Sparse geometric graphs with small dilation

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position

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

Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Influences on the formation and evolution of Physarum polycephalum inspired emergent transport networks

The single-celled organism Physarum polycephalum efficiently constructs and minimises dynamical nutrient transport networks resembling proximity graphs in the Toussaint hierarchy. We present a particle model which collectively approximates the behaviour of Physarum. We demonstrate spontaneous transport network formation and complex network evolution using the model and show that the model collectively exhibits quasi-physical emergent properties, allowing it to be considered as a virtual computing material. This material is used as an unconventional method to approximate spatially represented geometry problems by representing network nodes as nutrient sources. We demonstrate three different methods for the construction, evolution and minimisation of Physarum-like transport networks which approximate Steiner trees, relative neighbourhood graphs, convex hulls and concave hulls. We extend the model to adapt population size in response to nutrient availability and show how network evolution is dependent on relative node position (specifically inter-node angle), sensor scaling and nutrient concentration. We track network evolution using a real-time method to record transport network topology in response to global differences in nutrient concentration. We show how Steiner nodes are utilised at low nutrient concentrations whereas direct connections to nutrients are favoured when nutrient concentration is high. The results suggest that the foraging and minimising behaviour of Physarum-like transport networks reflect complex interplay between nutrient concentration, nutrient location, maximising foraging area coverage and minimising transport distance. The properties and behaviour of the synthetic virtual plasmodium may be useful in future physical instances of distributed unconventional computing devices, and may also provide clues to the generation of emergent computation behaviour by Physarum. © Springer Science+Business Media B.V. 2010