2018/10/25 by Karl Bringmann, Bringmann, Karl, Marvin Künnemann +3
Computer Science · #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #Digital Image Processing Techniques #FOS: Computer and information sciences #Image Processing and 3D Reconstruction
paper · pdf · doi:10.48550/arxiv.1810.10982
openalex publication_date 2018/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The discrete Fréchet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fréchet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length n in the plane, the fastest known algorithm runs in time \cal O(n5) [Ben Avraham, Kaplan, Sharir '15]. This is achieved by constructing an arrangement of disks of size \cal O(n4), and then traversing its faces while updating reachability in a directed grid graph of size N := \cal O(n2), which can be done in time \cal O(√(N)) per update [Diks, Sankowski '07]. The contribution of this paper is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than \cal O(√(N)), we improve this part of the algorithm: We observe that an offline variant of dynamic s-t-reachability in directed grid graphs suffices, and we solve this variant in amortized time \cal O(N1/3) per update, resulting in an improved running time of \cal O(n4.66...) for the discrete Fréchet distance under translation. Second, we provide evidence that constructing the arrangement of size \cal O(n4) is necessary in the worst case, by proving a conditional lower bound of n4 - o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.