 
Summary: Spanning Trees Crossing Few Barriers
Tetsuo Asano 1 Mark de Berg 2 Otfried Cheong 2
Leonidas J. Guibas 3 Jack Snoeyink 4 Hisao Tamaki 5
Abstract
We consider the problem of nding lowcost spanning trees for sets of n points in the plane,
where the cost of a spanning tree is dened as the total number of intersections of tree edges
with a given set of m barriers. We obtain the following results:
(i) if the barriers are possibly intersecting line segments, then there is always a spanning
tree of cost O(min(m 2 ; m p
n));
(ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost
O(m);
(iii) if the barriers are disjoint convex objects, then there is always a spanning tree of cost
O(n +m).
All our bounds are worstcase optimal.
1 Introduction
Consider the problem of batched point location, where the goal is to eÆciently locate n given points
in a planar subdivision dened by m line segments. This problem arises in many applications, and
in particular in the lineartime reconstruction of common geometric structures such as Voronoi and
Delaunay diagrams, or convex hulls [9]. In these applications the desired diagram is constructed
