The dynamic clusters method in nonhierarchical clustering

Given a finite setE R ⁿ, the problem is to find clusters (or subsets of similar points inE) and at the same time to find the most typical elements of this set. An original mathematical formulation is given to the problem. The proposed algorithm operates on groups of points, called samplings (samplings may be called multiple centers or cores); these samplings adapt and evolve into interesting clusters. Compared with other clustering algorithms, this algorithm requires less machine time and storage. We provide some propositions about nonprobabilistic convergence and a sufficient condition which ensures the decrease of the criterion. Some computational experiments are presented.     

Shallow circuits and concise formulae for multiple addition and multiplication

A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several different representation standards.The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {, , }-circuits of depth 3.48 log₂ n and {, , }-circuits of depth 4.95 log₂ n for the carry-save addition ofn numbers of arbitrary length. As a consequence we get multiplication circuits of the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases respectively to 4.48 log₂ n and 5.95 log₂ n.We also get {, , }-formulae of sizeO (n ^3.13) and {, }-formulae of sizeO (n ^4.57) for all the output bits of a carry-save addition ofn numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

Parallel integer sorting using small operations

We consider the problem of sortingn integers in the range [0,n ^c -1], wherec is a constant. It has been shown by Rajasekaran and Sen [14] that this problem can be solved optimally inO(logn) steps on an EREW PRAM withO(n) n -bit operations, for any constant >O. Though the number of operations is optimal, each operation is very large. In this paper, we show thatn integers in the range [0,n ^c -1] can be sorted inO(logn) time withO(nlogn)O(1)-bit operations andO(n) O(logn)-bit operations. The model used is a non-standard variant of an EREW PRAMtthat permits processors to have word-sizes ofO(1)-bits and (logn)-bits. Clearly, the speed of the proposed algorithm is optimal. Considering that the input to the problem consists ofO (n logn) bits, the proposed algorithm performs an optimal amount of work, measured at the bit level.This work was partially supported by The Northeast Parallel Architectures Center (NPAC) at Syracuse University, Syracuse, NY 13244 and The Rome Air Development Center, under contract F30602-88-D-0027.     

A geometric view of parametric linear programming

We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem () = min{c ^t x¦Ax =b + ¯b,x 0}. We then show that an optimality interval consists either of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function (). As a consequence, the optimality intervals form a partition of the closed interval {; ¦()¦ < }. Based on these optimality intervals, we also introduce an algorithm for solving the parametric RHS LP problem which requires an LP solver as a subroutine. If a polynomial-time LP solver is used to implement this subroutine, we obtain a substantial improvement on the complexity of those parametric RHS LP instances which exhibit degeneracy. When the number of breakpoints of () is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.This research was partially funded by the United States Navy-Office of Naval Research under Contract N00014-87-K-0202. Its financial support is gratefully acknowledged.     

A Delay-Averaged Logistic Model

For the equation x(t) = x(t) (1-(1/) _t-- ^t- x(u)du), > 0, > 0, > 0, conditions for the stability of a nonzero stationary solution under small perturbations are determined.     

Conditions for the Light-Traffic Insensitivity to Loss Probability in a GI/G/m/0 Queueing System

A loss queueing system GI/G/m/0 is considered. Let a(x) be a p.d.f. of interarrival intervals. Assume that this function behaves like cx^-1 for small x. Further let B(x) be a d.f. of service time; (1/) be the mean service time. Conditions are derived for the light-traffic insensitivity of the loss probability to the form of B(x) as (/ ) 0. In particular, the condition = 1 is necessary. Estimates for the loss probability are obtained.     

Undirecteds-t connectivity in polynomial time and sublinear space

Thes-t connectivity problem for undirected graphs is to decide whether two designated vertices,s andt, are in the same connected component. This paper presents the first known deterministic algorithms solving undirecteds-t connectivity using sublinear space and polynomial time. Our algorithms provide a nearly smooth time-space tradeoff between depth-first search and Savitch's algorithm. Forn vertex,m edge graphs, the simplest of our algorithms uses spaceO(s),n ^1/2log₂ nsnlog₂ n, and timeO(((m+n)n ² log² n)/s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space (nlogn), its time bound isO((m+n)logn), close to the optimal time for the problem. Another generalization uses less space, but more time: spaceO(n ^1/logn), for 2log₂ n, and timen ^O(). For constant the time remains polynomial.     

Solution of Systems of Linear Equations by the p-Adic Method

A new algorithm for solving systems of linear equations Ax = b in an Euclidean domain is suggested. In the case of the ring of integers, the complexity of this algorithm is O (n ³ mlog² ||A||), where n)$$ " align="middle" border="0"> is a matrix of rank n and , if standard algorithms for the multiplication of integers and matrices are used. Under the same conditions, the best algorithm of this kind among those published earlier, which was suggested by Labahn and Storjohann in [1], has complexity O (n ⁴ mlog² ||A||). True, when using fast algorithms for the multiplication of numbers and matrices, the theoretical complexity estimate for the latter algorithm is O (n mlog² ||A||), which is better than the similar estimate O (n ³ mlog||A||) for the new algorithm.     

Duality in Quantum Information Geometry

Let H be a separable Hilbert space. We consider the manifold M consisting of density operators on H such that ^p is of trace class for some p (0, 1). We say M is nearby if there exists C > 1 such that C ^–1C. We show that the space of nearby points to can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish M with a norm making it into a Banach manifold.     

Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered: in a set of admissible feedback controls v, with range inE ⁿ , find one minimizing the expectationE _sx { _s ^T L(t, (t), (t, (t)))dt + W _T ((T))} for all (s, x) [0,T) E ⁿ , whereL(t, x, ) = (/12)m ² – U(t, x) is the classical action integrand and is an-dimensional diffusion process in the weak sense, (see Bensoussan, 1982) with drift and diffusion coefficientD constant > 0.W _T andU are given real functions. Sufficiency conditions for the existence of such an optimal feedback control are given. Dedicated to George Leitmann Recommended by G.J. Olsder Presented at the Third Workshop on Control Mechanics in honor of George Leitmann, January 22–24, 1990, University of Southern California, Los Angeles, California (USA).     

Parallel general prefix computations with geometric,algebraic, and other applications

We introduce a generic problem component that captures the most common, difficult kernel of many problems. This kernel involves general prefix computations (GPC). GPC's lower bound complexity of (n logn) time is established, and we give optimal solutions on the sequential model inO(n logn) time, on the CREW PRAM model inO(logn) time, on the BSR (broadcasting with selective reduction) model in constant time, and on mesh-connected computers inO(n) time, all withn processors, plus anO(log² n) time solution on the hypercube model. We show that GPC techniques can be applied to a wide variety of geometric (point set and tree) problems, including triangulation of point sets, two-set dominance counting, ECDF searching, finding two-and three-dimensional maximal points, the reconstruction of trees from their traversals, counting inversions in a permutation, and matching parentheses.work partially supported by NSF IRI/8709726work partially supported by NSERC.     

The Kreisel length-of-proof problem

A first-order system F has theKreisel length-of-proof property if the following statement is true for all formulas(x): If there is ak1 such that for alln0 there is a proof of(¯n) in F with at mostk lines, then there is a proof of x(x) in F. We consider this property for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata (including individual axioms) and a finite number of rules of inference. We prove that any usual Parikh system formulation of Peano arithmetic has the Kreisel length-of-proof property if the underlying logic of the system is formulated without a schema for universal instantiation in either one of two ways. (In one way, the formula to be instantiated is built up in steps, and in the other way, the term to be substituted is built up in steps.) Our method of proof uses techniques and ideas from unification theory.     

Complexity classes and theories of finite models

LetL ^* be accepted in timef(n) by a nondeterministic Turing machine. Then there is a monadic existential second-order sentence in the language of + such that for everyx^*,xL if and only if a certain structureU _x ^f of cardinalityf(|x|) satisfies . It follows that ifL is accepted in nondeterministic timen ^d, d a natural number, then there is a sentence whose relational symbols ared-ary or less, whose finite spectrum isL. This research was partially supported by NSF Grants MCS78-01832 and MCS-8002695.     

On problem transformability in VLSI

The two basic performance parameters that capture the complexity of any VLSI chip are the area of the chip,A, and the computation time,T. A systematic approach for establishing lower bounds onA is presented. This approach relatesA to the bisection flow, . A theory of problem transformation based on , which captures bothAT ² andA complexity, is developed. A fundamental problem, namely, element uniqueness, is chosen as a computational prototype. It is shown under general input/output protocol assumptions that any chip that decides ifn elements (each with (1+)lognbits) are unique must have =(nlogn), and thus, AT²=(n ²log² n), andA= (nlogn). A theory of VLSI transformability reveals the inherentAT ² andA complexity of a large class of related problems.This work was supported in part by the Semiconductor Research Corporation under contract RSCH 84-06-049-6.     

A faster divide-and-conquer algorithm for constructing delaunay triangulations

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its (n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn2¹⁶, the range of the experiments. It is conjectured that the average number of edges it creates—a good measure of its efficiency—is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theL _p metric for 1<p.This research was supported by National Science Foundation Grants DCR-8352081 and DCR-8416190.     

Specification and verification of database dynamics

Summary A framework is proposed for the structured specification and verification of database dynamics. In this framework, the conceptual model of a database is a many sorted first order linear tense theory whose proper axioms specify the update and the triggering behaviour of the database. The use of conceptual modelling approaches for structuring such a theory is analysed. Semantic primitives based on the notions of event and process are adopted for modelling the dynamic aspects. Events are used to model both atomic database operations and communication actions (input/output). Nonatomic operations to be performed on the database (transactions) are modelled by processes in terms of trigger/reaction patterns of behaviour. The correctness of the specification is verified by proving that the desired requirements on the evolution of the database are theorems of the conceptual model. Besides the traditional data integrity constraints, requirements of the form Under condition W, it is guaranteed that the database operation Z will be successfully performed are also considered. Such liveness requirements have been ignored in the database literature, although they are essential to a complete definition of the database dynamics.

Notation

Classical Logic Symbols (Appendix 1) for all (universal quantifier) - exists at least once (existential quantifier) - ¬ no (negation) - implies (implication) - is equivalent to (equivalence) - and (conjunction) - or (disjunction) Tense Logic Symbols (Appendix 1) G always in the future - G ₀ always in the future and now - F sometime in the future - F ₀ sometime in the future or now - H always in the past - H ₀ always in the past and now - P sometime in the past - P ₀ sometime in the past or now - X in the next moment - Y in the previous moment - L always - M sometime Event Specification Symbols (Sects. 3 and 4.1) (x) means immediately after the occurrence of x - (x) means immediately before the occurrence of x - (x) means x is enabled, i.e., x may occur next - { } ({w ₁} x{w ₂}) states that if w ₁ holds before the occurrence of x, then w ₂ will hold after the occurrence of x (change rule) - [ ] ([oa₁, ..., oa_n]x) states that only the object attributes oa₁, ..., oa _n are modifiable by x (scope rule) - {{ }} ({{w}}x) states that if x may occur next, then w holds (enabling rule) Process Specification Symbols (Sects. 5.3 and 5.4) :: for causal rules - for behavioural rules Transition-Pattern Composition Symbols (Sects. 5.2 and 5.3) ; sequential composition - ¦ choice composition - parallel composition - :| guarded alternative composition Location Predicates (Sect. 5.2) (z) means immediately after the occurrence of the last event of z (after) - (z) means immediately before the occurrence of the first event of z (before) - (z) means after the beginning of z and before the end of z (during) - ( z) means before the occurrence of an event of z (at)     

The liberalized δ-rule in free variable semantic tableaux

In this paper we have a closer look at one of the rules of the tableau calculus presented by Fitting [4], called the -rule. We prove that a modification of this rule, called the ⁺-rule, which uses fewer free variables, is also sound and complete. We examine the relationship between the ⁺-rule and variations of the -rule presented by Smullyan [9]. This leads to a second proof of the soundness of the ⁺-rule. An example shows the relevance of this modification for building tableau-based theorem provers.     

Randomness in interactive proofs

This paper initiates a study of the quantitative aspects of randomness in interactive proofs. Our main result, which applies to the equivalent form of IP known as Arthur-Merlin (AM) games, is a randomness-efficient technique for decreasing the error probability. Given an AM proof system forL which achieves error probability 1/3 at the cost of Arthur sendingl(n) random bits per round, and given a polynomialk=k(n), we show how to construct an AM proof system forL which, in the same number of rounds as the original proof system, achieves error 2 ^–k(n) at the cost of Arthur sending onlyO(l+k) random bits per round.Underlying the transformation is a novel sampling method for approximating the average value of an arbitrary functionf:{0,1} ^l [0,1]. The method evaluates the function onO(^–2 log ^–1) sample points generated using onlyO(l + log ^–1) coin tosses to get an estimate which with probability at least 1- is within of the average value of the function.