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A counterexample for Steiner triangulation
I’ve been hesitating in writing up a blog post about my latest preprint, “Minimum-weight Steiner triangulation of convex polygons requires interior Steiner points” (with my student Zahra Hadizadeh, arXiv:2606.25302, to appear at CCCG) because, despite being a completely concrete two-dimensional construction, its exponential scale makes it difficult to visualize. In the paper we included an illustration using logarithmic coordinates but I didn’t find that entirely satisfactory so I’m trying again in a different way here.
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Ceramic orthogonal polyhedra
David Richter, a mathematician at Western Michigan University, recently found himself with a surfeit of ceramic orthogonal polyhedra and, knowing of my own interest in orthogonal polyhedra, generously offloaded two of them to me. They fit nicely in my office together with the paper and crochet orthogonal polyhedra I already had:
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Impossible patterns of sphere tangencies
My latest preprint, “Tangent spheres and integer distances” (arXiv:2606.18569, to appear at CCCG), involves the patterns of external tangencies of circles, spheres or higher-dimensional hyperspheres. You can make a graph whose vertices are a given set of spheres and whose edges are pairs of externally-tangent spheres, and I’d like to understand which graphs are possible. By the circle packing theorem, any planar graph can be represented by interior-disjoint circles in this way, but here I’m not requiring disjointness. So, for instance, you can represent any unit distance graph in the plane (such as the Petersen graph below) by expanding each vertex of the graph into a unit-diameter circle.
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Linkage
- Biomedical papers with fabricated references blew up by more than a factor of ten from 2023 to 2026 (\(\mathbb{M}\), via). Relatedly, arXiv has added a new policy banning author of papers with hallucinated citations or other artifacts of unchecked LLM use for a year (and after that permanently requiring publication prior to allowing a preprint, kind of problematic for publication in journals whose submission process depends on having an arXiv preprint. Unfortunately I can’t find a direct reference to this in the actual posted arXiv policies and I’m not going to link to X/twitter where Tom Dietterich posted the only semi-official communication I can find on it.
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Congratulations, Dr. Sridhar!
This past week was the final exam week for our spring term, and in it we also held the doctoral defense of another Ph.D. student, Vinesh Sridhar (advised by Mike Goodrich). Vinesh organized his thesis in an unusually symmetric way, with two parts on computing with noisy primitives, two parts on in-place algorithms, two parts on sequential algorithms, and two parts on parallel algorithms, in four combinations (noisy sequential, noisy parallel, in-place sequential, in-place parallel).
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Linkage
- Real-world application of square packing: Packing ten quart bottles in a milk crate slightly larger than a \(3\times 3\) packing.
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Congratulations, Dr. Patris!
I participated today in the successful Ph.D. defense of Nikolas Patris, a student of my newly-tenured colleague Ioannis Panageas. Nikolas has been working on problems of learning Nash equilibria of games and more generally finding saddle points of smooth functions, a crossover area between theoretical computer science and machine learning. His results are theoretical but his papers on them are in machine learning conferences:
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Linkage
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The degree of Fibonacci heaps
The Fibonacci heap is a priority queue data structure in which finding and removing the top-priority item takes logarithmic time (like a binary heap) but reprioritizing items to have higher priority takes only constant time, under amortized time analysis. This property is useful for speeding up the time complexity of algorithms including Dijkstra’s algorithm for shortest paths and Jarník’s algorithm for minimum spanning trees, so that they take logarithmic time per vertex and constant time per edge.
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Linkage
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