d edge-disjoint perfect matchings, it is proved that the ratio of the number of almost perfect matchings to the number of perfect matchings is at most n/sup 3n/d/. For any constant alpha >0 this yields a a fully polynomial randomized algorithm for approximating the number of perfect matchings in bipartite graphs with d>or= alpha n. Moreover, for some constant c>0 it is the fastest known approximation algorithm for bipartite graphs with d>or= clog n.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130864953","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}
It was conjectured by A. Borodin et al. that to solve the element distinctness problem requires TS= Omega (n/sup 2/) on a comparison-based branching program using space S and time T, which, if true, would be close to optimal since TS=O(n/sup 2/ log n) is achievable. They showed recently (1987) that TS= Omega (n/sup 3/2/(log n)/sup 1/2/). The author shows a near-optimal tradeoff TS= Omega (n/sup 2- epsilon (n)/), where epsilon (n)=O(1/(log n)/sup 1/2/).<>
{"title":"Near-optimal time-space tradeoff for element distinctness","authors":"A. Yao","doi":"10.1109/SFCS.1988.21925","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21925","url":null,"abstract":"It was conjectured by A. Borodin et al. that to solve the element distinctness problem requires TS= Omega (n/sup 2/) on a comparison-based branching program using space S and time T, which, if true, would be close to optimal since TS=O(n/sup 2/ log n) is achievable. They showed recently (1987) that TS= Omega (n/sup 3/2/(log n)/sup 1/2/). The author shows a near-optimal tradeoff TS= Omega (n/sup 2- epsilon (n)/), where epsilon (n)=O(1/(log n)/sup 1/2/).<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"264 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131299109","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}
An algorithm for solving linear programming problems is given. The expected number of arithmetic operations required by the algorithm is given. The expectation is with respect to the random choices made by the algorithm, and the bound holds for any given input. The technique can be extended to other convex programming problems.<>
{"title":"A Las Vegas algorithm for linear programming when the dimension is small","authors":"K. Clarkson","doi":"10.1109/SFCS.1988.21961","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21961","url":null,"abstract":"An algorithm for solving linear programming problems is given. The expected number of arithmetic operations required by the algorithm is given. The expectation is with respect to the random choices made by the algorithm, and the bound holds for any given input. The technique can be extended to other convex programming problems.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121656434","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}
A formal study of three-dimensional topological graph theory is initiated. The problem of deciding whether a graph can be embedded in 3-space so that no collection of vertex-disjoint cycles is topologically linked is considered first. The Robertson-Seymour Theory of Graph Minors is applicable to this problem and guarantees the existence of an O(n/sup 3/) algorithm for the decision problem. However, not even a finite-time decision procedure was known for this problem. A small set of forbidden minors for linkless embeddable graphs is exhibited, and it is shown that any graph with these minors cannot be embedded without linked cycles. It is further established that any graph that does not contain these minors is embeddable without linked cycles. Thus, an O(n/sup 3/) algorithm for the decision problem is demonstrated. It is believed that the proof technique will lead to an algorithm for actually embedding a graph, provided it does not contain the forbidden minors.<>
{"title":"Constructive results from graph minors: linkless embeddings","authors":"R. Motwani, A. Raghunathan, H. Saran","doi":"10.1109/SFCS.1988.21956","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21956","url":null,"abstract":"A formal study of three-dimensional topological graph theory is initiated. The problem of deciding whether a graph can be embedded in 3-space so that no collection of vertex-disjoint cycles is topologically linked is considered first. The Robertson-Seymour Theory of Graph Minors is applicable to this problem and guarantees the existence of an O(n/sup 3/) algorithm for the decision problem. However, not even a finite-time decision procedure was known for this problem. A small set of forbidden minors for linkless embeddable graphs is exhibited, and it is shown that any graph with these minors cannot be embedded without linked cycles. It is further established that any graph that does not contain these minors is embeddable without linked cycles. Thus, an O(n/sup 3/) algorithm for the decision problem is demonstrated. It is believed that the proof technique will lead to an algorithm for actually embedding a graph, provided it does not contain the forbidden minors.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"21 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116685208","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}
The decision problem for the existential theory of the reals is the problem of deciding if the set (x in R/sup n/; P(x) is nonempty, where P(x) is a predicate which is a Boolean function of atomic predicates either of which is a Boolean function of atomic predicates either of the form f/sub i/(x)>or=0 or f/sub j/(x)>, the f's being real polynomials. An algorithm is presented for deciding the existential theory of the reals that simultaneously achieves the best known time and space bounds. The time bound for the algorithm is slightly better than any previous bound.<>
实数存在论的判定问题是判定集合(x in R/sup n/;P(x)是非空的，其中P(x)是一个谓词，它是原子谓词的布尔函数，其中任何一个都是原子谓词的布尔函数，形式为f/下标i/(x)>或=0或f/下标j/(x)>， f是实多项式。提出了一种确定实数存在理论的算法，该算法同时达到了已知的最佳时间和空间界限。该算法的时间限制比之前的任何时间限制都略好。
{"title":"A faster PSPACE algorithm for deciding the existential theory of the reals","authors":"J. Renegar","doi":"10.1109/SFCS.1988.21945","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21945","url":null,"abstract":"The decision problem for the existential theory of the reals is the problem of deciding if the set (x in R/sup n/; P(x) is nonempty, where P(x) is a predicate which is a Boolean function of atomic predicates either of which is a Boolean function of atomic predicates either of the form f/sub i/(x)>or=0 or f/sub j/(x)>, the f's being real polynomials. An algorithm is presented for deciding the existential theory of the reals that simultaneously achieves the best known time and space bounds. The time bound for the algorithm is slightly better than any previous bound.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131273562","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}
Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets.<>
{"title":"The influence of variables on Boolean functions","authors":"J. Kahn, G. Kalai, N. Linial","doi":"10.1109/SFCS.1988.21923","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21923","url":null,"abstract":"Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets.<<ETX>>","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128314081","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}
The following problem, is considered: given a robot system find a minimal-time trajectory from a start position and velocity to a goal position and velocity, while avoiding obstacles and respecting dynamic constraints on velocity and acceleration. The simplified case of a point mass under Newtonian mechanics together with velocity and acceleration bounds is considered. The point must be flown from a start to a goal, amid 2-D or 3-D polyhedral obstacles. While exact solutions to this problem are not known, the first provably good approximation algorithm is given and shown to run in polynomial time.
{"title":"On the complexity of kinodynamic planning","authors":"J. Canny, B. Donald, J. Reif, P. Xavier","doi":"10.1109/SFCS.1988.21947","DOIUrl":"https://doi.org/10.1109/SFCS.1988.21947","url":null,"abstract":"The following problem, is considered: given a robot system find a minimal-time trajectory from a start position and velocity to a goal position and velocity, while avoiding obstacles and respecting dynamic constraints on velocity and acceleration. The simplified case of a point mass under Newtonian mechanics together with velocity and acceleration bounds is considered. The point must be flown from a start to a goal, amid 2-D or 3-D polyhedral obstacles. While exact solutions to this problem are not known, the first provably good approximation algorithm is given and shown to run in polynomial time.","PeriodicalId":113255,"journal":{"name":"[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134338182","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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