TREE-WEIGHTED NEIGHBORS AND GEOMETRIC k SMALLEST SPANNING TREES

We compute the k smallest spanning trees of a point set in the planar Euclidean metric in time O(n log n log k + k min (k, n)1/2 log (k/n)), and in the rectilinear metric in time O(n log n + n log log n log k + k ( min {k, n})1/2 log (k/n)). In three or four dimensions our time bound is O(n4/3 + ∊ + k( min {k, n})1/2 log (k/n)), and in higher dimensions the bound is O(n2−2/(⌈d/2⌉+1)+∊ + kn1/2 log n). Our technique involves a method of computing nearest neighbors using a modified set of distances formed by subtracting tree path lengths from the Euclidean distance.

The Steiner problem: a survey

For the past decade the Steiner minimal tree problem has attracted the attention of researchers in discrete optimization. A brief survey of the main results concerning the properties and algorithms of the Steiner problem in the Euclidean plane, the Steiner problem in the plane with rectilinear metric and the Steiner problem in networks is done in this paper. The main attention is paid to the recent results concerning the last problem.