Основні поняття
Investigating the Fréchet edit distance problem for polygonal curves and its variants.
Анотація
The content delves into defining and exploring the Fréchet edit distance problem, focusing on polygonal curves. It discusses the minimum number of edits required to maintain a specified distance threshold between two curves. The article covers various cases such as deletion, insertion, or both operations. Algorithms for discrete and continuous variants are provided along with complexity analysis. Funding sources and related works are also highlighted.
Introduction:
Shape matching between polygonal curves in computational geometry.
Use of continuous and discrete Fréchet distances for curve similarity.
Motivation:
Continuous vs. discrete Fréchet distances.
Sensitivity to outliers in real-world data.
The Fréchet Edit Distance:
Modifications and alternatives to address sensitivity issues.
Similarity measures based on standard definitions of Fréchet distance.
Our Results:
Polynomial time algorithms for various variants of Fréchet edit distance.
Complexity analysis for deletion-only, insertion-only, and combined operations.
DAG Complexes:
Definition and application in computing minimum link chains.
Topological ordering for efficient reachability determination.
Continuous Fréchet Distance:
Algorithms for deletion-only scenarios with complexity analysis.
Minimum Vertex Curves:
Definition and computation methods for minimum vertex curves.
Insertion Only:
Consideration of inserted subcurves between consecutive vertices.
Canonical Inserted Subcurves:
Definition of the set of canonical inserted subcurves based on specific conditions.
Статистика
Given two DAG complexes C1 and C2, start vertices s1 ∈ V (C1) , s2 ∈ V (C2), end vertices t1 ∈ V (C1) , t2 ∈ V (C2), determine if there exists two polygonal curves π1, π2 such that dF(π1, π2) ≤ δ can be computed in O(|C1||C2|) time by considering the free space of the product complex of C1 and C2.
For strong discrete Fréchet distance with deletions limited to deletions, an O(mn) time algorithm is described for any pair of curves in Rd.