Algebra & Algorithms
This course explores the interplay between algebra and algorithms, demonstrating how algebraic structures and theorems can be used to develop algorithms for various tasks. Students will gain an understanding of how computers use algebraic computations and how algebra can be applied to computer science. Programming assignments are included to reinforce the concepts. ▼
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Course Feature
Cost:
Free
Provider:
Coursera
Certificate:
Paid Certification
Language:
English
Start Date:
7th Mar, 2022
Course Overview
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Updated in [March 06th, 2023]
Algebra & Algorithms is a course that combines two of the most important branches of mathematics and computer science. Learners will gain an understanding of the fundamentals of algebra and how it is used to design efficient algorithms. They will learn how to use algebraic structures and theorems to develop algorithms for various tasks, such as graph algorithms. The course will also provide learners with programming assignments to help them apply their knowledge. Additionally, the course will provide an overview of the development paths for both algebra and algorithms, as well as related learning suggestions. This course is suitable for both math and computer science students, and will help them gain a better understanding of the interplay between the two disciplines.
[Applications]
The application of this course can be seen in the development of algorithms for arithmetic and algebraic computations, as well as graph algorithms. Additionally, the course can be used to gain a better understanding of the relationship between algebra and algorithms, and how they can be used together. Finally, the programming assignments can be used to gain a better understanding of how to implement algorithms in code.
[Career Paths]
1. Algorithm Developer: Algorithm developers are responsible for designing, developing, and testing algorithms for computer programs. They must have a strong understanding of mathematics, computer science, and software engineering principles. Algorithm developers must be able to think logically and analytically, and have excellent problem-solving skills. As algorithms become increasingly complex, the demand for algorithm developers is expected to grow.
2. Data Scientist: Data scientists are responsible for analyzing large amounts of data and using it to make predictions and develop insights. They must have a strong understanding of mathematics, statistics, and computer science. Data scientists must be able to think logically and analytically, and have excellent problem-solving skills. As data becomes increasingly important, the demand for data scientists is expected to grow.
3. Software Engineer: Software engineers are responsible for designing, developing, and testing software applications. They must have a strong understanding of computer science, software engineering principles, and programming languages. Software engineers must be able to think logically and analytically, and have excellent problem-solving skills. As software applications become increasingly complex, the demand for software engineers is expected to grow.
4. Artificial Intelligence Engineer: Artificial intelligence engineers are responsible for designing, developing, and testing artificial intelligence systems. They must have a strong understanding of mathematics, computer science, and software engineering principles. Artificial intelligence engineers must be able to think logically and analytically, and have excellent problem-solving skills. As artificial intelligence becomes increasingly important, the demand for artificial intelligence engineers is expected to grow.
[Education Paths]
1. Bachelor of Science in Computer Science: This degree path focuses on the fundamentals of computer science, including programming, algorithms, data structures, and software engineering. It also covers topics such as artificial intelligence, computer networks, and computer graphics. This degree path is becoming increasingly popular as technology advances and more businesses rely on computer systems.
2. Bachelor of Science in Mathematics: This degree path focuses on the fundamentals of mathematics, including calculus, linear algebra, and probability. It also covers topics such as abstract algebra, number theory, and topology. This degree path is becoming increasingly popular as technology advances and more businesses rely on mathematical models and algorithms.
3. Master of Science in Algorithms and Complexity: This degree path focuses on the design and analysis of algorithms, as well as the complexity of algorithms. It also covers topics such as graph theory, computational geometry, and randomized algorithms. This degree path is becoming increasingly popular as technology advances and more businesses rely on efficient algorithms.
4. Doctor of Philosophy in Algorithms and Computation: This degree path focuses on the design and analysis of algorithms, as well as the complexity of algorithms. It also covers topics such as computational complexity, approximation algorithms, and quantum computing. This degree path is becoming increasingly popular as technology advances and more businesses rely on efficient algorithms and computation.
Course Syllabus
Arithmetics in the Realm of Circuits
In this module we will study a mathematical model of hardware: the Boolean circuits. They provide the birds eye view of how one builds a complex logical circuit from elementary building blocks by linking them with wires. We will learn efficient ways to construct circuits for all four arithmetic operations on integers represented in binary form. Among other things we will see how to efficiently calculate all carries while adding two numbers (much faster than just computing these carries sequentially) and how Newton’s approximation method for solving equations helps with implementing integer division.Boolean Circuits for Arbitrary Functions
In this second and last module devoted to Boolean circuits we move from arithmetics to the problem of constructing efficient circuits for arbitrary Boolean functions. On the way, among other things, we will see how to estimate the number of trees via DFS traversals and how to construct efficiently a circuit that computes all functions of given small number of variables.More on Multiplication of Integers and Polynomials
Multiplication of integers is among the first things people learn to do with integers at school, later moving on to higher spheres: multiplying matrices, polynomials, permutations etc. Multiplication is one of the central things in algebra. This week we focus on classical algorithms for efficient integer multiplication first, and then move on to matrix and polynomial multiplication. On the way we will see how closely integer and polynomial multiplication are really intertwined and what polynomial interpolation has to do with multiplication.Graph Reachability and Distances via Matrix Multiplication
This module is the first one to feature application pf efficient algorithms for algebraic operations to something outside algebra. Currently we turn to distances in graphs. We also study a non-typical way of multiplying matrices motivated by applications to graph reachibiilty, namely, Boolean matrix multiplication, and consider a corresponding rather general speedup technique.More on Matrix Computations
This week we first learn how to compute determinant and invert a matrix on a parallel computer, and how working with seemingly “computationally inconvenient” infinite series helps deriving quite a efficient formulas for the determinant. The second part of this module is devoted to a more common single-threaded computational model and there we prove that essentially the complexity of inverting a matrix is the same as that of matrix multiplication.Matchings in Graphs via Matrix Determinants
In the last module of the course we learn some tools for efficient use of randomization and algebra in algorithm design and use our knowledge about the existence of fast algorithms for parallel determinant computation to find perfect matchings in bipartite graphs unbelievably quickly on parallel computers.Course Provider
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