## Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications (1996)

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Venue: | SIAM J. Comput |

Citations: | 54 - 13 self |

### BibTeX

@ARTICLE{Agarwal96verticaldecomposition,

author = {Pankaj K. Agarwal and Alon Efrat and Micha Sharir},

title = {Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications},

journal = {SIAM J. Comput},

year = {1996},

volume = {29},

pages = {39--50}

}

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### Abstract

Let 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 k-level of the arrangement A(F) is O(k 3+" /(n=k)), for any " ? 0, where /(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F . This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [51], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-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...