Terrain visibility graphs are a well-known graph class in computational geometry. They are closely related to polygon visibility graphs, but a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted considerable attention in the context of time series analysis (there called time series visibility graphs) with various practical applications in areas such as physics, geography, and medical sciences. Computing shortest paths in visibility graphs is a common task, for example, in line-of-sight communication. For time series analysis, graph characteristics involving shortest paths lengths (such as centrality measures) have proven useful. In this paper, we devise a fast output-sensitive shortest path algorithm on a superclass of terrain visibility graphs called terrain-like graphs (including all induced subgraphs of terrain visibility graphs). Our algorithm runs in $$O(d^*\log \varDelta )$$ time, where $$d^*$$ is the length (that is, the number of edges) of the shortest path and $$\varDelta $$ is the maximum vertex degree. Alternatively, with an $$O(n^2)$$ -time preprocessing our algorithm runs in $$O(d^*)$$ time.

中文翻译：

---

类地形图上的一种快速最短路径算法

地形可见性图是计算几何中众所周知的图类。它们与多边形可见性图密切相关，但精确的图论表征仍然未知。在过去的十年中，地形可见性图在时间序列分析（称为时间序列可见性图）的背景下引起了相当大的关注，在物理、地理和医学等领域具有各种实际应用。计算可见性图中的最短路径是一项常见任务，例如，在视线通信中。对于时间序列分析，涉及最短路径长度（例如中心性度量）的图特征已被证明是有用的。在本文中，我们在地形可见性图的超类上设计了一种快速输出敏感的最短路径算法，称为类地形图（包括地形可见性图的所有诱导子图）。我们的算法在 $$O(d^*\log \varDelta )$$ 时间内运行，其中 $$d^*$$ 是最短路径的长度（即边数），$$\varDelta $ $ 是最大顶点度数。或者，使用 $$O(n^2)$$ -time 预处理我们的算法在 $$O(d^*)$$ 时间内运行。