Monotone Paths in Line Arrangements with a Small Number of Directions

Abstract We give subquadratic bounds on the maximum length of an x-monotone path in an arrangement of n lines with at most C log log n directions, where C is a suitable constant. For instance, the maximum length of an x-monotone path in an arrangement of n lines having at most ten slopes is O(n67/34). In particular, we get tight estimates for the case of lines having at most five directions, by showing that previous constructions—Ω(n3/2) for arrangements with four slopes and Ω(n5/3) for arrangements with five slopes—due to Sharir and Matousek, respectively, are (asymptotically) best possible.

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