Let any instance p of problem P be characterized by n integer-valued parameters: )) ( , ), ( ( ) ( 1 p x p x p x n ! = ; } {∞ = " Z Z , | { ) ( p y P = } ) ( y p x ≤ be the subproblem defined for an arbitrary vector n Z y∈ . To determine the complexity of all subproblems ) ( y P , n Z y∈ , a notion of the basis system of subproblems is introduced. It is shown that if such a basis system exists,...

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In a real life environment, spatial data is highly skewed. In general, there are two kinds of skews in spatial data. One is the placement skew and the other is the area skew. This paper introduces methods and the complexity of estimating the result sizes of the multi-way join for the area skewed spatial data. Especially, this paper describes the number and sort of the statistics which the optim...

This paper studies the descriptional complexity of multi-parallel grammars with respect to the number of nonterminals and selectors, and the length of these selectors. As a result, it proves that every recursively enumerable language is generated by a multi-parallel grammar with no more than seven nonterminals and four selectors of length five.

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard semantic constructions that turn simple quantifiers into complex ones, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provi...

The central objects of secure multiparty computation are the “multiparty functions” (or functionalities) that it seeks to securely realize. In this chapter we survey a set of results that constitute a Cryptographic Complexity Theory. This theory classifies and compares multiparty functions according to their secure computability and reducibility to each other. The basic questions studied, under...

We study the design of pricing mechanisms and auctions when the mechanism designer does not know the distribution of buyers’ values. Instead the mechanism designer receives a set of samples from this distribution and his goal is to use the sample to design a pricing mechanism or auction with high expected profit. We provide generalization guarantees which bound the difference between average pr...

. The growing role of data-driven approaches to scientific discovery has unveiled a large class of models that involve latent transformations with a rigid algebraic constraint. Among them, multi-reference alignment (MRA) is a simple model that captures fundamental aspects of the statistical and algorithmic challenges arising from this new paradigm. In this model, an unknown signal is subject to...

The traditional Karatsuba algorithm for the multiplication of polynomials and multi-precision integers has a time complexity of O(n) and a space complexity of O(n). Roche proposed an improved algorithm with the same O(n) time complexity but with a much reduced O(log n) space complexity. In Roche’s paper details were provided for multiplication of polynomials, but not for multi-precision integer...