A Comparison of Resource-Bounded Molecular Computation Models

The number of molecular strands used by a molecular algorithm is an important measure of the algorithm's complexity. This measure is also called the volume used by the algorithm. We prove that three important polynomial-time models of molecular computation with bounded volume are equivalent to models of polynomial-time Turing machine computation with bounded nondeterminism. Without any assumption, we show that the Split operation does not increase the power of polynomial-time molecular computation. Assuming a plausible separation between Turing machine complexity classes, the Amplify operation does increase the power of polynomial-time molecular computation. Received September 28, 1997; revised March 24, 1998.     

Structure and importance of logspace-MOD class

We refine the techniques of Beigelet al. [4] who investigated polynomial-time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes and demonstrate their significance by proving that all standard problems of linear algebra over the finite ringsZ/kZ are complete for these classes. We then define new complexity classes LogFew and LogFew and identify them as adequate logspace versions of Few and Few. We show that LogFew is contained in and that LogFew is contained in for allk. Also an upper bound for in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in.     

On balanced versus unbalanced computation trees

A great number of complexity classes between P and PSPACE can be defined via leaf languages for computation trees of nondeterministic polynomial-time machines. Jenner, McKenzie, and Thérien (Proceedings of the 9th Conference on Structure in Complexity Theory, 1994) raised the issue of whether considering balanced or unbalanced trees makes any difference. For a number of leaf-language classes, coincidence of both models was shown, but for the very prominent example of leaf-language classes from the alternating logarithmic-time hierarchy the question was left open. It was only proved that in the balanced case these classes exactly characterize the classes from the polynomial-time hierarchy. Here, we show that balanced trees apparently make a difference: In the unbalanced case, a class from the logarithmic-time hierarchy characterizes the corresponding class from the polynomial-time hierarchy with a PP-oracle. Along the way, we get an interesting normal form for PP computations. The first and third authors were supported by Deutsche Forschungsgemeinschaft, Grant No. Wa 847/1-1, “k-wertige Schaltkreise.” The second author was supported in part by an Alexander von Humboldt fellowship.     

Generalized theorems on relationships among reducibility notions to certain complexity classes

In this paper we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP many-one reductions and polynomial-time conjunctive reductions or (II) closed under coNP many-one reductions and polynomial-time disjunctive reductions. We prove that, for such a classK, (1) reducibility notions of sets toK under polynomial-time constant-round truth-table reducibility, polynomial-time log-Turing reducibility, logspace constant-round truth-table reducibility, logspace log-Turing reducibility, and logspace Turing reducibility are all equivalent and (2) every set that is polynomial-time positive Turing reducible to a set inK is already inK.From these results, we derive some observations on the reducibility notions to C₌P and NP.     

基于超图的求解FD集最优覆盖的算法研究

本文在文献「３」、「４」、「５」所讨论的超图及 的分类的基础上，分析了最优覆盖对应的超图的结构特点，用替换化简方法解决了最优覆盖的多项式时间算法。     

分布式存储的并行串匹配算法的设计与分析

并行串匹配算法的研究大都集中在PRAM(parallel random access machine)模型上,其他更为实际的模型上的并行串匹配算法的研究相对要薄弱得多.该文采用将最优串行算法并行化的技术,利用模式串的周期性质,巧妙地将改进的KMP(Knuth-Morris-Pratt)算法并行化,提出了一个简便、高效且具有良好可扩放性的分布式串匹配算法,其计算复杂度为O(n/p+m),通信复杂度为O(ulogp相似文献   

Classes of bounded nondeterminism

We study certain language classes located betweenP andNP that are defined by polynomial-time machines with a bounded amount of nondeterminism. We observe that these classes have complete problems and find a characterization of the classes using robust machines with bounded access to the oracle, obtaining some other results in this direction. We also study questions related to the existence of complete tally sets in these classes and closure of the classes under different types of polynomial-time reducibilities.The research of this author was supported by CIRIT Grant EE87/2.     

The saga of minimum spanning trees

This article surveys the many facets of the Minimum Spanning Tree problem. We follow the history of the problem since the first polynomial-time solution by Bor˚uvka to the modern algorithms by Chazelle, Pettie and Ramachandran. We study the differences in time complexity dependent on the model of computation chosen and on the availability of random bits. We also briefly touch the dynamic maintenance of the MST and other related problems.     

Computational complexity of auditing finite attributes in statistical databases

We study the computational complexity of auditing finite attributes in databases allowing statistical queries. Given a database that supports statistical queries, the auditing problem is to check whether an attribute can be completely determined or not from a given set of statistical information. Some restricted cases of this problem have been investigated earlier, e.g. the complexity of statistical sum queries is known by the work of Kleinberg et al. (J. Comput. System Sci. 66 (2003) 244–253). We characterize all classes of statistical queries such that the auditing problem is polynomial-time solvable. We also prove that the problem is coNP-complete in all other cases under a plausible conjecture on the complexity of constraint satisfaction problems (CSP). The characterization is based on the complexity of certain CSP problems; the exact complexity for such problems is known in many cases. This result is obtained by exploiting connections between auditing and constraint satisfaction, and using certain algebraic techniques. We also study a generalization of the auditing problem where one asks if a set of statistical information imply that an attribute is restricted to K or less different values. We characterize all classes of polynomial-time solvable problems in this case, too.     

Polynomial time quantum computation with advice

Advice is supplementary information that enhances the computational power of an underlying computation. This paper focuses on advice that is given in the form of a pure quantum state and examines the influence of such advice on the behaviors of an underlying polynomial-time quantum computation with bounded-error probability.     

Preprocessing of Intractable Problems

Some computationally hard problems, e.g., deduction in logical knowledge bases– are such that part of an instance is known well before the rest of it, and remains the same for several subsequent instances of the problem. In these cases, it is useful to preprocess off-line this known part so as to simplify the remaining on-line problem. In this paper we investigate such a technique in the context of intractable, i.e., NP-hard, problems. Recent results in the literature show that not all NP-hard problems behave in the same way: for some of them preprocessing yields polynomial-time on-line simplified problems (we call them compilable), while for other ones their compilability implies some consequences that are considered unlikely. Our primary goal is to provide a sound methodology that can be used to either prove or disprove that a problem is compilable. To this end, we define new models of computation, complexity classes, and reductions. We find complete problems for such classes, “completeness” meaning they are “the less likely to be compilable.” We also investigate preprocessing that does not yield polynomial-time on-line algorithms, but generically “decreases” complexity. This leads us to define “hierarchies of compilability,” that are the analog of the polynomial hierarchy. A detailed comparison of our framework to the idea of “parameterized tractability” shows the differences between the two approaches.     

A Note on Learning from Multiple-Instance Examples

We describe a simple reduction from the problem of PAC-learning from multiple-instance examples to that of PAC-learning with one-sided random classification noise. Thus, all concept classes learnable with one-sided noise, which includes all concepts learnable in the usual 2-sided random noise model plus others such as the parity function, are learnable from multiple-instance examples. We also describe a more efficient (and somewhat technically more involved) reduction to the Statistical-Query model that results in a polynomial-time algorithm for learning axis-parallel rectangles with sample complexity Õ(d²r/²) , saving roughly a factor of r over the results of Auer et al. (1997).     

A note on measuring in P

We revisit the problem of generalising Lutz's resource-bounded measure to small complexity classes, and propose a definition of a random-based on , which we argue as being a good generalisation to of Lutz's . We cannot unconditionally prove the existence of such a measure, but we give sufficient and necessary conditions for its existence. We also revisit μ_τ, an for defined by Strauss [Inform. Comput. 136(1) (1997) 1], and correct an erroneous claim concerning the relations between μ_τ and random sets. A correction to this mistake is then proposed, which is a less powerful but accurate relation between μ_τ and random sets.

In order to obtain these results, we introduce a mathematical structure called a measuring system, which is a general setting that can be used to compare different s on any fixed complexity class through a partial ordering relation.     

Mutation systems

We propose mutation systems as a model of the evolution of a string subject to the effects of mutations and a fitness function. One fundamental question about such a system is whether knowing the rules for mutations and fitness, we can predict whether it is possible for one string to evolve into another. To explore this issue, we define a specific kind of mutation system with point mutations and a fitness function based on conserved strongly k-testable string patterns. We show that for any k greater than 1, such systems can simulate computation by both finite state machines (FSMs) and asynchronous cellular automata. The cellular automaton simulation shows that in this framework, universal computation is possible and the question of whether one string can evolve into another is undecidable. We also analyse the efficiency of the FSM simulation assuming random point mutations.     

Parameterized Intractability of Distinguishing Substring Selection

A central question in computational biology is the design of genetic markers to distinguish between two given sets of (DNA) sequences. This question is formalized as the NP-complete Distinguishing Substring Selection problem (DSSS for short) which asks, given a set of "good" strings and a set of "bad" strings, for a solution string which is, with respect to the Hamming metric, "away" from the good strings and "close" to the bad strings. More precisely, given integers d_g, d_b, and L, we ask for a length-L string s such that s has Hamming distance at least d_g to every length-L substring of the good strings and such that every bad string has a length-L substring with Hamming distance at most d_b to s. Studying the parameterized complexity of DSSS, we show that, already for binary alphabet, DSSS is W[1]-hard with respect to its natural parameters. This, in particular, implies that a recently given polynomial-time approximation scheme (PTAS) by Deng et al. cannot be replaced by a so-called efficient polynomial-time approximation scheme (EPTAS) unless an unlikely collapse in parameterized complexity theory occurs. This is seemingly the first computational biology problem for which such a border between PTAS (which exists) and EPTAS (which is unlikely to exist) could be established. By way of contrast, for a special case of DSSS, we present an exact fixed-parameter algorithm solving the problem efficiently. In this way we also exhibit a sharp border between fixed-parameter tractability and intractability results.     

Space bounded computations: review and new separation results

In this paper we review the key results about space bounded complexity classes, discuss the central open problems and outline the prominent proof techniques. We show that, for a slightly modified Turing machine model, low level deterministic and nondeterministic space bounded complexity classes are different. Furthermore, for this computation model, we show that Savitch's theorem and the Immerman-Szelepcsényi theorem do not hold in the range lg lg n to lg n. We also present other changes in the computation model which bring out and clarify the importance of space constructibility. We conclude by enumerating open problems which arise out of the discussion.     

Computing dominators in parallel

We present a fast parallel algorithm for computing the dominators of a directed acyclic graph. The model of computation used in a parallel random access machine that allows simultaneous reads but prohibits simultaneous writes into the same memory location. Let P_t(n) be the processor complexity of computing the transitive closure of an n-vertex directed graph on this model. The only known parallel algorithm for dominators requires O(log²n) time and uses O(nP_t(n)) processors. Our algorithm for dominators has the same time complexity but uses O(P_t(n)) processors, thereby improving the processor complexity of the previously known algorithm by a factor of n.     

On the power of generalized Mod-classes

We investigate the computational power of the new counting class ModP which generalizes the classes Mod _p P,p prime. We show that ModP is polynomialtime truth-table equivalent in power to #P and that ModP is contained in the class AmpMP. As a consequence, the classes PP, ModP, and AmpMP are all Turing equivalent, and thus AmpMP and ModP are not low for MP unless the counting hierarchy collapses to MP. Furthermore, we show that every set in C=P is reducible to some set in ModP via a random many-one reduction that uses only logarithmically many random bits. Hence, ModP and AmpMP are not closed under polynomial-time conjunctive reductions unless the counting hierarchy collapses.     

Dimension Is Compression

Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomial-space and constructive dimension can be precisely characterized in terms of Kolmogorov complexity, while analogous results for polynomial-time dimension haven??t been found. In this paper we remedy the situation by using the natural concept of reversible time-bounded compression for finite strings. We completely characterize polynomial-time dimension in terms of polynomial-time compressors.