Lecture

Matrix Chain Multiplication

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In course

CS-250: Algorithms I

The students learn the theory and practice of basic concepts and techniques in algorithms. The course covers mathematical induction, techniques for analyzing algorithms, elementary data structures, major algorithmic paradigms such as dynamic programming, s ...

Description

This lecture covers the dynamic programming algorithm for matrix chain multiplication, explaining the bottom-up approach to find the optimal solution. It discusses the algorithm for recovering an optimal solution and provides examples of matrix dimensions and instances. The instructor also explains the choice of where to make the outermost parenthesis and the overlapping subproblems. Additionally, the lecture introduces the concept of the longest common subsequence, defining it as a non-consecutive subsequence common to two sequences.

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Instructors (3)

Mikhail Kapralov

Teaching & PhD PhD Students Weronika Wrzos-Kaminska, Guy Weissenberg, Davide Mazzali, Ekaterina Kochetkova, Kshiteej Jitesh Sheth Past EPFL PhD Students Amir Zandieh, Aidasadat Mousavifar, Navid Nouri, Jakab Tardos, Grzegorz Adam Gluch, Makarov Mikhail Past EPFL PhD Students as codirector Paritosh Garg Courses Algorithms II CS-450 The students learn the theory and practice of basic concepts and techniques in algorithms. The course covers mathematical induction, techniques for analyzing alg

Ola Nils Anders Svensson

Alessandro Chiesa

Alessandro Chiesa is a faculty member in computer science. He conducts research in complexity theory, cryptography, and security, with a focus on the theoretical foundations and practical implementations of cryptographic proofs that are short and easy to verify. He is a co-author of several zkSNARK libraries, and is a co-inventor of the Zerocash protocol. He has co-founded Zcash and StarkWare Industries. He is a recipient of a Sloan Research Fellowship (2021), an Okawa Foundation Research Grant (2020), and Google Faculty Research Awards (2018 and 2017). He was included in MIT Technology Review's "35 Innovators Under 35" list in 2018.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Matrix multiplication algorithm

Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in seemingly unrelated problems such as counting the paths through a graph. Many different algorithms have been designed for multiplying matrices on different types of hardware, including parallel and distributed systems, where the computational work is spread over multiple processors (perhaps over a network).

Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, , adding two matrices would have the geometric effect of applying each matrix transformation separately onto , then adding the transformed vectors. However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. Two matrices must have an equal number of rows and columns to be added.

Rotation matrix

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae.

Computational complexity of matrix multiplication

In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the right amount of time it should take is of major practical relevance. Directly applying the mathematical definition of matrix multiplication gives an algorithm that requires n3 field operations to multiply two n × n matrices over that field (Θ(n3) in big O notation).

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