I perform in open mics in my spare time. When I decide to perform a cover of a song, I usually play the guitar and sing. Therefore, I prepare a backing track without the tracks I use as background music during the performance.

In this article, I show how to create a backing track using yt-dlp, Demucs, and FFmpeg.

When designing a system, it’s important to consider the limitations of the technologies chosen. Making some approximate calculations when the system is designed helps us decide on the tradeoffs of the different approaches. These approximations include:

Kafka is a distributed event streaming platform designed for building high-throughput, fault-tolerant, and scalable data streaming applications.

The pattern of batching data in memory, tracking it in a write-ahead log, and periodically flushing it to disk is ubiquitous today. Open-source examples include LevelDB, Cassandra, InfluxDB, and HBase.

Cassandra is a highly scalable, distributed NoSQL (non-relational) database management system designed for handling large amounts of data across multiple commodity servers.

Data partitioning refers to the process of dividing a system’s data into smaller, more manageable subsets, which are distributed across multiple storage locations or nodes.

Non-functional requirements refer to quality attributes or system characteristics that describe how well a software system or solution performs or behaves.

Let $a,b \in \mathbb{Z}$. We say that $a$ divides $b$, written $a \given b$, if there’s an integer $n$ such that $b = na$. If $a$ divides $b$, then $b$ is divisible by $a$, and $a$ is a divisor or factor of $b$. Also, $b$ is called a multiple of $a$.

Make is a build automation tool commonly used in software development to compile source code and create executable programs or other output files. It automates the process of building complex software projects, including compiling source code, linking object files, and creating executable files or other types of output.

A vector space is a set whose elements are called “vectors” (denoted as $\v{v}$ or $\mathbf{v}$), which have two operations defined on them: addition of vectors and multiplication of a scalar by a vector.

Quaternions are a set of 4-dimensional vectors that are used to represent rotations in computer graphics. They were discovered by William Hamilton as an extension of 2D complex numbers to a 3D equivalent.

Imaginary numbers were invented to solve problems for equations with no real roots. Complex numbers extend imaginary numbers by adding a real number.

This article discusses Eulerian circuits and trails in graphs. An Eulerian circuit is a closed trail that contains every edge of a graph, and an Eulerian trail is an open trail that contains all the edges of a graph but doesn’t end in the same start vertex.

This article also explains the Königsberg Bridge Problem and why it’s impossible to find a trail in it. Finally, there are two implementations in C++ to find Eulerian trails in directed and undirected graphs.

Given a weighted graph $G$ with $V$ vertices and $E$ edges, where all the weights are non-negative, and a source vertex $s$, the single-source shortest path problem consists of finding the distance from $s$ to all other vertices.

A strongly connected component of a directed graph is a subgraph in which there exists a path from every vertex to every other vertex in the subgraph.

This article covers the minimum spanning tree (MST). MSTs have important applications; for example, they can be used to minimize the cost of building a communication network or to identify important features or patterns in a dataset.

I implement the Prim and Kruskal algorithms to find the minimum spanning tree in a graph, with different implementations for sparse and dense graphs. With the theory covered, I also implement an algorithm to find the number of minimal spanning trees in a graph.

A vertex $v$ in a connected graph $G$ is called a cut-vertex if $G - v$ results in a disconnected graph. Note that $G - v$ is an induced subgraph of $G$ (meaning that $G - v$ contains all the vertices of $G$ except $v$ and a set of edges $G - E_v$, where $E_v$ consists of all the edges incident to $v$).

An edge $e = uv$ of a connected graph $G$ is called a bridge if $G - e$ is disconnected (it increases the number of components).

Topological sorting is a linear ordering of the vertices of a directed acyclic graph (DAG) such that for every directed edge (u, v), vertex u comes before vertex v in the ordering. In other words, it is a way to order the vertices of a DAG such that there are no directed cycles.

There are many ways to traverse a graph. For example, through breadth-first search and depth-first search. Exploring it with a breadth-first search has interesting properties, like implicitly computing the distance from a source $s$ to all the reachable vertices. Exploring it with a depth-first search has properties related to edges, like finding back edges, forward edges, and cross edges.

Graph theory has numerous applications in real life. It can be used in problems found in social networks, transportation networks, the internet, chemistry, computer science, and electrical networks, among others.

Integer factorization is the process of decomposing a composite number into a product of smaller integers, if these integers are restricted to be prime numbers then the process is called prime factorization.

A prime number is a natural number greater than $1$ which has no positive divisors other than $1$ and itself.

The Chinese Remainder Theorem (CRT) is a theorem that deals with finding a solution to a system of congruences.

Modular arithmetic is a type of arithmetic that deals with integers and remains within a fixed range of values. It involves performing arithmetic operations such as addition, subtraction, multiplication, and division, but with the added concept of a “modulus” or a “mod” value.

The derivative is a concept that represents the rate of change or the slope of a function at a particular point. It is a fundamental concept in calculus and is used to analyze how a function changes with respect to its input as the input changes very slightly.