ASSTRACT. Let U1 , U2, . . . , Ud be totally ordered sets and let V be a set of n d-dimensional vectors In U~ X Us. . X Ud . A partial ordering is defined on V in a natural way The problem of finding all maximal elements of V with respect to the partial ordering ~s considered The computational complexity of the problem is defined to be the number of required comparisons of two components and is denoted by Cd(n). It is tnwal that C~(n) = n -1 and C,~(n) < O(n 2) for d _~ 2 In this paper we show: (1) C2(n) = O(n logan) for d = 2, 3 and Cd(n) ~ O(n(log2n) ~-~) for d ~ 4, (2) C,t(n) >_ flog2 n!l for d _> 2 KEY WORDS AND PHRASES: maxima of a set of vectors, computattonal complexity, number of comparisons, algorithm, recurrence CR CATEaOmES. 5.25, 5,31, 5.39

Consider an ordered, static tree T where each node has a label from alphabet Σ. Tree T may be of arbitrary degree and shape. Our goal is designing a compressed storage scheme of T that supports basic navigational operations among the immediate neighbors of a node (i.e. parent, ith child, or any child with some label, . . .) as well as more sophisticated path-based search operations over its labeled structure.We present a novel approach to this problem by designing what we call the XBW-transform of the tree in the spirit of the well-known Burrows-Wheeler transform for strings [1994]. The XBW-transform uses path-sorting to linearize the labeled tree T into two coordinated arrays, one capturing the structure and the other the labels. For the first time, by using the properties of the XBW-transform, our compressed indexes support navigational and path-search operations over labeled trees within (near)-optimal time bounds and entropy-bounded space.Using the properties of the XBW-transform, we go beyond the information-theoretic lower bound. For the first time, our compressed indexes support navigational operations and path search operations within (near)-optimal time bounds and entropy-bounded space.Our XBW-transform is simple and likely to spur new results in the theory of tree compression and indexing, as well as interesting application contexts. As an example, we use the XBW-transform to design and implement a compressed index for XML documents whose compression ratio is significantly better than the one achievable by state-of-the-art tools, and its query time performance is order of magnitudes faster.

Consider an ordered, static tree T on t nodes where each node has a label from alphabet set Σ.

No abstract

No abstract

No abstract

We exploit the "regularity" of Boolean functions with the purpose of decreasing the time for constructing minimal threelevel expressions, in the sum of pseudoproducts (SPP) form recently developed. The regularity of a Boolean function of variables can be expressed by an autosymmetry degree (with 0). = 0 means no regularity, that is we are not able to provide any advantage over standard synthesis. For 1 the function is said to be autosymmetric, and a new function depending on variables only, called the restriction of , is identified in time polynomial in the number of points of. The relation between and is discussed in depth to show how a minimal SPP form for can be build in linear time from a minimal SPP form for. The concept of autosymmetry is then extended to functions with don't care conditions, and the SPP minimization technique is duly extended to such functions. A large set of experimental results is presented, showing that 61% of the outputs for the functions in the classical ESPRESSO benchmark suite are autosymmetric. The minimization time for such functions is critically reduced, and cases otherwise intractable are solved. The quality of the corresponding circuits, measured with some well established cost functions, is also improved. Finally, we discuss the role and meaning of autosymmetric functions, and why a great amount of functions of practical interest fall in this class.

No abstract