# Algorithms Seminar

## Removing Depth-Order Cycles Among Triangles: An Efficient Algorithm Generating Triangular Fragments

Speaker: Mark de Berg Thursday, October 19, 2017 12:00pm - 1:00pm D344 LSRC, Duke

Abstract

More than 25 years ago Chazelle et al. (FOCS 1991) studied the following question: Is it possible to cut any set of n lines in R^3 into a subquadratic number of fragments such that the resulting fragments admit a depth order? They proved an O(n^{9/4}) bound for the very special case of bipartite weavings of lines. Since then only little progress was made, until a recent breakthrough by Aronov and Sharir (STOC 2016) who showed that O(n^{3/2} polylog n) fragments suffice for any set of lines. In a follow-up paper Aronov, Miller and Sharir (SODA 2017) proved an O(n^{3/2+ε}) bound for triangles, but their method results in pieces with curved boundaries. Moreover, their method uses polynomial partitions, for which currently no algorithm is known. Thus the most natural version of the problem is still wide open: Can we cut any collection of n disjoint triangles in R^3 into a subquadratic number of triangular fragments that admit a depth order? And if so, can we compute the cuts efficiently? We answer this question by presenting an algorithm that cuts any set of n disjoint triangles in R^3 into O(n^{7/4} polylog n) triangular fragments that admit a depth order. The running time of our algorithm is O(n^3.69). As a byproduct of our approach we obtain a faster algorithm than the one by Aronov and Sharir to cut a set of lines into O(n^{3/2} polylog n) fragments that admit a depth order.

Biography

Mark de Berg received an M.Sc. in computer science from Utrecht University (the Netherlands) in 1988, and he received a Ph.D. from the same university in 1992. Currently he is a full professor at the TU Eindhoven. His main research interest is in algorithms and data structures, in particular for spatial data. He is (co-)author of two books on computational geometry and he has published over 175 papers in journals and conferences. Mark de Berg was on the program committee of many international conferences in the field, and he serves on the editorial board of three journals. He is currently the chair of the Steering Committee of the European Symposium on Algorithms, and the secretary of the international Computational-Geometry Steering Committee.