Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications

Let ${\cal F}$ be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the $({\le}k)$-level of the arrangement $\A({\cal F})$ is $O(k^{3+\varepsilon}\psi({n/k}))$ for any $\varepsilon>0$, where $\psi (r)$ is the maximum complexity of the lower envelope of a subset of at most $r$ functions of ${\cal F}$. This bound is nearly optimal in the worst case and implies the existence of shallow cuttings, in the sense of [J. Matousek, Comput. Geom., 2 (1992), pp. 169--186], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including (i) data structures for several generalized 3-dimensional range-searching problems; (ii) dynamic data structures for planar nearest- and farthest-neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.