Tommy Reddad

Pretty Brownian motion simulations

Fri, 17 Apr 2026 00:00:00 +0000

As I went through a laptop upgrade and migration recently, I discovered remnants of my old academic webpage. I found this post which has some pretty diagrams for simulations of Brownian motion and related mathematical objects. I thought it could be interesting to reproduce so that these pretty diagrams don’t get lost forever! Incidentally, I really enjoyed learning about Brownian motion, and I revisit my lecture notes from time to time.

In this page, you will find a few small simulations I ran about Brownian motion and related processes. I just think it looks cool. Don’t expect any kind of real explanation of what is going on. I learned most of what I know about Brownian motion from Louigi Addario-Berry’s Fall 2016 course on Brownian motion, largely based on the free book by Peter Mörters, and Yuval Peres.

A gender-swap EarthBound for my daughter

Thu, 04 Dec 2025 00:00:00 +0000

I had some fun over the last week preparing a gender-swapped version of EarthBound for the SNES. I made use of the excellent tool CoilSnake, which can be used to patch and rebuild an EarthBound ROM with modified assets and scripts.

The full set of changes includes things like:

gender swapping all Ness sprites, replacing his baseball cap with a red bow. Includes changes to Ness robot sprites, cycling sprites, climbing sprites, etc. There were maybe a dozen or so sprites that needed adjustment.

Coming soon: a few exploratory projects

Sat, 28 Sep 2024 00:00:00 +0000

I’ve been messing around in my free time doing a bit of coding for fun and to explore areas of interest. It’s a bit hard to motivate myself to code during my free time considering coding is also part of my day job. But my day job is not very technically interesting, so this is how I get to explore the fun of coding. I’ll briefly describe a few of these projects which I have in mind.

Rebuilt this website with Hugo

Fri, 09 Aug 2024 00:00:00 +0000

I decided yesterday to rebuild this website using Hugo, while it was originally built using Jekyll. In this post I’ll discuss my experience.

galtonwatson, a Go module for efficient manipulation of Galton-Watson trees

Sat, 21 Nov 2020 00:00:00 +0000

galtonwatson is a Go module in early development which implements efficient algorithms for the generation and manipulation of Galton-Watson trees. By extension, this tool can be used to generate uniformly random samples from many classes of rooted trees, including:

uniformly random binary trees of a given size,
uniformly random d-ary trees of a given size,
uniformly random Cayley trees, i.e., unordered labeled trees of a given size,
uniformly random ordered trees of a given size,
etc.

The Burrows-Wheeler transform, revisited

Sun, 27 Sep 2020 00:00:00 +0000

In my previous post, I discussed the Burrows-Wheeler transform, and embedded an HTML form on the page which dynamically computed the forward and inverse Burrows-Wheeler transforms with an additional move-to-front compression step. My initial implementation of this form was done in pure JavaScript with no particular care for efficiency, and in particular used \(O(n^2 \log n)\) space and time. The new implementation on the same page reduces this complexity to \(O(n \log n)\) time and \(O(n)\) space, making the form usable on larger texts. It is possible, as noted in the paper of Gil and Scott, that this time complexity can be reduced to \(O(n)\) by using linear time suffix array construction algorithms.

The Burrows-Wheeler transform and move-to-front compression

Sat, 08 Aug 2020 00:00:00 +0000

Alphabet:

Bit count:

Alphabet:

Bit count:

The above form implements algorithms to code and decode from the move-to-front compression scheme, with or without an additional application of the Burrows-Wheeler transform. These extremely slick algorithms blew my mind when I first heard of them. They are currently in use, among other places, in [bzip2](https://en.wikipedia.org/wiki/Bzip2) file compression.

Splay trees and optimality

Sat, 27 Jul 2019 00:00:00 +0000

The splay tree is probably my favourite data structure. Is it useful in practice? Probably not, but its remarkable optimality properties coupled with its bare simplicity are so tantalizing that I’ve fallen in love with splaying. In the rest of this post, I’ll describe the splay tree structure, and present some of my favourite splay tree properties. You will also find an instructive D3 visualization of a splay tree in motion.

A uniform sum threshold problem

Mon, 22 Jul 2019 00:00:00 +0000

Let \(U_1, U_2, \dots\) be an infinite sequence of independent \(\mathrm{Uniform}[0, 1]\) random variables. Let \(N\) be the minimum index for which \[ U_1 + U_2 + \dots + U_N > 1 . \] What is the expected value \(\mathbf{E}\{N\}\)?

Generating spherical points without complex operations

Mon, 08 Jul 2019 00:00:00 +0000

These days, most of everyone’s favourite languages and libraries for scientific computing come ready-equipped with random number generators for most common univariate distributions: the uniform, binomial, normal, geometric, exponential, beta, etc. In my experience, multivariate generation is comparatively hit-or-miss. But in any case, since documentation usually doesn’t specify implementation methods or running time, you usually can’t be sure of the efficiency of one of these functions without personally examining some source code, or being lucky and finding that someone else on StackExchange already did. Thankfully, when in doubt, one can always refer to the excellent and totally free book Non-Uniform Random Variate Generation by my old PhD supervisor Luc Devroye. In fact, it seems this book is even more than free, as stated in this plea posted by the author on the book’s webpage.

I give anyone the permission, even without asking me, to take these PDF files to a printer, print as many copies as you like, and sell them for profit. If you would like me to advertise the sales points of the hard copies, please let me know. To the libraries: Please do not charge patrons for copying this book. I grant everyone the right to copy at will, for free.

Discrete minimax estimation with trees

Thu, 27 Jun 2019 00:00:00 +0000

This morning, I submitted the final version of my paper Discrete minimax estimation with trees (Devroye and Reddad, 2019), which is to appear in the Electronic Journal of Statistics. I think this paper is conceptually quite interesting, and I’m very happy with the final result, so in this post I’ll describe some of the main ideas present in the work.

Some conditioned Galton-Watson trees never grow

Wed, 26 Jun 2019 00:00:00 +0000

When programmers hear the phrase “random tree,” they most likely think of a random binary search tree, i.e., a binary search tree built from the insertion of a uniformly random permutation of \(n\) keys—denote such a tree by \(\mathrm{BST}_n\). A mathematician might instead think that a ``random tree’’ is more likely to be a uniformly random tree taken from some class, like the set of all ordered binary trees with \(n\) nodes—denote such a tree by \(\mathrm{UNIF}_n\). Of course, neither would be wrong. It should be clear, though, that these two distributions on the space of binary trees are quite different. In particular, most undergraduate students of computer science learn, through the study of comparison-based sorting algorithms, that \[ \mathbf{E}\{\mathrm{height}(\mathrm{BST}_n)\} = \varTheta(\log n) , \] and some will learn that \[ \mathbf{E}\{\mathrm{height}(\mathrm{UNIF}_n)\} = \varTheta(\sqrt{n}) . \] Though random binary search trees might seem more immediately relevant to programmers, uniformly random binary trees are part of a bigger picture which is comparatively more versatile in the probabilistic analysis of algorithms. To this end, we introduce the concept of Galton-Watson trees.

About

Mon, 01 Jan 0001 00:00:00 +0000

Resumé (pdf)

I’m a software developer at Google and computer scientist based in Albuquerque, New Mexico. My previous academic work concerned applied probability theory, statistics, and the analysis of algorithms. I’m interested in deep learning especially concerning natural language understanding. Interested in games, math, sourdough bread, and jiu-jitsu.

Ex-PhD student of Luc Devroye at McGill University, where I worked on many different problems concerning random graphs, neural networks, minimax density estimation, and properties of Galton-Watson trees. Completed a Master of Computer Science at the Computational Geometry Lab at Carleton University in Ottawa, under the supervision of Pat Morin, Vida Dujmović, and Jit Bose, where I produced a thesis on Encoding Arguments.

Publications