Spanning Trees Crossing Few Barriers (Algorithm Engineering as a New Paradigm)

We consider the problem of finding low-cost spanning trees for sets of $n$ points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of $m$ barriers. We obtain the following results: 1. if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost $O( \min(m^{2}, m\sqrt{n}))$ ; 2. if the barriers are disjoint line segments, then there is always a spanning tree of cost $O(m)$ ; 3. if the barriers are disjoint fat objects, discs for example, then there is always a spanning tree of cost $O(n+m)$ . All our bounds are worst-case optimal.