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People Keep Inventing Prolly Trees
A novel data structure has been invented four times now, with three different names. We think that's a good sign.
Topics /
Math & Algorithms
The most popular blog posts about math & algorithms on Hacker News. 2,851 posts.
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Visualizing the Math Behind Logistic Regression and Newton's Method
The Laziest Programmer - Because someone else has already solved your problem.
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Who Can Understand the Proof? A Window on Formalized Mathematics
Stephen Wolfram discusses understanding proofs discovered using automated theorem proving. Based on Wolfram’s proof of the simplest axioms of logic of Boolean algebra.
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The 8000th Busy Beaver number eludes ZF set theory (2016)
I’ve supervised a lot of great student projects in my nine years at MIT, but my inner nerdy teenager has never been as personally delighted by a project as it is right now. To…
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Why cans of soup are shaped the way they are
What is special about cylinders, and why do they have specific aspect ratios?
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Using k-d trees to efficiently calculate nearest neighbors in 3D vector space
What would the world look like if there were only 16 colors? I use k-d trees to perform nearest neighbor search to find out.
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Time-Series Compression Algorithms
These algorithms aren't magic – but combined they can save over 90% of storage costs and speed up queries. Here’s how 5 common ones work.
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Relaxed Radix Balanced Trees (2024)
I’m adding immutable vectors to my language, Ivan, and needed to pick a suitable data structure to implement them. Clojure uses Persistent Vectors (PVs) which support lookups, updates, and appends all in ‘effectively’ constant time. However, it doesn’t have efficient insert or merge operations. Relaxed Radix Balanced…
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The 2,500 year old history of why Python’s all([]) returns True
How a Python function relates to a two thousand year old debate in philosophy
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Memory access is O(N^[1/3])
In computer science, we often compare the efficiency of algorithms by describing their runtime as a function of the size of the input. Sorting is O(n * log(n)), meaning that sorting a list of N items takes an amount of time proportional to the number of items multiplied by its logarithm. Matrix multiplication is…
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What if AI doesn't need more RAM but better math?
How TurboQuant compresses the KV cache without losing accuracy, and what that could mean for memory stocks
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What number is halfway between 1 and 9?
A new information-theoretical model of human sensory perception and memory sheds light on some peculiarities of the nervous system.
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The bunkbed conjecture is false
What follows is an unusual story of perseverance. We start with a conjecture and after some plot twists end up discussing the meaning of truth. While the title is a spoiler, you might not be able t…
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Introduction to Clifford Algebra (2006)
Clifford algebra gives a unified view of many things that otherwise would need to be understood separately.
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Gödel's Incompleteness Theorem in Bash
Gödel’s first incompleteness theorem states that there are mathematical statements that can neither be proven or disproven. His proof is fairly difficult to ...
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Primitive Recursive Functions for a Working Programmer
Programmers on the internet often use Turing-completeness terminology. Typically, not being Turing-complete is extolled as a virtue or even a requirement in specific domains. I claim that most such discussions are misinformed --- that not being Turing complete doesn't actually mean what folks want it to mean, and is…
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Timsort (2019)
Timsort: A very fast , O(n log n), stable sorting algorithm built for the real world — not constructed in academia. Timsort is a sorting algorithm that is efficient for real-world data and not created in an academic laboratory. Tim Peters created Timsort for the Python programming language in 2001.
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Algorithm for Drawing Trees
I recently wanted to take a hierarchy of items and draw them in a nice tree structure. For example, a Family Tree. At the time, I thought “This will be easy, I’ll just Google for an alg…
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A Basic Introduction to Neural Networks (2007)
Although the mathematics involved with neural networking is not a trivial matter, a user can rather easily gain at least an operational understanding of their structure and function.
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City street orientations around the world
This post is adapted from this research paper that you can read/cite for more info. It analyzes and visualizes 100 cities around the world.
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Exact Polygonal Filtering: Using Green's Theorem and Clipping for Anti-Aliasing
Efficient filtering of polygonal shapes with closed-form solutions based on Green’s theorem with piecewise-polynomial filters (e.g. box/bilinear/bicubic).
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Are polynomial features the root of all evil? (2024)
High-degree polynomial features aren’t inherently unstable: the power basis and unscaled inputs are. This post shows how regularization and Bernstein polynomials make polynomial regression behave.
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A new Sudoku layout with 81 uniquely shaped cells
My journey making a Sudoku variant that doesn’t have rows or columns
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Dissecting Lemire's nearly divisionless random number generator
A very late blog post announcing the readability contest winners, a new (very) annotated implementation, and a surprising sort-of security issue found along the way.
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Some fun with π in Julia
A Pi Day tour of π in Julia: how Julia represents π, computing it with inline assembly and Taylor series (Madhava and Machin), and finding guaranteed bounds with interval arithmetic.
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