On counting pairs of intersecting segments and off-line triangle range searching

We describe a method for decomposing planar sets of segments and points. Using this method we obtain new efficientdeterministic algorithms for counting pairs of intersecting segments, and for answering off-line triangle range queries. In particular we obtain the following results:Givenn segments in the plane, the number of pairs of intersecting segments is counted in timeO(n1+ɛ+K1/3n2/3+ɛ), whereK is the number of intersection points among the segments, and ɛ>0 is an arbitrarily small constant.Givenn segments in the plane which are colored with two colors, the number of pairs ofbichromatic intersecting segments is counted in timeO(n1+ɛ+Km1/3n2/3+ɛ), whereKm is the number ofmonochromatic intersection points, and ɛ>0 is an arbitrarily small constant.Givenn weighted points andn triangles on a plane, the sum of weights of points in each triangle is computed in timeO(n1+ε+ϰ1/3n2/3+ε), where ϰ is the number of vertices in the arrangement of the triangles, and ɛ>0 is an arbitrarily small constant.