Weakly independent database schemes

Independent database schemes are best known to have constraints enforced globally by enforcing them locally in individual relations, and therefore are highly desirable in an updates-busy and distributed environment. However, independence excludes all database schemes with interrelational functional dependencies and therefore is too strong a condition. In this paper, a natural generalization of independent schemes along this line, called weakly independent schemes, is defined. A database scheme is weakly independent with respect to a set of embedded functional dependencies if the constraints can be enforced globally by enforcing local as well as interrelational functional dependencies. (a) A characterization of the weak independence and an exponential time test (in the number of functional dependencies) are presented. (b) Two polynomial time conditions are also presented, one is necessary and the other is sufficient for a database scheme to be weakly independent. (c) It is shown that a certain acyclicity of the database scheme implies weak independence. Extension to the case of embedded functional dependencies plus the join dependency is also considered. Received: 18 July 1991 / 28 September 1995     

用于XML模式和DTD规范化设计的层次模式设计

XML模式和DTD(document type definition)规范化设计是给出一个很好地表示数据间依赖关系并消除了冗余的XML模式或DTD的集合.目前在这一方面开展的研究还不多,而且才刚起步.Provost提出将关系数据库理论应用于XML模式规范化设计的思想,这一思想还没有付诸实施.在Provost思想的基础上给出用于XML模式和DTD规范化设计的层次模式设计的算法.首先分析了基于Provost思想的层次分解;然后给出用于消除冗余模式的分解树设计算法;最后给出用于XML模式和DTD规范化设计的层次     

Characterizing the MVDs That Hold in a Connected Subtree of a Join Tree: A Correction

Given an acyclic database scheme R and a connected subtree T of a join tree for R, this paper provides a characterization of the set of multivalued dependencies (MVDs) that hold over T with respect to infinR, the corresponding join dependency (JD) of R. This result corrects an error in one of my previous papers "A Comparative Study of Various Nested Normal Forms," which was published in vol. 14, no. 2, pp. 369-385, Mar./Apr. 2002.     

An algebraic theory of functional and multivalued dependencies in relational databases

Computation of the dependency basis is the fundamental step in solving the membership problem for functional dependencies (FDs) and multivalued dependencies (MVDs) in relational database theory. We examine this problem from an algebraic perspective. We introduce the notion of the inference basis of a set M of MVDs and show that it contains the maximum information about the logical consequences of M. We propose the notion of a dependency-lattice and develop an algebraic characterization of inference basis using simple notions from lattice theory. We also establish several interesting properties of dependency-lattices related to the implication problem. Founded on our characterization, we synthesize efficient algorithms for (a): computing the inference basis of a given set M of MVDs; (b): computing the dependency basis of a given attribute set w.r.t. M; and (c): solving the membership problem for MVDs. We also show that our results naturally extend to incorporate FDs also in a way that enables the solution of the membership problem for both FDs and MVDs put together. We finally show that our algorithms are more efficient than existing ones, when used to solve what we term the ‘generalized membership problem’.     

A note on lossless database decompositions

It is known that under a wide variety of assumptions a database decomposition is lossless if and only if the database scheme has a lossless join. Biskup, Dayal and Bernstein (1979) have shown that when the given dependencies are functional, the database scheme has a lossless join if and only if one of the relation schemes is a key for the universal scheme. In this note we supply an alternative proof of that characterization. The proof uses tools from the theory of embedded join dependencies and the theory of tuple and equality generating dependencies, but is, nevertheless, much simpler than the previously published proof.     

XML多值依赖及其消除冗余模式的算法

从消除XML文档内数据冗余的角度出发研究了文档的规范化问题.首先引入XML上的数据冗余及其消除处理示例,同时基于函数依赖,提出了规范化的DTD概念和XML DTD 规范化处理规则;其次通过XML多值依赖的定义,给出用于消除冗余模式的算法;最后给出用于XML模式及其消除冗余模式的算法.该算法相应于其他XML模式的研究,在算法产生的层次模式中,完全MVD和嵌入MVD的集合由给出的MVD集合导出;并且产生的XML模式具有消除冗余模式和满足无损连接的特性.     

Membership problems for data dependencies in relational expressions

In relational databases, a query can be formulated in terms of a relational algebra expression using projection, selection, restriction, cross product and union. In this paper, we consider a problem, called the membership problem, of determining whether a given dependency d is valid in a given relational expression E over a given database scheme R that is, whether every instance of the view scheme defined by E satisfies d (assuming that the underlying constraints in R are always satisfied).Consider the case where each relation scheme in R is associated with functional dependencies (FDs) as constraints, and d is an FD. Then the complement of the membership problem is NP-complete. However, if E contains no union, then the membership problem can be solved in polynomial time. Furthermore, if E contains neither a union nor a projection, then we can construct in polynomial time a cover for valid FDs in E, that is, a set of FDs which implies every valid FD in E.Consider the case where each relation scheme in R is associated with multivalued dependencies (MVDs) as well as FDs, and d is an FD or an MVD. Even if E consists of selections and cross products only, the membership problem is NP-hard. However, if E contains no union, and each relation scheme name in R occurs in E at most once, then the membership problem can be solved in polynomial time. As a corollary of this result, it can be determined in polynomial time whether a given FD or MVD is valid in $\text{R}{}_1\text{???R}{}_{\mathrm{s}}$, where R₁,…,R_s are relation schemes with FDs and MVDs, and $\text{R}{}_{\mathrm{i}}\text{?R}{}_{\mathrm{j}}$ is the natural join of R_i and R_j.     

On characterizing boundedness of database schemes with bounded dependencies

We investigate the effect of bounded dependencies on the boundedness of database schemes. The following results are proved. A database scheme with only bounded equality-generating dependencies is always bounded with respect to dependencies; a lossless database scheme with bounded full implicational dependencies is bounded w.r.t. dependencies if and only if the implicational dependencies are equivalent to a single join dependency and some equality-generating dependencies. By a known method, this condition can be tested effectively. These results are relevant in database theory in that they determine in a rather general case whether queries under the representative instance approach can be expressed in relational algebra.     

Compact scheme forests in nested normal form

In the context of data base design for nested relational structures, update anomalies can be avoided if the nested scheme forest composed of scheme trees is in nested normal form (NNF) with respect to the associated set of data dependencies. In practice, minimizing the number of normal scheme trees in the nested scheme forest, which is in NNF, will also be an important design goal. This is because the number of computationally expensive join operations that are required in order to answer a given query is related to the number of normal scheme trees that must be used in the query expression.

We prove that the problem of finding a succinct NNF scheme forest is NP-complete for a subclass of the class of split-free sets of multivalued dependencies.     

An Efficient Algorithm for Processing Multi-Relation Queries in Relational Databases

After a relation scheme R is decomposed into the set of schemes ρ={R1，…，Rn},we may pose queries as if Rexisted in the database,taking a join of Ri‘s,when it is necessary to implement the query,Suppos a query involves a set of attributes S R，we want to find the smallest subset of ρ whose union includes.S.We prove that the problem is NP-complete and present a polynomial-bounded approximation algorithm.A subset of ρ whose union includes S and has a decomposition into 3NF with a lossless join and preservation of dependencies in given in the paper.     

Polynomial Tests of Normal Forms and Some Related Results

The following problem is called the everywhere-cover problem:“Given a set of dependencies over a database scheme,is he set of dependencies explicitly given for each relation scheme equivalent to the dependencies implied for that relation sheme?”It is shown that when the everywhere-cover problem has a yes answer,examining only the dependencies explicitly given will suffice to test 3NF, BCNF and 4NF of a database scheme.But this does not hold for 2NF.Consequently,in such cases,tests of BCNF and 4NF all take polynomial time.Then a proof is given that test of 3NF of a database scheme is Co-NP-complete,and from this result it is shown that everywhere-cover is also Co-NP-complete when only functional dependencies are allowed.These results lead to doubt the truth of the well believed conjecture that no polynomial time algorithm for designing a lossless BCNF database scheme is likely to exist.     

Full hierarchical dependencies in fixed and undetermined universes

Full hierarchical dependencies (FHDs) constitute a large class of relational dependencies. A relation exhibits an FHD precisely when it is the natural join over at least two of its projections that all share the same join attributes. Therefore, FHDs generalise multivalued dependencies (MVDs) in which case the number of these projections is precisely two. The implication of FHDs has originally been defined in the context of some fixed finite universe. This paper identifies a sound and complete set of inference rules for the implication of FHDs. This axiomatisation is very reminiscent of that for MVDs. Then, an alternative notion of FHD implication is introduced in which the underlying set of attributes is left undetermined. The first main result establishes a finite axiomatisation for FHD implication in undetermined universes. It is then formally clarified that the complementation rule is only a mere means for database normalisation. In fact, the second main result establishes a finite axiomatisation for FHD implication in fixed universes which allows to infer FHDs either without using the complementation rule at all or only in the very last step of the inference. This also characterises the expressiveness of an incomplete set of inference rules in fixed universes. The results extend previous work on MVDs by Biskup.     

Load-Balanced Join Processing in Shared-Nothing Systems

In a shared-nothing parallel database system, a join operation is split into a set of tasks that are allocated to the nodes in the system to be executed concurrently and independently. While parallel processing could greatly reduce the completion time of a join operation, the system performance may degrade because of load imbalance across the nodes caused by data skewness in the relations. Load-balanced join processing uses various techniques to evenly distribute the load among nodes in a system and hence improves the overall system performance. In this paper, the basic issues in designing load-balanced parallel join algorithms are identified. From the solutions to those issues, a large set of load-balanced join algorithms can be constructed. Performance of four representative algorithms-two dynamic load-balancing algorithms proposed in this paper and two static load-balancing algorithms adapted from similar algorithms in the literature-is studied and compared with that of a parallel join algorithm that does not balance the join load. The results of our study clearly show the benefits of load-balancing. This study also demonstrates that the dynamic load-balancing techniques proposed in this paper not only are feasible but also provide good system performance.     

Spatial Reasoning About Points in a Multidimensional Setting

For n 1, we consider the possible relations between two points of the Euclidean space of dimension n. We define the n-point algebra on the pattern of the point algebra and the cardinal algebra. Generalizing the concept of convexity just as the one of preconvexity, we prove that the consistency problem of convex n-point networks is polynomial for n 1, whereas the consistency problem of preconvex n-point networks is NP-complete for n 3. We characterize a subset of the set of all preconvex relations: the set of all strongly preconvex relations, which contains the set of all convex relations. We demonstrate that the consistency problem of strongly preconvex n-point networks can be decided in polynomial time by means of the weak path-consistency method for all n 1. For n = 3 the set of all strongly preconvex relations is a maximal tractable subclass of the set of all n-point relations. Finally, we prove that the concept of strong preconvexity corresponds to the one of ORD-Horn representability.     

Maximum <Emphasis Type="Italic">k</Emphasis>-Splittable <Emphasis Type="Italic">s</Emphasis>,<Emphasis Type="Italic">t</Emphasis>-Flows

Given a graph with a source and a sink node, the NP-hard maximum k-splittable s,t-flow (M k SF) problem is to find a flow of maximum value from s to t with a flow decomposition using at most k paths. The multicommodity variant of this problem is a natural generalization of disjoint paths and unsplittable flow problems. Constructing a k-splittable flow requires two interdepending decisions. One has to decide on k paths (routing) and on the flow values for the paths (packing). We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step. Usually the routing is considered before the packing. Our main contributions are as follows: (i) We show that for constant k a polynomial number of packing alternatives containing at least one packing used by an optimal M k SF solution can be constructed in polynomial time. If k is part of the input, we obtain a slightly weaker result. In this case we can guarantee that, for any fixed ε>0, the computed set of alternatives contains a packing used by a (1−ε)-approximate solution. The latter result is based on the observation that (1−ε)-approximate flows only require constantly many different flow values. We believe that this observation is of interest in its own right. (ii) Based on (i), we prove that, for constant k, the M k SF problem can be solved in polynomial time on graphs of bounded treewidth. If k is part of the input, this problem is still NP-hard and we present a polynomial time approximation scheme for it.     

模糊关系无损连接分解的实现及检验

本文基于一个模糊关系数据库系统的实验性原型FRDB1中定义的模糊关系,首先定义了模糊等价性,描述并验证了模糊函数依赖的理论,然后基于模糊关系代数和模糊等价性的附加约束条件实现了模糊关系的无损连接分解,最后证明了追踪算法检验模糊关系的无损连接分解。     

Testing a dependency-preserving decomposition for losslessness

It is shown that when a decomposition of a relation scheme preserves the associated set of functional dependencies, the problem of testing the decomposition for the lossless join property reduces to the problem of computing the closures of the relation schemes in the decomposition. As a consequence of this result an efficient algorithm for losslessness is proposed.     

Exact Complexity of the Winner Problem for Young Elections

In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for \p _|| ^NP , the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carroll's 1876 voting scheme were recently established by Hemaspaandra et al. In contrast, we prove that the winner and ranking problems in Fishburn's homogeneous variant of Carroll's voting scheme can be solved efficiently by linear programming.     

Sublinear Algorithms for Approximating String Compressibility

We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and a variant of Lempel-Ziv (LZ77), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly. Our investigation of LZ77 yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to LZ77 to the number of distinct short substrings contained in it (its ?th subword complexity , for small ?). In addition, we show that approximating the compressibility with respect to LZ77 is related to approximating the support size of a distribution.     

On the synthetic factorization of projectively invariant polynomials

We prove that, after multiplication with a suitable monomial, every homogeneous bracket polynomial of rank r≥3 can be factored into a meet and join expression in the Cayley algebra. The main tool in our construction is an explicit algorithm for rewriting polynomial functions in terms of synthetic constructions in projective geometry. We also discuss applications of Cayley factorization to automated geometry theorem proving.