Numerical stability of algorithms for line arrangements

Abstract

We analyze the behavior of two line arrangement algorithms, a sweepline algorithm and an incremental algorithm, in approximate arithmetic. The algorithms have running times O(n2 log n) and O(n2) respectively. We show that each of these algorithms can be implemented to have O(n∈) relative error. This means that each algorithm produces an arrangement realized by a set of pseudolines so that each pseudoline differs from the corresponding line relatively by at most O(n∈). We also show that there is a line arrangement algorithm with O(n2 log n) running time and O(∈) relative error.