Some maths problems are easy to state and hard to speed up. Here is one: given a list of numbers, are there three that add up to zero? It is called 3SUM. The textbook method takes time roughly proportional to the square of the number of items — written as n squared, or O(n²). For decades, nobody had found a method that was better in the technical sense that counts in this area of computer science.
A preprint posted to arXiv on 5 October 2026 reports that the textbook limit has finally been beaten, if only by a whisker. Josh Alman of Columbia University and Virginia Vassilevska Williams of MIT describe a deterministic algorithm for 3SUM that runs in O(n^1.9992) time for integers of polynomial size. In the same paper, they describe an algorithm for All-Pairs Shortest Paths — finding the shortest route between every pair of places in a network — that runs in O(n^2.9995) time, just below the textbook O(n³).
Those numbers look almost identical to the old ones. 1.9992 is barely below 2, and 2.9995 is barely below 3. So why does anyone care? Because in fine-grained complexity theory, the question is not whether an algorithm is a little faster in practice, but whether the exponent itself can be pushed below the textbook value at all. The 3SUM and APSP hypotheses — the working assumptions that no such improvement exists — have been used as foundations for many other results about what is hard to compute. The authors say their results refute those hypotheses, along with several related ones, using known reductions from their new algorithms.
The engine of the result, in plain terms, is a smarter way to multiply certain thin matrices and find triangles in sparse, lopsided graphs — graphs where one part is much smaller than the other two. By computing only a carefully chosen set of entries of a matrix product, rather than the whole product, the authors save enough work for the saving to cascade through to 3SUM and shortest paths.
Three cautions belong in the same breath as the headline. First, this is a preprint, not yet peer-reviewed or published in a journal, so it should be read as a reported result that other researchers will check. Second, a simplified proof of the same improvements, with weaker exponents, has already been posted by other researchers, which is how checking often begins in this field. Third, nobody should expect their satnav or spreadsheet to get faster tomorrow: the gain only shows up for enormous inputs, and the method is a theoretical breakthrough, not a practical shortcut.
For students, the lesson is about how maths moves. Progress is not always a brand-new idea that changes everything overnight. Sometimes it is proving that a wall everyone assumed was solid has a crack in it — and once the crack is there, other mathematicians have a reason to keep chipping at it. A problem that looks finished in a textbook may not be finished at all.
If your child likes puzzles where a small change in approach opens up a whole problem, a first lesson in maths can give them that kind of thinking — working step by step, checking each claim, and learning when a tiny improvement is actually a big deal, with booster and mock-exam help in maths along the way. Book your first lesson
Sources:
- Alman, J. and Vassilevska Williams, V., "Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs", arXiv:2610.06783, submitted 5 October 2026: https://arxiv.org/abs/2610.06783
- Karingula, S. R. and Lovett, S., "A simplified proof of polynomial improvements for 3SUM and APSP", arXiv:2610.15663: https://arxiv.org/html/2610.15663v1