C++

Duality: The Hidden Structure of Opposites

Many structures come in pairs: forward/reverse AD, push/pull iteration, encode/decode. Recognizing duality lets you transfer theorems and insights between domains.

Seeing Structure First

A reflection on eleven explorations in generic programming—how algorithms arise from algebraic structure.

Alexander Stepanov: Efficient Programming with Components (A9 Lectures)

Notes

18-part lecture series on efficient programming. Covers the intellectual foundations behind STL.

Elements of Programming

Notes

Rigorous foundations of generic programming. Connects algebra and algorithms. Stepanov’s magnum opus.

Sean Parent: C++ Seasoning (That's a Rotate)

Notes

Classic talk on recognizing algorithmic patterns. ‘No raw loops’ - shows how rotate solves many problems elegantly.

Alga: Algebraic Text Processing with Fuzzy Matching

A mathematically elegant C++20 library for algebraic text processing and compositional parsing with fuzzy matching capabilities.

CBT: Computational Basis Transforms - A Unified Framework

A C++17 header-only library implementing Computational Basis Transforms - a unified framework for understanding how FFT, logarithmic arithmetic, and Bayesian inference are all instances of the same pattern.

libdis: Disjoint Interval Sets as a Complete Boolean Algebra

A modern C++ header-only library implementing disjoint interval sets as first-class mathematical objects with rigorous Boolean algebra operations.

Alea: A Modern C++ Library for Algebraic Random Elements

Alga: Algebraic Parser Composition through Monadic Design A C++20 Template Library for Type-Safe Text Processing

ZeroIPC: Transforming Shared Memory into an Active Computational Substrate

ZeroIPC transforms shared memory from passive storage into an active computational substrate, enabling functional and reactive programming paradigms across process boundaries with zero-copy performance.

Algebraic Composition for Streaming Data Reduction: A Type-Safe Framework with Numerical Stability

Algebraic Hashing A Modern C++20 Library for Composable Hash Functions Version 2.0

Cryptographic perfect hash functions: A theoretical analysis on space efficiency, time complexity, and entropy

maph: Maps Based on Perfect Hashing for Sub-Microsecond Key-Value Storage

PFC: Zero-Copy Data Compression Through Prefix-Free Codecs and Generic Programming

Differentiation: Three Ways

Three approaches to computing derivatives—forward-mode AD, reverse-mode AD, and finite differences—each with different trade-offs. Understanding when to use each is essential for numerical computing and machine learning.

maph: Maps Based on Perfect Hashing for Sub-Microsecond Key-Value Storage

A high-performance key-value storage system achieving sub-microsecond latency through memory-mapped I/O, approximate perfect hashing, and lock-free atomic operations. 10M ops/sec single-threaded, 98M ops/sec with 16 threads—12× faster than Redis, 87× …

PFC: Zero-Copy Data Compression Through Prefix-Free Codecs

A header-only C++20 library that achieves 3-10× compression with zero marshaling overhead. PFC makes compression an intrinsic type property through prefix-free codes (Elias Gamma/Delta, Fibonacci, Rice), algebraic types, and Stepanov's generic …

Accumux: Compositional Online Statistical Reductions in C++

A modern C++20 library for compositional online data reductions with numerically stable algorithms and algebraic composition.

Runtime Polymorphism Without Inheritance

Sean Parent's type erasure technique provides value-semantic polymorphism without inheritance. Combined with Stepanov's algebraic thinking, we can type-erase entire algebraic structures.

Numerical Integration with Generic Concepts

Numerical integration demonstrates both classical numerical analysis and Stepanov's philosophy: by identifying the minimal algebraic requirements, our quadrature routines work with dual numbers for automatic differentiation under the integral.

Reverse-Mode Automatic Differentiation

Reverse-mode automatic differentiation powers modern machine learning. Understanding how it works demystifies PyTorch, JAX, and TensorFlow—it's just the chain rule applied systematically.

Algebraic Hashing: Composable Hash Functions Through XOR

Most hash libraries treat hash functions as black boxes. Algebraic Hashing exposes their mathematical structure, letting you compose hash functions like algebraic expressions—with zero runtime overhead.

The Core Insight

Hash functions form an abelian …

Numerical Differentiation

The art of numerical differentiation lies in choosing step size h wisely—small enough that the approximation is good, but not so small that floating-point errors dominate.

Forward-Mode Automatic Differentiation

Dual numbers extend our number system with an infinitesimal epsilon where epsilon^2 = 0. Evaluating f(x + epsilon) yields f(x) + epsilon * f'(x)—the derivative emerges automatically from the algebra.

Teaching Linear Algebra with C++20 Concepts

elementa is a pedagogical linear algebra library where every design decision prioritizes clarity over cleverness—code that reads like a textbook that happens to compile.

Polynomials as Euclidean Domains

The same GCD algorithm works for integers and polynomials because both are Euclidean domains. This profound insight shows how algebraic structure determines algorithmic applicability.

Exact Rational Arithmetic

Rational numbers give us exact arithmetic where floating-point fails. The implementation reveals deep connections to GCD, the Stern-Brocot tree, and the algebraic structure of fields.

How Iterators Give You N+M Instead of N×M

The problem is combinatorial. You have N algorithms (sort, search, find, copy) and M containers (array, list, tree, hash table). The naive approach: implement each algorithm for each container. That’s N×M implementations.

The insight is to …

Is It Prime?

The Miller-Rabin primality test demonstrates how probabilistic algorithms can achieve arbitrary certainty, trading absolute truth for practical efficiency.

Modular Arithmetic as Rings

Integers modulo N form a ring—an algebraic structure that determines which algorithms apply. Understanding this structure unlocks algorithms from cryptography to competitive programming.

One Algorithm, Infinite Powers

The Russian peasant algorithm teaches us that one algorithm can compute products, powers, Fibonacci numbers, and more—once we see the underlying algebraic structure.