https://cs.lozic.me/areas/combinatorics/Recent content in Combinatorics on CS TheoremsHugoen-usFri, 04 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t005-adversary-arguments/Fri, 04 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t005-adversary-arguments/<h2 id="symptom">Symptom</h2> <p>You need the largest and second-largest element of an array. The obvious way is two passes: $n-1$ comparisons for the max, then $n-2$ for the max of the rest. That is $2n - 3$.</p>https://cs.lozic.me/posts/t003-comparison-sort-lower-bound/Fri, 20 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t003-comparison-sort-lower-bound/<h2 id="symptom">Symptom</h2> <p>Every general-purpose sort in every standard library is $O(n \log n)$. Timsort, introsort, pdqsort, merge sort, heapsort. Decades of work by extremely motivated people, an enormous amount of money riding on it, and they all land on the same exponent.</p>https://cs.lozic.me/posts/t004-no-universal-lossless-compressor/Fri, 04 Sep 2026 12:00:00 +0200https://cs.lozic.me/posts/t004-no-universal-lossless-compressor/<h2 id="symptom">Symptom</h2> <p>Someone sends you a pitch deck. The claim is a compression algorithm that reduces <em>any</em> file by a guaranteed ratio, and — this is always the tell — that it can be applied repeatedly. Four gigabytes down to one, then one down to two fifty, and so on until the whole film is a few hundred bytes and a clever decoder.</p>https://cs.lozic.me/posts/t001-pigeonhole-principle/Fri, 28 Aug 2026 12:00:00 +0200https://cs.lozic.me/posts/t001-pigeonhole-principle/<h2 id="symptom">Symptom</h2> <p>You are asked to build a cache key. You have a 64-bit hash, and someone on the team says: &ldquo;collisions are basically impossible, there are eighteen quintillion values.&rdquo;</p> <p>Or: your service assigns short IDs to uploads, six characters of base-36, and you are wondering when you need to worry.</p>