Cracovians: The Twisted Twins of Matrices

Linear algebra is typically explained using matrices. But matrix theory is just one possible perspective. Below, I describe an alternative approach to linear algebra. Tadeusz Banachiewicz (1882–195…

posit: thin triangle other tricks (REVEALED!)

First post in breaking down why the posits presentations are egregiously disingenuous.

Last fifty years of integer linear programming: a focus on recent practical advances

<div><p>Mixed-integer linear programming (MILP) has become a cornerstone of operations research. This is driven by the enhanced efficiency of modern solvers, which can today find globally optimal solutions within seconds for problems that were out of reach a decade ago. The versatility of these solvers allowed successful applications in many areas, such as transportation, logistics, supply chain management, revenue management, finance, telecommunications, and manufacturing. Despite the impressive success already obtained, many challenges remain, and MILP is still a very active field.</p><p>This article provides an overview of the most significant results achieved in advancing the MILP solution methods. Given the immense literature on this topic, we made deliberate choices to focus on computational aspects and recent practical performance improvements, emphasizing research that reports computational experiments. We organize our survey into three main parts, dedicated to branch-and-cut methods, Dantzig-Wolfe decomposition, and Benders decomposition. The paper concludes by highlighting ongoing challenges and future opportunities in MILP research.</p></div>

The Hat, the Spectre and SAT solvers

Introduction In this blog post you are going to read about two things:

Solving LinkedIn Queens with SMT

For sure easier than solving it in SAT!

The Hashtable Packing Problem

Modern Minimal Perfect Hashing: A Survey

Given a set $S$ of $n$ keys, a perfect hash function for $S$ maps the keys in $S$ to the first $m \geq n$ integers without collisions. It may return an arbitrary result for any key not in $S$ and is called minimal if $m = n$. The most important parameters are its space consumption, construction time, and query time. Years of research now enable modern perfect hash functions to be extremely fast to query, very space-efficient, and scale to billions of keys. Different approaches give different trade-offs between these aspects. For example, the smallest constructions get within 0.1% of the space lower bound of $\log_2(e)$ bits per key. Others are particularly fast to query, requiring only one memory access. Perfect hashing has many applications, for example to avoid collision resolution in static hash tables, and is used in databases, bioinformatics, and stringology. Since the last comprehensive survey in 1997, significant progress has been made. This survey covers the latest developments and provides a starting point for getting familiar with the topic. Additionally, our extensive experimental evaluation can serve as a guide to select a perfect hash function for use in applications.

The New Godel Prize Winner Tastes Great and is Less Filling

David Zuckerman The 2025 Gödel Prize has been awarded to Eshan Chattopadhyay and David Zuckerman for their paper Explicit two-source extrac...

TPDE: A Fast Adaptable Compiler Back-End Framework

Fast machine code generation is especially important for fast start-up just-in-time compilation, where the compilation time is part of the end-to-end latency. However, widely used compiler frameworks like LLVM do not prioritize fast compilation and require an extra IR translation step increasing latency even further; and rolling a custom code generator is a substantial engineering effort, especially when targeting multiple architectures. Therefore, in this paper, we present TPDE, a compiler back-end framework that adapts to existing code representations in SSA form. Using an IR-specific adapter providing canonical access to IR data structures and a specification of the IR semantics, the framework performs one analysis pass and then performs the compilation in just a single pass, combining instruction selection, register allocation, and instruction encoding. The generated target instructions are primarily derived code written in high-level language through LLVM's Machine IR, easing portability to different architectures while enabling optimizations during code generation. To show the generality of our framework, we build a new back-end for LLVM from scratch targeting x86-64 and AArch64. Performance results on SPECint 2017 show that we can compile LLVM-IR 8--24x faster than LLVM -O0 while being on-par in terms of run-time performance. We also demonstrate the benefits of adapting to domain-specific IRs in JIT contexts, particularly WebAssembly and database query compilation, where avoiding the extra IR translation further reduces compilation latency.

The radix 2^51 trick