The Fréchet distance is a well-studied distance measure between two curves. In this work, we demonstrate that the merit of Fréchet distance extends beyond evaluating similarity, and introduce a new setting in which it proves useful. Consider a situation where two agents are required to visit a given set of sites, while staying close to each other throughout their traversal. In this paper, we study problems where the goal is to construct two curves whose vertices are from a given set of points, under the constraint that the Fréchet distance between the curves is kept as small as possible. This problem can be viewed as a variant of the Traveling Salesman Problem (TSP), and thus may be of interest in routing, network planning and more. We present a near-linear algorithm for this problem under the discrete Fréchet distance, and explore several variants of the problem, including minimizing the lengths of the curves and balancing the number of sites assigned to each agent. Lastly, we prove that the problem is NP-hard under the continuous Fréchet Distance.

翻译：Fréchet距离是衡量两条曲线相似性的经典距离度量。本文证明Fréchet距离的价值不仅限于相似性评估，并引入了一个新的应用场景。考虑两个智能体需访问给定站点集，且在整个遍历过程中彼此保持接近的情形。我们研究的目标是构造两条顶点均来自给定点集的曲线，同时尽可能减小曲线之间的Fréchet距离。该问题可视为旅行商问题（TSP）的变体，因此可能对路径规划、网络布局等领域具有重要价值。我们针对离散Fréchet距离提出了近线性算法，并探讨了问题的多种变体，包括最小化曲线长度以及平衡分配给每个智能体的站点数量。最后，我们证明了该问题在连续Fréchet距离下是NP难的。