Abstract. An Armstrong relation for a set of functional dependencies (FDs) is a relation that satisfies each FD implied by the set but no FD that is not implied by it. The structure and size (number of tuples) of Armstrong relatsons are investigated. Upper and lower bounds on the size of minimal-sized Armstrong relations are derived, and upper and lower bounds on the number of distinct entries that must appear m an Armstrong relation are given. It is shown that the time complexity of finding an Armstrong relation, gwen a set of functional dependencies, is precisely exponential in the number of attributes. Also shown ,s the falsity of a natural conjecture which says that almost all relations obeying a given set of FDs are Armstrong relations for that set of FDs. Finally, Armstrong relations are used to generahze a result, obtained by Demetrovics using quite complicated methods, about the possible sets of keys for a relauon.

A periodic sorting network consists of a sequence of identical blocks. In this paper, the periodic balanced sorting network, which consists of logn blocks, is introduced. Each block, called a balanced merging block, merges elements on the even input lines with those on the odd input lines.The periodic balanced sorting network sorts n items in @[lognI') time using (n/2)(logn)' comparators. Although these bounds are comparable to many existing sorting networks, the periodic structure enables a hardware implementation consisting of only one block with the output of the block recycled back as input until the output is sorted. An implementation of our network on the shuffle exchange interconnection model in which the direction of the comparators are all identical and fixed is also presented.An earlier version of this paper appeared as DOWD, M., PERL, Y., RUWLPH, L., AND SAKS, M. The balanced sorting network. value on the first port, the second smallest on the second port, etc. The time required for the values to appear at the output ports, that is, the sorting time, is usually determined by the number of phases of the network, where a phase is a set of comparators that are all active at the same time. The unordered set of items are input to the first phase; the output of the ith phase becomes the input to the i + 1 phase. We describe the actions of a sorting network in terms of the comparisons performed on a vector of size n. Initially the vector x contains the unordered items, and at the end, x(i) contains the ith item in nondecreasing order. A phase of the network compares and possibly rearranges pairs of elements of x. It is often useful to group a sequence of connected phases together into a block. In addition to simplifying the network description, blocks often perform important functions such as merging lists of sorted subsets of the inputs. A periodic sorting network is defined as a network composed of a sequence of identical blocks.A very simple example of a periodic sorting network is the well-known oddeven transposition sorting network [6]. This network consists of fn/21 identical blocks, where each block contains two phases. The first phase compares x(i) with x(i + 1) for all even values of i, 0 5 i < n, and the second phase compares x(i -1) with x(i) for all even values of i, 0 < i 5 n. This network is based on the bubble sort algorithm and requires O(n) time.An example of a nonperiodic sorting network is the well-known Bitonic Sort of Batcher [2]. As described below, the network consists of ensembles of merging blocks and is based on a merge-sort: The output of two n/2 sorting networks are input to a size-n merge block. Bitonic sort requires the same time and space bounds as our periodic balanced sorting network, even the constants are nearly identical.Periodic sorting networks have more efficient hardware implementations since only one block needs to be built and a sorting network can be realized by recirculating the output of a block back as its input. In addition to a savings in hardware, they may some...

| Binary tree structures have been very useful in solving divide-and-conquer type of problems. Embedding binary trees into another network|the host network|helps in designing solutions for the host network using the known solutions on binary trees. Embedding arbitrary binary trees into networks, in particular into the hypercube, has been addressed in the literature. The latter was achieved with load 1 and constant dilation. The n-star graph is a recently introduced interconnection network for massively parallel systems. It enjoys symmetry and fault tolerance properties that make it a viable alternative to the hypercube. In this paper, we address the problem of embedding arbitrary binary trees into the n-star graph. This work is the rst to present such an embedding. The tree has be(n ? 1)!c or fewer vertices. The embedding leads to load 1 and constant dilation for all values of n. It therefore enables the star graph to e ciently simulate an arbitrary binary tree with only a constant factor of communication delay. Keywords| Star graph, interconnection network, massively parallel, binary tree, embedding, load, dilation, tree partitioning.

Appropriate restrictions may be placed on a Σ 1 1 well-order on V κ where κ is an inaccessible cardinal, so that it gives rise to a function chain when κ is a Mahlo cardinal. Set chains may be defined for all second order formulas. Postulating that the sets in the resulting set chain are stationary yields a powerful new axiom.

No abstract