Related papers: Integer multiplication is at least as hard as matr…
The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82},…
We show, for the input vectors $(a_0, a_1, ..., a_{n-1})$ and $(b_0, b_1, ..., b_{n-1})$, where $a_i$'s and $b_j$'s are real numbers, after $O(n\log^4 n)$ time preprocessing for each of them, the vector multiplication $(a_0, a_1, ...,…
Matrix multiplication is a fundamental kernel in high performance computing. Many algorithms for fast matrix multiplication can only be applied to enormous matrices ($n>10^{100}$) and thus cannot be used in practice. Of all algorithms…
Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic…
We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra…
The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when…
Constant bit-size Transformers are known to be Turing complete, but existing constructions require $\Omega(s(n))$ chain-of-thought (CoT) steps per simulated Turing machine (TM) step, leading to impractical reasoning lengths. In this paper,…
The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is…
Two-dimensional constrained coding is a problem that is much more difficult than its one-dimensional counterpart. Indeed, in two dimensions, obtaining the answers to very natural questions becomes uncomputable. In particular, it is…
Many useful tasks in data science and machine learning applications can be written as simple variations of matrix multiplication. However, users have difficulty performing such tasks as existing matrix/vector libraries support only a…
We consider the problem of secure distributed matrix multiplication in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. In this paper, we answer the following question: Is it…
In this paper we consider the time and the crossing sequence complexities of one-tape off-line Turing machines. We show that the running time of each nondeterministic machine accepting a nonregular language must grow at least as n\log n, in…
Designing problems using matrices is very important in Computer Science. Fields like graph computer, graphs theory, and machine learning use matrices very often to solve their own problems. The most often matrix operation is the…
Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a…
Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer…
This note modifies the reference encoding of Turing machines in the $\lambda$-calculus by Dal Lago and Accattoli, which is tuned for time efficiency, as to accommodate logarithmic space. There are two main changes: Turing machines now have…
Distribution networks with periodically repeating events often hold great promise to exploit economies of scale. Joint replenishment problems are a fundamental model in inventory management, manufacturing, and logistics that capture these…
We describe the Turing Machine, list some of its many influences on the theory of computation and complexity of computations, and illustrate its importance.
An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…
This paper deals with circulant matrices. It is shown that a circulant matrix can be multiplied by a vector in time O(n log(n)) in a ring with roots of unity without making use of an FFT algorithm. With our algorithm we achieve a speedup of…