Related papers: Lower Bounds for Multiplication via Network Coding
We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $\Sigma\Pi\Sigma$…
The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of…
Assuming a conjectural upper bound for the least prime in an arithmetic progression, we show that n-bit integers may be multiplied in O(n log n 4^(log^* n)) bit operations.
Any monotone Boolean circuit computing the $n$-dimensional Boolean convolution requires at least $n^2$ and-gates. This precisely matches the obvious upper bound.
We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…
We prove that $n$-bit integers may be multiplied in $O(n \log n \, 4^{\log^* n})$ bit operations. This complexity bound had been achieved previously by several authors, assuming various unproved number-theoretic hypotheses. Our proof is…
We present a quantum algorithm for multiplying two $n$-bit integers with overall circuit depth and $T$-depth both bounded by $O(\log^{2} n)$, while using $O(n^{2})$ gates and ancillary qubits. Our construction generates partial products via…
A tight lower bound for required I/O when computing an ordinary matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of segments needed…
It is shown that the counting function of n Boolean variables can be implemented with the formulae of size O(n^3.06) over the basis of all 2-input Boolean functions and of size O(n^4.54) over the standard basis. The same bounds follow for…
We present an algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations. Previously, the best known bound was O(n log n 6^{log^* n}). We also prove that for a fixed prime p, polynomials in…
A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. H{\aa}stad's celebrated switching lemma yields exponential lower bounds for the…
In this paper, we investigate computational power of threshold circuits and other theoretical models of neural networks in terms of the following four complexity measures: size (the number of gates), depth, weight and energy. Here the…
Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers $n$ and $d$ such that $d\geq \omega(\log{n})$, any syntactic depth four circuit of bounded individual degree $\delta = o(d)$ that computes the…
We provide accurate upper bounds on the Boolean circuit complexity of the standard and the Karatsuba methods of integer multiplication
Boolean circuits of McCulloch-Pitts threshold gates are a classic model of neural computation studied heavily in the late 20th century as a model of general computation. Recent advances in large-scale neural computing hardware has made…
We revisit the hardness of approximating the diameter of a network. In the CONGEST model of distributed computing, $ \tilde \Omega (n) $ rounds are necessary to compute the diameter [Frischknecht et al. SODA'12], where $ \tilde \Omega…
In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…
We give a new proof of F\"urer's bound for the cost of multiplying n-bit integers in the bit complexity model. Unlike F\"urer, our method does not require constructing special coefficient rings with "fast" roots of unity. Moreover, we prove…
Bit addition arises virtually everywhere in digital circuits: arithmetic operations, increment/decrement operators, computing addresses and table indices, and so on. Since bit addition is such a basic task in Boolean circuit synthesis, a…
We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n \times n \times…