CS531 Program 2 (Due October 26, 2009)

Tower of Corvallis

You all have probably heard of Tower of Hanoi and how long it takes to solve it. In this assignment you will be solving a much easier puzzle called Tower of Corvallis! Tower of Corvallis is played with m pegs and n disks. The nice thing is that any disk can go on any other. The goal simply is to get from an initial position to a goal position in the smallest number of steps. Remember you can only remove the top disk from the peg at any time and place it on the top of the disks on another peg!

Suppose m=3 and n=10. Let us number the disks 0-9 so that we can denote a state with 3 numbers or lists or arrays or whatever. For example, (0123456789,-,-) is a state and (012,3456,789) is another state, etc. The leftmost digit in the number is at the top. Let us fix the goal (final) state to be (9876543210,_,_) if there are 10 disks (and similarly for n disks). Also fix the initial state to be some configuration of disks, all of which are on the first peg. Call the three pegs A,B, and C.

You will write a search program to solve this puzzle using A* search . and a memory-bounded heuristic search algorithm called RBFS. You will also design and implement one admissible heuristic for this problem and one non-admissible heuristic.

You may use any programming language, but it must be modularized as much as possible. The code must be developed completely by your team.

NMAX is the maximum number of nodes you have the time to search. Set this so that the search takes less that 10 minutes with your least informed heuristic and slowest algorithm on the hardest problems on your computer. 

 
     For the two algorithms A* and RBFS 
        For each heuristic function h, 
          For at least 4 different sizes (number of disks)
             For each of the 20 problems p, 
		Solve p using h or until NMAX nodes are expanded. 
                Record the solution length if successful, the number of nodes 
	               expanded, and the total CPU time spent on evaluating
                       the heuristic and on solving the whole problem.
A list of several problems with numbers of disks 4 , 5 , 6 , 7 , 8 , 9 , 10 , are available on the web. Only the starting state is described for each problem and all the disks are assumed to be on peg A. The final state for all problems is the same. It is to get all disks to peg A in the order 9876543210. The top disk is the leftmost. As described in the pseudocode, you should at least try 4 different sizes (number of disks) of problems.

Plot the number of nodes searched against the problem size (number of disks) for the two algorithms and the two heuristics. Plot similar graphs for CPU time vs number of disks.

Show in a table, the average solution lengths found by the different heuristics for each problem size (number of disks), heuristic, and search algorithm.

For the report, let us follow the same format you used for the first programming assignment. Your report should look like a research paper with abstract, introduction (a summary of the goals and results), pseudocode of the algorithms, experimental setup, result tables and plots, and the discussion. You should include answers to following questions in your discussion of results. Please feel free to add any other important observations.

  1. Show an example solution sequence for each algorithm for the largest size you tested in the following format: 
     
[0123,_,_], [123,0,_], [23,0,1], [23,_,01] ...
  2. Is there a clear preference ordering among the heuristics you tested considering the number of nodes searched and the total CPU time taken to solve the problems for the two algorithms?
  3. Can a small sacrifice in optimality give a large reduction in the number of nodes expanded? What about CPU time?
  4. How did you come up with your heuristic evaluation function?
  5. How do the two algorithms compare in the amount of search involved and the cpu-time?
  6. Do you think that either of these algorithms scale to even larger problems? What is the largest problem you could solve with the best algorithm+heuristic combination? Report the wall-clock time, CPU-time, and the number of nodes searched.
  7. Is there any tradeoff between how good a heuristic is in cutting down the number of nodes and how long it took to compute? Can you quantify it?
  8. Is there anything else you found that is of interest?
Important Note: It is very important for this assignment to write efficient code.

Submit the source code using the ENGR assignment submission system. Click "Login to ENGR". Login, and click the "Submit an Assignment" button. Submit the report in the class on Oct, 26.