A min-wise independent permutation family is known to be an efficient tool to estimate similarity of documents. Toward good understanding of min-wise independence, we present a characterization of exactly min-wise independent permutation families by size uniformity, which represents certain symmetry of the string representation of a family. Also, we present a general construction strategy which produce any exactly min-wise independent permutation family using this characterization.

R-regular Mth band filters are an important class of digital filters and are used in constructing Mth-band wavelet filter banks, where the regularity is essential. But this kind of filter has larger stopband peak errors compared with a minimax filter of the same length. In this paper, peak errors in stopband of R-regular 4th-band filters are reduced by means of superimposing two filters with successive regularities. Then the stopband peak errors in the resulting filters are compared with the original ones. The results show that the stopband peak errors are reduced significantly in the synthesized filter that has the same length as the longer one of the two original filters, at the cost of regularity.

An important problem in mathematics and data science, given two or more metric spaces, is obtaining a metric of the product space by aggregating the source metrics using a multivariate function. In 1981, Borsík and Doboš solved the problem, and much progress has subsequently been made in generalizations of the problem. The triangle inequality is a key property for a bivariate function to be a metric. In the metric aggregation, requesting the triangle inequality of the resulting metric imposes the subadditivity on the aggregating function. However, in some applications, such as the image matching, a relaxed notion of the triangle inequality is useful and this relaxation may enlarge the scope of the aggregators to include some natural superadditive functions such as the harmonic mean. This paper examines the aggregation of two semimetrics (i.e. metrics with a relaxed triangle inequality) by the harmonic mean is studied and shows that such aggregation weakly preserves the relaxed triangle inequalities. As an application, the paper presents an alternative simple proof of the relaxed triangle inequality satisfied by the robust Jaccard-Tanimoto set dissimilarity, which was originally shown by Gragera and Suppakitpaisarn in 2016.

For the multi-objective time series search problem, Hasegawa and Itoh [Theoretical Computer Science, Vol.78, pp.58-66, 2018] presented the best possible online algorithm balanced price policy for any monotone function f:Rk→R. Specifically the competitive ratio with respect to the monotone function f(c1,...,ck)=(c1+…+ck)/k is referred to as the arithmetic mean component competitive ratio. Hasegawa and Itoh derived the explicit representation of the arithmetic mean component competitive ratio for k=2, but it has not been known for any integer k≥3. In this paper, we derive the explicit representations of the arithmetic mean component competitive ratio for k=3 and k=4, respectively. On the other hand, we show that it is computationally difficult to derive the explicit representation of the arithmetic mean component competitive ratio for arbitrary integer k in a way similar to the cases for k=2, 3, and 4.

The notion of k-wise independent permutations has several applications. From the practical point of view, it often suffices to consider almost (i.e., ε-approximate) k-wise independent permutation families rather than k-wise independent permutation families, however, we know little about how to construct families of ε-approximate k-wise independent permutations of small size. For any n > 0, let Sn be the set of all permutations on {0,1,..., n - 1}. In this paper, we investigate the size of families of ε-approximate k-wise independent permutations and show that (1) for any constant ε 0, if a family Sn of permutations is ε-approximate k-wise independent, then || n(n - 1) (n - k + 1) if ε< 1; || {n(n - 1) (n - k + 1)}/(1 +ε) otherwise; (2) for any constant 0< ε 1, there exists a family Sn of ε-approximate k-wise independent permutations such that || = ; (3) for any constant ε> 0 and any n = pm - 1 with p prime, it is possible to construct a polynomial time samplable family Sn of ε-approximate pairwise independent permutations such that || = O(n(n - 1)/ε); (4) for any constant ε> 0 and any n = pm with p prime, it is possible to construct a polynomial time samplable family Sn of ε-approximate 3-wise independent permutations such that || = O(n(n - 1)(n - 2)/ε). Our results are derived by combinatorial arguments, i.e., probabilistic methods and linear algebra methods.

As pseudorandom number generators for Monte Carlo simulations, inversive linear congruential generators (ICG) have some advantages compared with traditional linear congruential generators. It has been shown that a sequence generated by an ICG has a low discrepancy even if the length of the sequence is far shorter than its period. In this paper, we formulate fractional linear congruential generators (FCG), a generalized concept of the inversive linear congruential generators. It is shown that the sequence generated by an FCG is a geometrical shift of a sequence from an ICG and satisfies the same upper bounds of discrepancy. As an application of the general formulation, we show that under certain condition, "Leap-Frog technique," a way of splitting a random number sequence to parallel sequences, can be applied to the ICG or FCG with no extra cost on discrepancy.

A family C of min-wise independent permutations is known to be a useful tool of indexing replicated documents on the Web. For any integer n>0, a family C of permutations on [n]={1,2,. . . ,n} is said to be min-wise independent if for any (nonempty) X [n] and any x X, Pr ( min {π(X)} = π(x))= ||X||-1 when π is chosen uniformly at random from C, where ||A|| is the cardinality of a finite set A. For any integer n>0, it has been known that (1) ||C|| lcm(n,n-1,. . . ,2,1) = en-o(n) for any family C of min-wise independent permutations on [n]; (2) there exists a polynomial time samplable C family of min-wise independent permutations on [n] such that ||C|| 4n. However, it has been unclear whether there exists a min-wise independent family C such that ||C|| = lcm(n,n-1,. . . ,2,1) for each integer n>0 and how to construct such a family C of min-wise independent permutations for each integer n>0 if it exists. In this paper, we shall construct a family Fn of permutations for each integer n>0 and show that Fn is min-wise independent and ||Fn|| = lcm(n,n-1,. . . ,2,1). Moreover, we present a polynomial time sampling algorithm for the family. Thus the family Fn of min-wise independent permutations is optimal in the sense of family size and is easy to implement because of its polynomial time samplability.