This lab gives you practice in the use of recursion to solve problems. All three checkpoints addressed in
this lab deal with finding and counting the number of paths between points on a rectilinear grid. A starting
point (x, y) with non-negative integer coordinates is provided. You are only allowed to move horizontally
and vertically along the grid. Hence, from (x, y) you may move to (x + 1, y), (x − 1, y), (x, y − 1), (x, y + 1).
Your goal is to return to the origin (0, 0) in such a way that you never increase the distance to the origin.
The distance is counted as the minimum number of total vertical and horizontal steps to reach the origin. In
the first checkpoint the grid will be “free”. In the second and third checkpoints, some of the grid locations
will be “blocked” in the sense that you can not move to that point.
Stop now and play with small examples. Draw pictures of a grid. Think about the implications of the rules
before you proceed with the checkpoints.
Did you notice that the rules prevent certain moves from occurring? What are they in particular? If you
don’t get them right you will not be able to do the lab correctly. Confirm your understanding with one of
the lab TAs.
Now, write a simple recursive program (from scratch) that reads a starting location, (x, y) and returns the
total number of different paths back to the origin when the entire grid is “free”. Two paths are different if
there is at least one step on the path that is different even if most of the steps are the same. As a hint, when
x == 2 and y == 1 there are 3 paths and when x == 2 and y == 2 there are 6.
To complete this checkpoint show a TA your program to count all paths back to the origin.
Please download the files:
Starting from the supplied program, start.cpp, write a program to count the number of paths when some
of the grid locations are blocked. When a location is blocked, no path can go through it.
Before writing your own code, compile and run start.cpp. Use the files grid1.txt and grid2.txt as input.
These have a sequence of blocked locations, ending with the point location (0, 0) (which isn’t blocked, but
signals the end of the input). The next location is the location for the initial (x, y). By changing this location
you will be able to test your program in different ways.
To complete this checkpoint show a TA your code to find and count legal (unblocked) paths.