Visibility graph

In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents a point location, and each edge represents a visible connection between them. That is, if the line segment connecting two locations does not pass through any obstacle, an edge is drawn between them in the graph. When the set of locations lies in a line, this can be understood as an ordered series. Visibility graphs have therefore been extended to the realm of time series analysis. Visibility graphs may be used to find Euclidean shortest paths among a set of polygonal obstacles in the plane: the shortest path between two obstacles follows straight line segments except at the vertices of the obstacles, where it may turn, so the Euclidean shortest path is the shortest path in a visibility graph that has as its nodes the start and destination points and the vertices of the obstacles. Therefore, the Euclidean shortest path problem may be decomposed into two simpler subproblems: constructing the visibility graph, and applying a shortest path algorithm such as Dijkstra's algorithm to the graph. For planning the motion of a robot that has non-negligible size compared to the obstacles, a similar approach may be used after expanding the obstacles to compensate for the size of the robot. attribute the visibility graph method for Euclidean shortest paths to research in 1969 by Nils Nilsson on motion planning for Shakey the robot, and also cite a 1973 description of this method by Russian mathematicians M. B. Ignat'yev, F. M. Kulakov, and A. M. Pokrovskiy. Visibility graphs may also be used to calculate the placement of radio antennas, or as a tool used within architecture and urban planning through visibility graph analysis. The visibility graph of a set of locations that lie in a line can be interpreted as a graph-theoretical representation of a time series. This particular case builds a bridge between time series, dynamical systems and graph theory.