Spanning Trees Crossing Few Barriers

On Spanning Trees with Low Crossing Numbers

Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O( √ n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic...

Approximating Spanning Trees with Low Crossing Number

We present a linear programming based algorithm for computing a spanning tree T of a set P of n points in IR, such that its crossing number is O(min(t log n, n1−1/d)), where t the minimum crossing number of any spanning tree of P . This is the first guaranteed approximation algorithm for this problem. We provide a similar approximation algorithm for the more general settings of building a spann...

Enumerating Constrained Non-crossing Geometric Spanning Trees

In this paper we present an algorithm for enumerating without repetitions all non-crossing geometric spanning trees on a given set of n points in the plane under edge constraints (i.e., some edges are required to be included in spanning trees). We will first prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the constrained smallest ...

Edge Operations on Non-Crossing Spanning Trees

On finding spanning trees with few leaves

The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a connection). Besides this application, the problem has its own theoretical importance as a generalization of the Hamiltonian path problem. Lu and Ravi showed that there is no constant f...