Maximum Gap Minimization in Polylines

Abstract

Given a polyline consisting of n segments, we study the problem of selecting k of its segments such that the maximum induced gap length without a selected segment is minimized. This optimization problem has applications in the domains of trajectory visualization (see Fig. 1) and facility location. We design several heuristics and exact algorithms for simple polylines, along with algorithm engineering techniques to achieve good practical running times even for large values of n and k. The fastest exact algorithm is based on dynamic programming and exhibits a running time of O(nk) while using space linear in n. Furthermore, we consider incremental problem variants. For the case where a given set of k segments shall be augmented by a single additional segment, we devise an optimal algorithm which runs in O(k+logn) on a suitable polyline representation. If not only a single segment but k′ segments shall be added, we can compute the optimal segment set in time O(nk′) by modifying the dynamic programming approach for the original problem. Experiments on large sets of real-world trajectories as well as artificial polylines show the trade-offs between quality and running time of the different approaches.