The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to handwriting recognition. More recently, the Frechet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Frechet distance, in an effort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are path-connected.

翻译：弗雷歇距离常用于测量路径之间的距离，应用领域涵盖地图匹配、GPS轨迹分析到手写识别等。近年来，弗雷歇距离已被推广至度量空间中嵌入或浸入的同一图的两个副本之间的距离；这一更通用的框架在图形分析领域开辟了更复杂的应用前景。本文首次系统研究在弗雷歇距离下映射至R^n的路径与图空间的基本拓扑性质，旨在为理解这些距离的实际应用奠定理论基础。特别地，我们证明了这些空间及其中的度量球是否具有路径连通性。