253 research outputs found
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
The Complexity of Separating Points in the Plane
We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles
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
The Complexity of Drawing Graphs on Few Lines and Few Planes
It is well known that any graph admits a crossing-free straight-line drawing
in and that any planar graph admits the same even in
. For a graph and , let denote
the minimum number of lines in that together can cover all edges
of a drawing of . For , must be planar. We investigate the
complexity of computing these parameters and obtain the following hardness and
algorithmic results.
- For , we prove that deciding whether for a
given graph and integer is -complete.
- Since , deciding is NP-hard for . On the positive side, we show that the problem
is fixed-parameter tractable with respect to .
- Since , both and
are computable in polynomial space. On the negative side, we show
that drawings that are optimal with respect to or
sometimes require irrational coordinates.
- Let be the minimum number of planes in needed
to cover a straight-line drawing of a graph . We prove that deciding whether
is NP-hard for any fixed . Hence, the problem is
not fixed-parameter tractable with respect to unless
Flip Distance Between Triangulations of a Simple Polygon is NP-Complete
Let T be a triangulation of a simple polygon. A flip in T is the operation of
removing one diagonal of T and adding a different one such that the resulting
graph is again a triangulation. The flip distance between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is
still open after over 25 years of intensive study. We show that computing the
flip distance between two triangulations of a simple polygon is NP-complete.
This complements a recent result that shows APX-hardness of determining the
flip distance between two triangulations of a planar point set.Comment: Accepted versio
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 CâC bonds at the C4âC5 or C5âC6 positions within the ring, as well as the presence of a ÎČ-keto group at the C4 position. Our results demonstrate that 2-pyrones undergo a range of reactions unique to their structure, such as retro-DielsâAlder 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
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