### Approximating the integral Fréchet distance

Anil Maheshwari, Jörg-Rüdiger Sack, Christian Scheffer
2018 Computational geometry
We present a pseudo-polynomial time (1+ε)-approximation algorithm for computing the integral and average Fréchet distance between two given polygonal curves T 1 and T 2 . The running time is in O(ζ 4 n 4 /ε 2 ) where n is the complexity of T 1 and T 2 and ζ is the maximal ratio of the lengths of any pair of segments from T 1 and T 2 . Furthermore, we give relations between weighted shortest paths inside a single parameter cell C and the monotone free space axis of C. As a result we present a
more » ... ple construction of weighted shortest paths inside a parameter cell. Additionally, such a shortest path provides an optimal solution for the partial Fréchet similarity of segments for all leash lengths. These two aspects are related to each other and are of independent interest. A. Maheshwari, J.-R. Sack, and C. Scheffer 26:3 that decreases the matched distance as quickly as possible."[2, p. 237]. The matching induced by π ab fulfils this requirement. S WAT 2 0 1 6 A. Maheshwari, J.-R. Sack, and C. Scheffer 26:11 the boundary of C r . Furthermore, let C q and C s be the parameter cells such that q ∈ C q and s ∈ C s . We denote the monotone free space axis of C q , C r , and C s by q , r , and s , respectively. Let u 1 := arg min a∈e1 w(a) and u 2 := arg min a∈e2 w(a). The proof of Lemma 15 is similar to the proof of Lemma 12. Lemma 16. If d 1 (u 1 , u 2 ) ≥ 6 max{w(u 1 ), w(u 2 )}, then there is a path π qs ⊂ G 2 between q and s such that | π qs | w ≤ (1 + ε)|π qs | w .