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Alex Towell6d ago • 12 min read
Bits Follow Types
Codecs are not ad-hoc bit formats. They are constructions on the algebraic structure of types.
#c#genericprogramming#codecs#informationtheory
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Alex Towell6d ago • 4 min read
Free Algebras: Why Lists and Polynomials Are Universal
The free monoid on a set is the type of lists over that set. The universal property says fold is the...
#c#genericprogramming#algorithms#monoids
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Alex Towell6d ago • 3 min read
Exact Rational Arithmetic
Rational numbers give exact arithmetic where floating-point fails. The implementation connects GCD, ...
#c#genericprogramming#numbertheory#fractions
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Alex Towell6d ago • 3 min read
Modular Arithmetic as Rings
Integers modulo N form a ring, an algebraic structure that determines which algorithms apply. Unders...
#c#genericprogramming#numbertheory#rings
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Alex Towell6d ago • 2 min read
How Iterators Give You N+M Instead of NxM
Iterators reduce the NxM algorithm-container problem to N+M by interposing an abstraction layer, fol...
#c#genericprogramming#stepanov#iterators
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Alex Towell6d ago • 5 min read
Semirings: One Algorithm, Six Graph Problems
A semiring has two monoidal operations linked by distributivity. Matrix multiplication over differen...
#c#genericprogramming#algorithms#semirings
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Alex Towell6d ago • 5 min read
Streaming Statistics, One Monoid at a Time
Online accumulators are monoids. Default construction is the identity, combination via += is the bin...
#c#genericprogramming#algorithms#monoids
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Alex Towell6d ago • 5 min read
Homomorphisms: The Maps Between Structures
A homomorphism preserves structure. fold is the universal homomorphism from the free monoid. This is...
#c#genericprogramming#algorithms#monoids
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Alex Towell6d ago • 14 min read
Seeing Structure First
A reflection on eleven explorations in generic programming, and how algorithms arise from algebraic ...
#c#genericprogramming#algorithms#monoids
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Alex Towell6d ago • 3 min read
Polynomials as Euclidean Domains
The same GCD algorithm works for integers and polynomials because both are Euclidean domains. One st...
#c#genericprogramming#algebra#polynomials
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Alex Towell6d ago • 3 min read
Packed Containers: Zero-Waste Bit-Level Storage in C++
What if containers wasted zero bits? A C++ library for packing arbitrary value types at the bit leve...
#c#datastructures#informationtheory#genericprogramming
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Alex Towell6d ago • 10 min read
When Lists Become Bits
Prefix-freeness is the property that lifts the free-monoid construction into bit space.
#c#genericprogramming#codecs#informationtheory
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Alex Towell6d ago • 3 min read
Lattices: Fixed Points and Iteration
A lattice has two operations, meet and join, satisfying absorption laws. Tarski's theorem gives a ge...
#c#genericprogramming#algorithms#lattices
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Alex Towell6d ago • 4 min read
One Algorithm, Infinite Powers
The Russian peasant algorithm computes products, powers, Fibonacci numbers, and more, once you see t...
#c#genericprogramming#algorithms#monoids
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Alex Towell6d ago • 4 min read
Choosing the Algebra
The Stepanov series showed that algorithms arise from algebraic structure. This post is about the fl...
#c#genericprogramming#algorithms#algebraicstructures
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