In many CAD systems for VLSI design the specification of a layout is internally represented by a set of geometric constraints that take the form of linear inequalities between pairs of layout components. Some of the constraints may be explicitly stated by the circuit designer. Others are internally generated by the CAD system, using the design rules of the fabrication process. Layout compaction is then equivalent to finding a minimum area layout satisfying all constraints. We discuss the complexity of the constraint resolution problem arising in this context. Hereby we allow circuits to be specified hierarchically. The complexity of the constraint resolution is then measured in terms of the length of the hierarchical specification. We show the following results: 1. It is decidable in polynomial (cubic) time whether a given hierarchical layout specification yields a consistent set of geometric constraints. The size of minimum area layouts satisfying the constraints can also be determined in cubic time. 2. For every layout specification that is consistent a hierarchical description L of a minimum area layout can be computed in polynomial time in the length of L. 3. There is a consistent layout specification with the following property: No layout satisfying the constraints is concise, i.e., every hierarchical layout description consistent with the specification has a length which grows exponentially in the length of the specification. 4. We define a subclass of so-called well-formed layout specifications. Each well-formed specification has a concise layout, which can be hierarchically described in linear space. Such a layout can be found in polynomial time. However, it is in general not a minimum area layout. Indeed, there is a consistent well-formed specification all of whose minimum area layouts are inconcise,.i.e., need exponential space to be described.
{"title":"The complexity of compacting hierarchically specified layouts of integrated circuits","authors":"Thomas Lengauer","doi":"10.1109/SFCS.1982.92","DOIUrl":"https://doi.org/10.1109/SFCS.1982.92","url":null,"abstract":"In many CAD systems for VLSI design the specification of a layout is internally represented by a set of geometric constraints that take the form of linear inequalities between pairs of layout components. Some of the constraints may be explicitly stated by the circuit designer. Others are internally generated by the CAD system, using the design rules of the fabrication process. Layout compaction is then equivalent to finding a minimum area layout satisfying all constraints. We discuss the complexity of the constraint resolution problem arising in this context. Hereby we allow circuits to be specified hierarchically. The complexity of the constraint resolution is then measured in terms of the length of the hierarchical specification. We show the following results: 1. It is decidable in polynomial (cubic) time whether a given hierarchical layout specification yields a consistent set of geometric constraints. The size of minimum area layouts satisfying the constraints can also be determined in cubic time. 2. For every layout specification that is consistent a hierarchical description L of a minimum area layout can be computed in polynomial time in the length of L. 3. There is a consistent layout specification with the following property: No layout satisfying the constraints is concise, i.e., every hierarchical layout description consistent with the specification has a length which grows exponentially in the length of the specification. 4. We define a subclass of so-called well-formed layout specifications. Each well-formed specification has a concise layout, which can be hierarchically described in linear space. Such a layout can be found in polynomial time. However, it is in general not a minimum area layout. Indeed, there is a consistent well-formed specification all of whose minimum area layouts are inconcise,.i.e., need exponential space to be described.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115927956","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 describe a data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time. Both the data structure and the query algorithm are easy to implement.
{"title":"Storing a sparse table with O(1) worst case access time","authors":"M. Fredman, J. Komlos, E. Szemerédi","doi":"10.1145/828.1884","DOIUrl":"https://doi.org/10.1145/828.1884","url":null,"abstract":"We describe a data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time. Both the data structure and the query algorithm are easy to implement.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116941664","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}
Algorithms are given that compute maximum flows in planar directed networks either in O((logn)3) parallel time using O(n4) processors or O((logn)2) parallel time using O(n6) processors. The resource consumption of these algorithms is dominated by the cost of finding the value of a maximum flow. When such a value is given, or when the computation is on an undirected network, the bound is O((logn)2) time using O(n4) processors. No efficient parallel algorithm is known for the maximum flow problem in general networks.
{"title":"Parallel algorithms for minimum cuts and maximum flows in planar networks","authors":"Donald B. Johnson, S. M. Venkatesan","doi":"10.1145/31846.31849","DOIUrl":"https://doi.org/10.1145/31846.31849","url":null,"abstract":"Algorithms are given that compute maximum flows in planar directed networks either in O((logn)3) parallel time using O(n4) processors or O((logn)2) parallel time using O(n6) processors. The resource consumption of these algorithms is dominated by the cost of finding the value of a maximum flow. When such a value is given, or when the computation is on an undirected network, the bound is O((logn)2) time using O(n4) processors. No efficient parallel algorithm is known for the maximum flow problem in general networks.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124656713","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}
Suppose a polynomial P(x), (where x is the column matrix of the indeterminates x1.~.x), has a symmetry analogous to those of the de~ermiRant, whereby taking a certain linear combination of the The purpose of the paper is to define the set of symmetries of a polynomial, explore its structure, and identify the computationally useful members; a method of computing the latter symmetries is presented. It is shown how the set of symmetries determines whether or not Gaussian style elimination or transformation style algorithms can aid computation. To this end a robust notion of the "dimen-sion" of a polynomial is defined, yielding a tech~ nique for proving negative results in complexity. Let ~'¥.. and z be n x 1 column matrices. z is the wrappeg convolution of ~ and ¥.. iff '1j z. = LX. Y.. d • z is the Hademard J i=l ~ ~+J mo n product (or pairwise product) of ~ and ¥.."iff '1j z. = x. • y j • An efficient technique for eval-J J uating a wrapped convolution [1 p.254] relies upon transforming the convolution into a Hademard product by means of the discrete "Fburier transform. The question "can the permanent be transformed analogously in a way that may assist faster computation?" is considered, and answered in part. In order to construct such a scheme whereby P can be evaluated at any point ~, it must be possible for the symmetry (T, t) to depend upon x, in order to introduce zeros into Tx + t. (In practice several successive transformations may be made, introducing successively more zeros whilst preserving those previously present. Such a scheme constitutes a Gaussian elimination style algorithm for evaluating P) • In order for this to be possible, it is necessary that some of the symmetries of P form a continuum: these continuous symmetries include all of the computationally useful symmetries of P. variables before evaluating P only alters the result by a constant factor plus a constant additive term, (where T is an n x n matrix of constants, t is an n X 1 matrix of constants and k,k' are constants) • Then P could be computed at ~ by computing P at Ta + t, multiplying by k and adding k'. If Ta + t has more components equal to zero than a then there may be some computational advantage in this scheme as compared to evaluating …
假设多项式P(x)(其中x是不定式x1.~.x的列矩阵)具有类似于不定式的对称，本文的目的是定义多项式的对称集，探讨其结构，并找出计算上有用的成员;提出了一种计算后一种对称性的方法。它显示了对称性集如何决定高斯式消去或变换式算法是否可以帮助计算。为此，定义了多项式“维数”的鲁棒概念，从而产生了一种证明复杂性负结果的技术。让~”¥. .z是n × 1列矩阵。Z是~和¥的包裹卷积。i ' 1jz . = LX。Y . .d•z是~和¥的哈德矩阵J i=l ~ ~+J mo n积(或成对积)。计算包裹卷积的一种有效技术[1 p.254]依赖于通过离散的“Fburier变换”将卷积变换成hademad积。“能否以一种有助于更快计算的方式类比地转换永久性对象?”这个问题得到了考虑，并得到了部分回答。为了构造这样一个P可以在任意点~求值的方案，对称(T, T)必须可能依赖于x，以便在Tx + T中引入零。(在实践中，可以进行几个连续的变换，在保留先前存在的同时相继引入更多的零。)这样的方案构成了评估P的高斯消除式算法)•为了使这成为可能，有必要使P的一些对称性形成一个连续体:这些连续的对称性包括所有在计算上有用的P变量的对称性，在计算P之前，P只会通过一个常数因子加上一个常数相加项来改变结果，(其中T是一个n x n的常数矩阵，T是一个n x 1的常数矩阵，k,k'是常数)•然后P可以通过在Ta + T处计算P，乘以k，加上k'来计算~。如果Ta + t等于0的分量比a多，那么在这个方案中可能有一些计算优势，与评估…
{"title":"Generalised symmetries of polynomials in algebraic complexity","authors":"Carl Sturtivant","doi":"10.1109/SFCS.1982.70","DOIUrl":"https://doi.org/10.1109/SFCS.1982.70","url":null,"abstract":"Suppose a polynomial P(x), (where x is the column matrix of the indeterminates x1.~.x), has a symmetry analogous to those of the de~ermiRant, whereby taking a certain linear combination of the The purpose of the paper is to define the set of symmetries of a polynomial, explore its structure, and identify the computationally useful members; a method of computing the latter symmetries is presented. It is shown how the set of symmetries determines whether or not Gaussian style elimination or transformation style algorithms can aid computation. To this end a robust notion of the \"dimen-sion\" of a polynomial is defined, yielding a tech~ nique for proving negative results in complexity. Let ~'¥.. and z be n x 1 column matrices. z is the wrappeg convolution of ~ and ¥.. iff '1j z. = LX. Y.. d • z is the Hademard J i=l ~ ~+J mo n product (or pairwise product) of ~ and ¥..\"iff '1j z. = x. • y j • An efficient technique for eval-J J uating a wrapped convolution [1 p.254] relies upon transforming the convolution into a Hademard product by means of the discrete \"Fburier transform. The question \"can the permanent be transformed analogously in a way that may assist faster computation?\" is considered, and answered in part. In order to construct such a scheme whereby P can be evaluated at any point ~, it must be possible for the symmetry (T, t) to depend upon x, in order to introduce zeros into Tx + t. (In practice several successive transformations may be made, introducing successively more zeros whilst preserving those previously present. Such a scheme constitutes a Gaussian elimination style algorithm for evaluating P) • In order for this to be possible, it is necessary that some of the symmetries of P form a continuum: these continuous symmetries include all of the computationally useful symmetries of P. variables before evaluating P only alters the result by a constant factor plus a constant additive term, (where T is an n x n matrix of constants, t is an n X 1 matrix of constants and k,k' are constants) • Then P could be computed at ~ by computing P at Ta + t, multiplying by k and adding k'. If Ta + t has more components equal to zero than a then there may be some computational advantage in this scheme as compared to evaluating …","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132843656","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}
Two different notions of Byzantine Agreement - immediate and eventually - are defined depending on whether the agreement involves an action to be performed synchronously or not. The lower bounds for time complexity depend on what kind of agreement has to be achieved. All previous algorithms to reach Byzantine Agreement ensure immediate agreement. We present two algorithms that in many cases reach the second type of agreement faster than previously known algorithms showing that there actually is a difference between the two notions: Eventual Byzantine Agreement can be reached earlier than Immediate.
{"title":"'Eventual' is earlier than 'immediate'","authors":"D. Dolev, R. Reischuk, H. Strong","doi":"10.1109/SFCS.1982.51","DOIUrl":"https://doi.org/10.1109/SFCS.1982.51","url":null,"abstract":"Two different notions of Byzantine Agreement - immediate and eventually - are defined depending on whether the agreement involves an action to be performed synchronously or not. The lower bounds for time complexity depend on what kind of agreement has to be achieved. All previous algorithms to reach Byzantine Agreement ensure immediate agreement. We present two algorithms that in many cases reach the second type of agreement faster than previously known algorithms showing that there actually is a difference between the two notions: Eventual Byzantine Agreement can be reached earlier than Immediate.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123128877","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 modification of the ellipsoid algorithm is shown to be capable of testing for satisfiability a system of linear equations and inequalities with integer coefficients of the form Ax = b, x ≥ 0. All the rational arithmetic is performed exactly, without losing polynomiality of the computing time. In case of satisfiability, the approach always provides a rational feasible point. The bulk of the computations consists of a sequence of linear least squares problems, each a rank one modification of the preceding one. The continued fractions jump is used to compute some of the coordinates of a feasible point.
{"title":"The ellipsoid algorithm for linear inequalities in exact arithmetic","authors":"S. Ursic","doi":"10.1109/SFCS.1982.44","DOIUrl":"https://doi.org/10.1109/SFCS.1982.44","url":null,"abstract":"A modification of the ellipsoid algorithm is shown to be capable of testing for satisfiability a system of linear equations and inequalities with integer coefficients of the form Ax = b, x ≥ 0. All the rational arithmetic is performed exactly, without losing polynomiality of the computing time. In case of satisfiability, the approach always provides a rational feasible point. The bulk of the computations consists of a sequence of linear least squares problems, each a rank one modification of the preceding one. The continued fractions jump is used to compute some of the coordinates of a feasible point.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122258515","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 study the family S of rational cones obtained by iterated substitutions from rational cones L1, .., Ln. This family is a semi-group and to every non empty word u defined on the alphabet {L1, ..., Ln}, corresponds a rational cone U of S. We give sufficient conditions for S to be free (U = U′ implies u = u′) and to verify the subpattern property (U ⊂ U′ implies u is a subpattern of u′). We study, more particularly, the case where L1, ..., Ln are bounded rational cones.
我们研究了由有理锥L1，…通过迭代替换得到的有理锥族S。Ln。这个族是一个半群，对于字母{L1，…， Ln}，对应于S的有理锥U，我们给出S是自由的充分条件(U = U '暗示U = U ')，并验证子模式的性质(U≠U '暗示U是U '的子模式)。更具体地说，我们研究L1，…， Ln是有界有理锥。
{"title":"Substitution of bounded rational cone","authors":"J. Beauquier, M. Latteux","doi":"10.1109/SFCS.1982.90","DOIUrl":"https://doi.org/10.1109/SFCS.1982.90","url":null,"abstract":"We study the family S of rational cones obtained by iterated substitutions from rational cones L1, .., Ln. This family is a semi-group and to every non empty word u defined on the alphabet {L1, ..., Ln}, corresponds a rational cone U of S. We give sufficient conditions for S to be free (U = U′ implies u = u′) and to verify the subpattern property (U ⊂ U′ implies u is a subpattern of u′). We study, more particularly, the case where L1, ..., Ln are bounded rational cones.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116712602","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}
Z. Galil, C. Hoffmann, E. Luks, C. Schnorr, M. Weber
The main results of this paper are an O(n3) probabilistic algorithm and an O(n3 log n) deterministic algorithm that test whether two given trivalent graphs are isomorphic. In fact, the algorithms construct the set of all isomorphisms of the two graphs. Variants of these algorithms construct the set of all automorphisms of a trivalent graph. The algorithms make use of some new improved permutation group algorithms that exploit the fact that the groups involved are 2-groups. A remarkable property of the probabilistic algorithm is that it computes Isoe,ei(X,Y), i = 1,...,m, m = O(n) (the set of all isomorhisms φ: X → Y with φ(e)=ei) for the cost of computing the single set Isoe,el(X,Y).
本文的主要结果是一个O(n3)概率算法和一个O(n3 log n)确定性算法，用于检验给定的两个三价图是否同构。实际上，这些算法构造了两个图的所有同构的集合。这些算法的变体构造了一个三价图的所有自同构的集合。该算法利用了一些新的改进的置换群算法，利用了所涉及的群是2群的事实。概率算法的一个显著特性是它计算Isoe,ei(X,Y)， i = 1，…，m, m = O(n) (φ(e)=ei的所有同构的集合φ: X→Y)，计算单个集合Isoe,el(X,Y)的代价。
{"title":"An O(n3 log n) deterministic and an O (n 3) probabilistic isomorphism test for trivalent graphs","authors":"Z. Galil, C. Hoffmann, E. Luks, C. Schnorr, M. Weber","doi":"10.1109/SFCS.1982.62","DOIUrl":"https://doi.org/10.1109/SFCS.1982.62","url":null,"abstract":"The main results of this paper are an O(n3) probabilistic algorithm and an O(n3 log n) deterministic algorithm that test whether two given trivalent graphs are isomorphic. In fact, the algorithms construct the set of all isomorphisms of the two graphs. Variants of these algorithms construct the set of all automorphisms of a trivalent graph. The algorithms make use of some new improved permutation group algorithms that exploit the fact that the groups involved are 2-groups. A remarkable property of the probabilistic algorithm is that it computes Isoe,ei(X,Y), i = 1,...,m, m = O(n) (the set of all isomorhisms φ: X → Y with φ(e)=ei) for the cost of computing the single set Isoe,el(X,Y).","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124763441","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}
Much effort has been devoted in the second half of this century to make precise the notion of Randomness. Let us informally recall one of these definitions due to Kolmogorov []. A sequence of bits A =all a2••.•• at is random if the length of the minimal program outputting A is at least k We remark that the above definition is highly non constructive and rules out the possibility of pseudo random number generators. Also. the length of a program, from a Complexity Theory point of view, is a rather unnatural measure. A more operative definition of Randomness should be pursued in the light of modern Complexity Theory.
{"title":"How to generate cryptographically strong sequences of pseudo random bits","authors":"M. Blum, S. Micali","doi":"10.1109/SFCS.1982.72","DOIUrl":"https://doi.org/10.1109/SFCS.1982.72","url":null,"abstract":"Much effort has been devoted in the second half of this century to make precise the notion of Randomness. Let us informally recall one of these definitions due to Kolmogorov []. A sequence of bits A =all a2••.•• at is random if the length of the minimal program outputting A is at least k We remark that the above definition is highly non constructive and rules out the possibility of pseudo random number generators. Also. the length of a program, from a Complexity Theory point of view, is a rather unnatural measure. A more operative definition of Randomness should be pursued in the light of modern Complexity Theory.","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134290692","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}
Kolmogorov-CoMplexity has turned out to be a very useful tool in proving lower bounds [5], [6], [7]. Here we will give further applications of Kolmogorov-Complexity. Firstly we will greatly simplify two well-known lower bound proofs: (a) the n(n logn/loglog n) lower bound for on-line-multiplication, originally proven in [2], [4], (b) the n(n 2 /log n) lower bound for a time-space trade-off in sorting [9]. We will also improve this bound to n(N 2 loglog N/log N). Secondly we will demonstrate how to use Kolmogorov-Complexity in analysing proba-bilistic algorithms: (c) We analyse in an elementary way the routing algorithm for n-dimensional cubes given in [10]. 1. The concept of Kolmogorov-Complexity Let C be the class of one-dimensional Turing-machines with tape-alphabet {O,I,B}. Let U be an universal machine in C. For w 1 ,w 2 E{O,I}* define the Kolmogorov-Complexity [3] by: K(w 1 {W 2 }: =the length of the shortest o/ 1 s t r i ng (" pro gram I' J p, s uc h t hat U with input pBw2 computes wI and stops. K(w): =K(wlempty string). 45 Because one program p can only generate one word w, we have Fact I : Let w 2 E{O,l}* , then i) *{wl E{O,1}*IK(w 1 IW 2) ~n} <2 n + 1 _1 i i) (Especially) there exists a string w E{O,l}n with K(wlw 2) ~n If K(wl the empty string) =n , then w i s called a random string. Two easy consequences of Fact 1 are Fact 2: (Strings with low complexity are improbable). Let w 2 E{O,l}* be fixed and determine wI E{O,l}n by tossinq a fair coin n times, then for all c Fact 3: (Random strings are locally n almost random). Let w =w 1 w 2 w 3 E{O,l} be random. Then, with w'w 2 : =w 1 w 3 ' We need the following notation. For w: =w 1 w 1 w 2 w2"" ,w n w n Ol ·
Kolmogorov-CoMplexity已经被证明是证明下界的一个非常有用的工具[5]，[6]，[7]。这里我们将给出kolmogorov复杂度的进一步应用。首先，我们将极大地简化两个众所周知的下界证明:(a)在线乘法的n(n logn/ logn)下界，最初在[2]，[4]中得到证明，(b)排序中时空权衡的n(n 2 /log n)下界[9]。我们还将把这个界限提高到n(n2 loglog n /log n)。其次，我们将演示如何在分析概率算法中使用Kolmogorov-Complexity: (c)我们以一种基本的方式分析了[10]中给出的n维立方体的路由算法。1. 设C为带字母{O,I,B}的一维图灵机类。设U是c语言中的一个通用机，对于w1, w2 E{O,I}*定义kolmogorov复杂度[3]:K(w1 {w2}: =最短的0 / 1的长度，即I(程序I' J p)，当U输入pBw2计算wI并停止时。K(w): =K(空字符串)。45由于一个程序p只能生成一个单词w，我们有了事实1:设w2e {O,l}*， then I) *{wl E{O,1}*IK(w1iw 2) ~n} < 2n + 1 _1 _1 I I)(特别地)存在一个字符串w E{O,l}n与K(wlw 2) ~n，如果K(空字符串wl) =n，则称wi为随机字符串。事实1的两个简单结果是事实2:(低复杂度的字符串是不可能的)。设w2e {O,l}*是固定的，并通过投掷n次均匀硬币来确定wI E{O,l}n，则对于所有c，事实3:(随机字符串局部n几乎是随机的)。设w =w 1 w 2 w 3 E{0, 1}是随机的。然后，用w'w 2: =w 1 w 3 '，我们需要下面的符号。对于w: = w1w1w2w2“”，w1w2w·
{"title":"Three applications of Kolmogorov-complexity","authors":"Stefan Reisch, G. Schnitger","doi":"10.1109/SFCS.1982.96","DOIUrl":"https://doi.org/10.1109/SFCS.1982.96","url":null,"abstract":"Kolmogorov-CoMplexity has turned out to be a very useful tool in proving lower bounds [5], [6], [7]. Here we will give further applications of Kolmogorov-Complexity. Firstly we will greatly simplify two well-known lower bound proofs: (a) the n(n logn/loglog n) lower bound for on-line-multiplication, originally proven in [2], [4], (b) the n(n 2 /log n) lower bound for a time-space trade-off in sorting [9]. We will also improve this bound to n(N 2 loglog N/log N). Secondly we will demonstrate how to use Kolmogorov-Complexity in analysing proba-bilistic algorithms: (c) We analyse in an elementary way the routing algorithm for n-dimensional cubes given in [10]. 1. The concept of Kolmogorov-Complexity Let C be the class of one-dimensional Turing-machines with tape-alphabet {O,I,B}. Let U be an universal machine in C. For w 1 ,w 2 E{O,I}* define the Kolmogorov-Complexity [3] by: K(w 1 {W 2 }: =the length of the shortest o/ 1 s t r i ng (\" pro gram I' J p, s uc h t hat U with input pBw2 computes wI and stops. K(w): =K(wlempty string). 45 Because one program p can only generate one word w, we have Fact I : Let w 2 E{O,l}* , then i) *{wl E{O,1}*IK(w 1 IW 2) ~n} <2 n + 1 _1 i i) (Especially) there exists a string w E{O,l}n with K(wlw 2) ~n If K(wl the empty string) =n , then w i s called a random string. Two easy consequences of Fact 1 are Fact 2: (Strings with low complexity are improbable). Let w 2 E{O,l}* be fixed and determine wI E{O,l}n by tossinq a fair coin n times, then for all c Fact 3: (Random strings are locally n almost random). Let w =w 1 w 2 w 3 E{O,l} be random. Then, with w'w 2 : =w 1 w 3 ' We need the following notation. For w: =w 1 w 1 w 2 w2\"\" ,w n w n Ol ·","PeriodicalId":127919,"journal":{"name":"23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1982-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124042151","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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