# Lab 8 — Recursion solution

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. Checkpoint 1 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. Checkpoint 2 Please download the ﬁles: http://www.cs.rpi.edu/academics/courses/spring15/csci1200/labs/08_recursion/grid1.txt http://www.cs.rpi.edu/academics/courses/spring15/csci1200/labs/08_recursion/grid2.txt http://www.cs.rpi.edu/academics/courses/spring15/csci1200/labs/08_recursion/start.cpp 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. Checkpoint 3 Modify your program from Checkpoint 2 so that it ﬁnds and outputs a legal path from the given (x,y) location to the origin. If there is more than one path, it does not matter which it outputs. Did you notice that all legal paths to the origin are the same length under the rules provided? To complete this checkpoint and the entire lab, show a TA or programming mentor your modiﬁed program.

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