A Fast Shortest Path Algorithm on Terrain-like Graphs

Abstract

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 $$O(d^*\log \varDelta )$$ O ( d ∗ log Δ ) time, where $$d^*$$ d ∗ is the length (that is, the number of edges) of the shortest path and $$\varDelta $$ Δ is the maximum vertex degree. Alternatively, with an $$O(n^2)$$ O ( n 2 ) -time preprocessing our algorithm runs in $$O(d^*)$$ O ( d ∗ ) time.

Social Graphs and Their Applications to Robotics

“…It is utilised in robot motion planning when the geometry of environment is known [118]. In addition, visibility graph is a main concept in calculational geometry, for a given group of geometrical objects (e.g., segments, polygons, points, rectangles) they encode which objects are visible to each other [136]. In contrast to the Voronoi diagram, the visibility graph (VG) uses the nodes of the obstacles including the starting and goal points in the C-space as the vertices [21].…”