Computing a Shortest Path on Euclidian Graphs

In this example we will demonstrate how Gato allows students to learn more about developing better algorithms by making use of problem-specific knowledge.

One fundamental problem in algorithmic graph theory is finding the shortest path between two vertices in a graph. That is, given a set of vertices and a set of weighted edges where the weight on an edge corresponds to the distance between the vertices it connects, find a path such that the sum of all weights of edges on the path is minimal.

If the graph is Euclidean (i.e., the distance between vertices is simply the Euclidean distance between their positions in a two-dimensional embedding), then the standard algorithm can be improved.

By running the standard algorithm on several different graphs, students observe that the shortest path is always "close" to a straight line between the two vertices. This observation can then be used to investigate how the standard algorithm selects which vertex to visit next by single-stepping through the algorithm. This may lead students to conclude that visiting vertices which are too "far away" from a straight line is unnecessary.

This conclusion can be formalized mathematically in terms of a simple inequality and directly implemented in Gato (basically by changing one line). The modified algorithm can then be run again to experimentally verify the student's argument and -- in case of subtle errors -- to provide insights, again by single-stepping through the algorithm, into the source of error.

Computing a shortest path

Loading a graph and an algorithm
	Both graphs and algorithms are loaded into Gato during run-time. The user may select the files he/she wants using dialogs typical for the specific platform.

Running an algorithm
	After a graph and an algorithm have been loaded, the Start button is used to let the algorithm run until termination, since we are only interested in looking at shortest paths.

Looking at shortest paths
	Here we see what a student would observe during several runs. In the case of Euclidean graphs, where the weight of an egde corresponds to the Euclidean distance between the vertices it connects, the shortest path between two vertices (here: top left and bottom right) is "close" to a straigt line between the two vertices. Note, that all vertices and all edges have been visited.

Examining choice of edges

Breakpoints

One feature for investigating an algorithm are breakpoints; i.e., to specify a line of the source code at which execution is halted. Users can set and remove breakpoints (displayed as red lines) by clicking on them with the mouse. Once started, the algorithm will halt, when it reaches the line. The user can then observe the state and either use the Stop button to quit or use Cont to continue execution.

Single-stepping
	Alternatively, the user can choose to step -- i.e., to execute one line at a time -- through the algorithm.

Observing choice

By single-stepping through the lines of code which choose the next vertex visited by the algorithm, the student can observe that in this instance several alternatives can be ruled out due to a simple geometric consideration (which is obvious to humans but not to algorithms): a straight line is the shortest connection between two points (here: of the neighbors of vertex 6, certainly vertex 7 will be on a shortest path to vertex 9, but not vertex 10)

Changing the algorithm

Editing algorithms
	The user can use an external editor of his/her choice for changing algorithms.

Comparing results

Running the modified algorithm
	By running the modified algorithm the student can heuristically examine the speed-up the changes produced. The modified algorithm visits a far smaller number of edges and vertices (traversed edges are displayed in grey).

Checking for errors
	In case of errors in the modified algorithm, students can use breakpoints and single-stepping and/or modified graphs to investigate under which circumstances errors occured.