Notes

I like dancing bachata so I made this page to record my bachata journey

Ordinary moments don’t have a value right now but they will value over time. I decided to document my life by journaling, printing pictures and making vlogs.

Kubernetes is an open-source container orchestration platform that automates the deployment, scaling, and management of containerized applications.

I try to improve my productivity every day to get better use of my time. This article summarizes the things I’m used to doing everyday at work. These are divided into: getting used to multitasking, having time for deep focus and for breaks, handling incoming emails, task management and development tools.

This is my interview preparation plan for a software engineering role, I share a summary of my preparation, a plan of problems to tackle for coding interviews, the topics to study for a system design interviews and the questions to prepare for the behavioral section.

When designing a system it’s important to consider the limitations of the technologies chosen, making some approximate calculations when the system is designed help us decide on the tradeoffs of the different approaches, these approximations include

This article has a table with latency comparison numbers, common numbers used in system design calculations and some challenges to put all of these info into practice.

A super brief introduction to what ML is for, the problems where it can be applied along with some example applications.

A glossary of terms used in machine learning.
For an up to date glossary also check https://developers.google.com/machine-learning/glossary

Hyperparameter tuning is a process of finding out by trial and error the hyperparemeteres (the parameters fed into the model for it to change its inner parameters) to converge efficiently.

The algorithms that we use every day to manipulate data assume that we have access to all the data we need. What if there’s more data that can fit in a single computer or if accessing the data itself to do searches is expensive? If so, we can use specialized data structures that can help us “estimate” the actual value without actually computing it, in some cases an estimate might be good enough. These data structures are: count-min sketch, bloom filters, and reservoir-sampling.

Expectation maximization is a method of finding maximum likelihood estimates of parameters of a model. The method alternates between making an expectation (E) step based on the current estimate of the parameters and a maximization (M) step which computes new parameters.

A Bayesian network is a directed graph in which each node is annotated with quantitative probability information. This article covers the definition of a bayesian network with a graphical representation, the determination of independence between variables and the problem of finding the probability distribution of a set of query values given some observed events.

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

This article covers key designs in kafka such as how messages for a topic are shared into partitions assigned to brokers. Then we see some guarantees about producers, consumers and consumer groups.

The pattern of batching data up in memory, tracked in a write ahead log, and periodically flushed to disk is ubiquitous today. OSS examples are LevelDB, Cassandra, InfluxDB, or HBase.

In this article I implement a tiny memtable for a timeseries database in golang and briefly talk about how it can be compressed into a sorted string table.

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

This article covers key design features of cassandra such as the usage of consistent hashing, the write pattern to a write ahead log and a memtable, the read pattern from the memtable and from sstables, and finally and most important, some examples about data modeling for different types of queries.

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.

This article covers some strategies for partitioning including random partitioning, by hash key, by range and a hybrid approach for skewed workloads. We also see strategies to rebalance partitions if there's a static or dynamic number of partitions.

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

This article covers well known non functional requirements such as reliability, availability, scalability, performance and durability.

Future/promises refer to constructs used to synchronize program execution. Learning how it works under the hood by implementing it is a great fundamental skill to know.

This article is about writing an A+ Promise implementation from scratch following the A+ promise spec in JavaScript the TDD way.

Let $a,b \in \mathbb{Z}$, we say that $a$ divides $b$, written $a \given b$, if there’s an integer $n$ so 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 $a$.

This article covers the greatest common divisor and how to find it using the euclidean algorithm, the extended euclidean algorithm to find solutions to the equation $ax + by = gcd(a, b)$ where $a, b$ are unknowns.

Flat shading is the simplest shading model, we cover the advantages/disadvantages and a simple implementation in GLSL.

Diffuse shading is a technique to render the surface of objects that are not shinny.

As an example on the picture (credits to Marc Kleen) we see a real life car with a matte coating which we want to emulate using the Lambertian shading model.

Surface shading is a process to color a surface, in computer graphic applications this is done to mimic how objects look in real life. This article covers the variables available in the rendering pipeline.

A first person camera captures objects from the viewpoint of a player’s character. Some aspects have to be considered like the characteristics of the camera (orbiting with the mouse and translation with keyboard keys) as well as how we could capture all these characteristics with math and linear algebra.

In this article I analyze the math needed to design and implement a 1st person shooter camera in C++.

Quaternions are an alternate way to describe orientation or rotations in 3d space using an ordered set of four numbers. They have the ability to uniquely describe any 3d rotation about an arbitrary axis and do not suffer from a problem using euler angles called gimbal lock.

GCC is a suite of compilers for various programming languages, including C, C++. In this article, I cover the compilation stages and the flags used to compile source code into a binary

“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.

In this article I cover the following: targets and prerequisites, variables, recipes to build an out of date target and finally an example of how to use it in a simple C++ project.

CMake is a cross-platform build system generator of Makefiles, projects specify their build process with platform-independent CMake listfiles included in each directory of a source tree with the name CMakeLists.txt. This article explains how to use CMake to build projects.

Refresher notes on C++, includes mechanics of a C++ program, program structure, pointers, functions, classes and misc operations

The math behind culling and clipping and how it’s related with the camera and with what it sees.

• Culling is a process where geometry that’s not visible from the camera is discarded to save processing time.
• Clipping is a process that removes parts of primitives that are outside the view volume (clipping against the six faces of the view volume).

An affine space is a generalization of the notion of a vector space, but without the requirement of a fixed origin or a notion of “zero”.

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 an scalar by a vector.

This article covers some examples of vector spaces, basis of vectores spaces and linear maps.

In an affine space any point can be represented by a sum of an origin point plus a set of scaled vector. This article covers defining all the points in a triangle in an affine space.

Different algorithms to test geometric properties like finding the intersection of two lines.

One matrix transformation in the 3D to a 2D transformation pipeline is the viewport transform where objects are transformed from normalized device coordinates (NDC) to screen coordinates (SC).

In short it's the transformation of numbers in the range [-1, 1] to numbers corresponding to pixels on the screen, which is a linear mapping computed with linear interpolation.
In this article I cover the math behind the generation of the viewport transformation matrix.

A normal vector to a curve at a particular point is a vector perpendicular to the tangent vector of the curve at that point (also called a gradient).

An eigenvalue represents how the object scales (or stretches/compresses) a particular direction (or eigenvector) when acted upon by the object. This article covers how to find these values in a square matrix as well as how it’s applicable in compute graphics.

In projective geometry unlike euclidean geometry, two parallel lines meet at a point. Desargues introduced the concept of a line at finity where a point at infinity can be defined. This article covers the need of a point at infinity in projective space, the line at infinity and the projective plane.

Ray tracing is the process to identify the color of all the pixels in a 2d screen by emitting rays from all the pixels simulating how light travels in real life. This article covers the math for the ray generation from each pixel for both orthographic and perspective cameras.

Rendering is a process that takes an input a set of objects and produces as its output an array of pixels (image) each of which stores information about the color of the image at a particular point in a grid (determined by the target width and height).

In Computer Graphics 3D objects created in an abstract 3D world will eventually need to be displayed in a screen, to view these objects in a 2D plane like a screen objects will need to be projected from the 3D space to the 2D plane with a transformation matrix.

In this article I cover two types of transformations: Orthographic projection and Perspective projection and analyze the math behind the transformation matrices.

One matrix transformation in the 3D to a 2D transformation pipeline is the view transform where objects are transformed from world space to view space. a transformation matrix.

In this article I cover the math behind the generation of this transformation matrix.

Taking multiple matrices each encoding a single transformations and combining them is how we transform vectors between different spaces. This article creating a transformation matrix that combines a rotation followed by a translation, a translation followed by a rotation and creating transformation matrices to transform between different coordinate systems.

Perspective projection is a fundamental projection technique that transforms objects in a higher dimension to a lower dimension. This transformation is usually used for objects in a 3d world to be rendered into a screen (a 2d surface), in the transformation these objects give the realistic impression of depth.

This article covers the math behind it and how to generate the transformation matrix to achieve the transformation.

Orthographic projection is a fundamental projection technique that transforms objects in a higher dimension to a lower dimension. This transformation is usually used for objects in a 3d world to be rendered into a screen (a 2d surface) and in the process keeps parallel lines parallel in the lower dimension.

This article covers the math behind it and how to generate the transformation matrix to achieve the transformation.

We build different types of transformation matrices to translate objects along cardinal axes, arbitrary axes in 2d and 3d with matrix multiplication!

Euler angles are a way to describe the orientation of a rigid body with 3 values, these values represent 3 angles:

• yaw - Rotation around the vertical axis
• pitch - Rotation around the side-to-side axis
• roll - Rotation around the front-to-back axis

Shearing is a transformation that skews the coordinate space, the idea is to add a multiple of one coordinate to the other

The basics of rotation in 2d and 3d for computer graphics with a focus on 3d rotation about cardinal axes and 3d rotation with quaternions.

For quaternions, please also look at https://eater.net/quaternions amazing animations!

We build different types of transformation matrices to scale objects along cardinal axes, arbitrary axes in 2d and 3d with matrix multiplication!

A linear transformation can be represented with a matrix which transforms vectors from one space to another. Transformation matrices allow arbitrary transformations to be displayed in the same format. Also matrices can be multiplied to enable composition. This article covers how to think and reason about these matrices and the way we can represent them (row vectors vs column vectors).

The position and orientation of an object in real life can be described with direction and magnitude e.g. the TV is 3 meters in front of me. While that description is good for me it might be that for someone else in a room the TV is 5 meters to the right of that person. Information about objects are given in the context of a reference frame. Usually in Computer Graphics objects need to be expressed with respect to the camera frame, this article covers why we need to have multiple reference frames as well as the math needed to express objects in a different reference frame.

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.

This article covers the definition of a quaternion, its notation and operations.

Imaginary numbers were invented to solve problems for equestions with no real roots, complex numbers extend imaginary numbers by adding a real number.

This article covers the definition of complex numbers, operations such as addition, product, norm, conjugate, inverse and square root. Finally, this article covers the geometric and polar representations of complex numbers.

Hamiltonian graphs and hamiltonian cycles.

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 how it’s impossible to find a trail on it. Finally there are two implementations in C++ to find Eulerian trails in directed and underected graphs.

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

In this article I describe the problem in a weighted and unweighted graph as well as implementations using BFS for unweighted graphs and Dijkstra's algorithm for weighted graphs using an array and a priority queue.

Introduction to trees in Graph Theory

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.

In this article I implement Tarjan's algorithm to find strongly connected components in a graph.

This article covers minimum spanning tree (MST), MSTs have important applications, MST can be used to minimize the cost of building a communication network, or it can be used 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, also with the theory covered I also implement an algorithm to find the number of nimal 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 $G$ but $v$ and a set of edges $G - V$ where $V$ consists of all the edges incident to $v$).

In this article I implement an algorthm to find the articulation points in an undirected graph, also I explain biconnected components in an undirected graph and explain concepts such as edge connectivity and vertex connectivity.

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

In this article I implement an algorthm to find the bridges of an undirected graph using DFS. Next I describe an algorithm to find strong bridges in directed graphs.

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.

In this article I implement the topological sorting algorithm as well as an example of how to use it to find the shortest path in a directed acyclic graph.

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 about edges like finding back edges, forward edges and cross edges.

This article has implementations for both BFS and DFS.

Graph Theory has numerous applications in real life, it can be used in problems found in social networks, transportation networks, the internet, chemistry, computer sciense, electrical networks among others.

In general, any problem that involves relationships between objects can be modeled as a graph.

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.

This article covers factorization using trial division and fermat factorization through Pollard's Rho algorithm and using the sieve of eratosthenes.

The divisor function returns the number of divisors of an integer. This article covers important relations of the divisor function and prime numbers.

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

This article covers different algorithms for checking if a number is prime or not including a naive test, the erathostenes sieve, the euler primality test and the miller-rabin primality test.

This article describes and implements a solution for the following problem, given two numbers $n$ and $k$ find the greatest power of $k$

Let $n!_{\%p}$ be a special factorial where $n!$ is divided by the maximum exponent of $p$ that divides $n!$. This article describes this problem and its solution with an implementation in C++.

The discrete logarithms finds a solution for $x$ in the congruence $a^x \equiv b \pmod{n}$ where $a$, $b$ and $n$ are integers, $a$ and $n$ are coprime. I cover two algorithms to solve this problem: by trial multiplication and using baby step giant step.

The chinese remainder theorem (CRT) is a theorem that deals with finding a solution to a system of congruences.

This article covers the defition of the CRT and an example implementation in C++.

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.

This article covers the definition a congruence relation, and some of its properties like addition, multiplication, exponentiation and inverse. Next I show how we can use the extended euclidean algorithm to find the modular multiplicative inverse in a general case and in the case of coprime numbers.

The extended euclidean algorithm finds solutions to the equation $ax + by = gcd(a, b)$ where $a, b$ are unknowns. This article covers a few applications of the extended euclidean algorithm like finding the modular multiplicative inverse of a number, and finding solutions for linear congruence equations.

Given two numbers $a$ and $n$ finding $a^n$ involves doing $n$ multiplications of $a$, however, it’s possible to do this in $log(n)$ operations by using binary exponentiation.

The erathostenes sieve is an algorithm to find prime numbers up to a positive number $n$ using $O(n)$ space.

The euclidean algorithm finds the greatest common divisor of two numbers. In this article I implement the algorithm from scratch in C++.

Euler’s phi function represented as $\phi(n)$ gives for a number $n$ the number of coprimes in the range $[1..n]$, in other words the quantity numbers in the range $[1..n]$ whose greatest common divisor with $n$ is the unity. In this article I try to explain how it works and implement it in C++.

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.

This article covers physical and geometric interpretation of the derivative as well as some applications like finding maxima and minima in a function and newton-raphson.

An integral is a mathematical concept that represents the accumulation or summing up of quantities over a certain interval or region. In this article we’ll discuss the properties of the integral by looking an example of antidifferentiation and some examples of evaluating definite integrals.

Taylor Series helps approximate the value of a definite integral for a function whose antiderivative is hard to find. This article explains the key ideas behind Taylor’s Theorem and an example of approximating its value with a polynomial function.

This article gives an introduction to calculus starting with the concept of a function and how calculus helps us solve problems related to determining tangents to curves (expressed as functions), finding the minim/maxima like determinining the maximum range of a projectile, and to find the length of curves areas and volumes.