Minimum-weight triangulation is NP-hard

A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures. This revision contains a few improvements in the expositio

Approximating Tverberg Points in Linear Time for Any Fixed Dimension

Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1), ..., conv(P_r) have non-empty intersection. A point in the intersection of the conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d+1), but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (2010).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 2012. This version removes an incorrect example at the end of Section 3.

A Static Optimality Transformation with Applications to Planar Point Location

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.Comment: 13 pages, 1 figure, a preliminary version appeared at SoCG 201

Cascade oxime formation, cyclization to a nitrone, and intermolecular dipolar cycloaddition.

Simple haloaldehydes, including enolisable aldehydes, were found to be suitable for the formation of cyclic products by cascade (domino) condensation, cyclisation, dipolar cycloaddition chemistry. This multi-component reaction approach to heterocyclic compounds was explored by using hydroxylamine, a selection of aldehydes, and a selection of activated dipolarophiles. Initial condensation gives intermediate oximes that undergo cyclisation with displacement of halide to give intermediate nitrones; these nitrones undergo in situ intermolecular dipolar cycloaddition reactions to give isoxazolidines. The cycloadducts from using dimethyl fumarate were treated with zinc/acetic acid to give lactam products and this provides an easy way to prepare pyrrolizinones, indolizinones, and pyrrolo[2,1-a]isoquinolinones. The chemistry is illustrated with a very short synthesis of the pyrrolizidine alkaloid macronecine and a formal synthesis of petasinecine

Mechanistic Insights into Ring-Opening and Decarboxylation of 2-Pyrones in Liquid Water and Tetrahydrofuran

2-Pyrones, such as triacetic acid lactone, are a promising class of biorenewable platform chemicals that provide access to an array of chemical products and intermediates. We illustrate through the combination of results from experimental studies and first-principle density functional theory calculations that key structural features dictate the mechanisms underlying ring-opening and decarboxylation of 2-pyrones, including the degree of ring saturation, the presence of C═C bonds at the C4═C5 or C5═C6 positions within the ring, as well as the presence of a β-keto group at the C4 position. Our results demonstrate that 2-pyrones undergo a range of reactions unique to their structure, such as retro-Diels–Alder reactions and nucleophilic addition of water. In addition, the reactivity of 2-pyrones and the final products formed is shown to depend on the solvent used and the acidity of the reaction environment. The mechanistic insights obtained here provide guidance for the selective conversion of 2-pyrones to targeted chemicals.Reprinted (adapted) with permission from Journal of American Chemical Society, 135(15); 5699-5708. Doi: 10.1021/ja312075r. Copyright 2013 American Chemical Society. </p