MPI-I-95-1-022

We describe a new method for decomposing planar sets of segments and points. Using this method we obtain new efficient {\em deterministic} algorithms for counting pairs of intersecting segments, and for answering off-line triangle range queries. In particular we obtain the following results: \noindent (1) Given $n$ segments in the plane, the number $K$ of pairs of intersecting segments is computed in time $O(n^{1+\epsilon} + K^{1/3}n^{2/3 + \epsilon})$, where $\epsilon >0$ an arbitrarily small constant. \noindent (2) Given $n$ segments in the plane which are coloured with two colours, the number of pairs of {\em bi-chromatic} intersecting segments is computed in time $O(n^{1+\epsilon} + K_m^{1/3}n^{2/3 +\epsilon})$, where $K_m$ is the number of {\em mono-chromatic} intersections, and $\epsilon >0$ is an arbitrarily small constant. \noindent (3) Given $n$ weighted points and $n$ triangles on a plane, the sum of weights of points in each triangle is computed in time $O(n^{1+\epsilon} + {\cal K}^{1/3}n^{2/3 +\epsilon})$, where ${\cal K}$ is the number of vertices in the arrangement of the triangles, and $\epsilon>0$ an arbitrarily small constant. The above bounds depend sub-linearly on the number of intersections among segments $K$ (resp. $K_m$, ${\cal K}$), which is desirable since $K$ (resp. $K_m$, ${\cal K}$) can range from zero to $O(n^2)$. All of the above algorithms use optimal $\Theta(n)$ storage. The constants of proportionality in the big-Oh notation increase as $\epsilon$ decreases. These results are based on properties of the sparse nets introduced by Chazelle.