Optimal aggregation algorithms for middleware

The authors develop a formal group-theoretic model, called the Cayley graph model, for designing, analyzing, and improving such networks. They show that this model is universal and demonstrate how interconnection networks can be concisely represented in this model. It is shown that this model enables the authors to design networks based on representations of finite groups. They can then analyze these networks by interpreting the group-theoretic structure graph theoretically, Using these ideas, and motivated by certain well-known combinatorial problems, they develop two classes of networks called star graphs and pancake graphs. These networks are shown to have better performance than previous networks. >

A group-theoretic model for symmetric interconnection networks

Assume that each object in a database has m grades, or scores, one for each of m attributes. For example, an object can have a color grade, that tells how red it is, and a shape grade, that tells how round it is. For each attribute, there is a sorted list, which lists each object and its grade under that attribute, sorted by grade (highest grade first). There is some monotone aggregation function, or combining rule, such as min or average, that combines the individual grades to obtain an overall grade.To determine objects that have the best overall grades, the naive algorithm must access every object in the database, to find its grade under each attribute. Fagin has given an algorithm (“Fagin's Algorithm”, or FA) that is much more efficient. For some distributions on grades, and for some monotone aggregation functions, FA is optimal in a high-probability sense.We analyze an elegant and remarkably simple algorithm (“the threshold algorithm”, or TA) that is optimal in a much stronger sense than FA. We show that TA is essentially optimal, not just for some monotone aggregation functions, but for all of them, and not just in a high-probability sense, but over every database. Unlike FA, which requires large buffers (whose size may grow unboundedly as the database size grows), TA requires only a small, constant-size buffer.We distinguish two types of access: sorted access (where the middleware system obtains the grade of an object in some sorted list by proceeding through the list sequentially from the top), and random access (where the middleware system requests the grade of object in a list, and obtains it in one step). We consider the scenarios where random access is either impossible, or expensive relative to sorted access, and provide algorithms that are essentially optimal for these cases as well.

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We show that any language recognized by an NC 1 circuit (fan-in 2, depth O (log n )) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC 1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC 1 . Further, following Ruzzo ( J. Comput. System Sci. 22 (1981), 365–383) and Cook ( Inform. and Control 64 (1985) 2–22) , if the branching programs are restricted to be ATIME(logn)-uniform, they recognize the same languages as do ATIME(log n )-uniform NC 1 circuits, that is, those languages in ATIME(log n ). We also extend the method of proof to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC 1 .

Bounded-width polynomial-size branching programs recognize exactly those languages in NC1

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.

/pdf/optimal-bounds-for-the-predecessor-problem-and-related-4l95864sxi.pdf

Optimal bounds for the predecessor problem and related problems

We survey results from distributed computing that show tasks to be impossible, either outright or within given resource bounds, in various models The parameters of the models considered include synchrony, fault-tolerance, different communication media, and randomization The resource bounds refer to time, space and message complexity These results are useful in understanding the inherent difficulty of individual problems and in studying the power of different models of distributed computing There is a strong emphasis in our presentation on explaining the wide variety of techniques that are used to obtain the results described

Hundreds of impossibility results for distributed computing

Shared memory models of parallel computation (e.g., parallel RAMs) that allow simultaneous read/write access are very natural and already widely used for parallel algorithm design. The various models differ from each other in the mechanism by which they resolve write conflicts. To understand the effect of these communication primitives on the power of parallelism, we extensively study the relationship between four such models that appear in the literature, and prove nontrivial separations and simulation results among them.

Relations between concurrent-write models of parallel computation

Shared-memory models for parallel computation (e.g. parallel RAMs) are very natural and already widely used for parallel algorithm design. The various models differ from each other mainly in the way they restrict simultaneous processor access to a shared memory cell. Understanding the relative power of these models is important for understanding the power of parallel computation.Two recent pioneering works shed some light in this question. Cook and Dwork [CD] (resp. Snir [S]) present problems that, for instances of size n, can be solved in O(1) time on an n-processor PRAM that allows simultaneous write (resp. read) access to shared memory, but require Ω(log n) time on a PRAM that forbids simultaneous write (resp. read) access, regardless of the number of processors.When allowing simultaneous write access, the model must include a write-conflict resolution scheme. Three such schemes were suggested in the literature, and in this paper we study their relative power. Here the situation is more sensitive, as a small increase in the number of processors allows constant time simulation of the strongest by the weakest. By fixing the number of processors and parametrizing the number of shared memory cells, we obtain tight separation results between the models, thereby partially answering open questions of Vishkin [V]. New lower bounds techniques are developed for this purpose.

/pdf/relations-between-concurrent-write-models-of-parallel-4b093d8cx8.pdf

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed efficiently stored set. Our algorithms are for the unit-cost word-level RAM with multiplication and extend to give optimal dynamic algorithms. The lower bounds are proved in a much stronger communication game model, but they apply to the cell probe and RAM models and to both static and dynamic predecessor problems.

/pdf/optimal-bounds-for-the-predecessor-problem-48xfi856mg.pdf

Faith E. Fich

Papers

Optimal bounds for the predecessor problem and related problems

Hundreds of impossibility results for distributed computing

Relations between concurrent-write models of parallel computation

Relations between concurrent-write models of parallel computation

Optimal bounds for the predecessor problem