Compass routing on geometric networks

Suppose that a traveler arrives in the city of Toronto, and wants to walk to the famous CN Tower, one of the tallest free-standing structures in the world. Our visitor, lacking a map of Toronto, is standing at an intersection fromwhich he can see the CN Tower, and several streets S1; : : : ; Sm from which he can choose to start his walk. A natural (and most likely safe) assumption is that our visitor should choose the street whose direction points closest to the CN Tower; see Figure 1. A close look at maps of numerous cities around the world shows us that using this method to explore a new and unknown city will, in general, yield walks that will be close enough to the optimal ones to travel from one location to another. In mathematical terms, we can model the maps of many cities by geometric graphs in which street intersections are represented by the vertices of our graphs, and streets by straight line segments. Compass routing on geometric networks, in its most elemental form yields the following algorithm: