Dylan Wulf David Shull Fernando Faria
Konane Game implementation
================ Overview ================
In this project, we implemented a program to play the game of Konane (Hawaiian Checkers) using the assigned language, Scheme. The game is typically played on an 8 × 8 board of light and dark pieces as shown (using X for dark and O for light).
To have a meaningful implementation, our program used the Minimax search algorithm with Alpha-Beta pruning. For each game, our program also provides the following information:
- The number of times a static evaluation was done.
- The number of cut offs that took place.
- The average branching factor.
================ How to Run ================
To run this project, open the file "PL_Final.rkt" using DrRacket version 6.7. Select the R5RS language in the list of languages. Next, press the keys Ctrl+R on a Windows/Linux machine or Cmd+R on an OS X machine. Type in "Yes" if the computer is going first or "No" if the computer is going second. Next, enter moves into the prompt and view the computer's response. Continue to do this until either player has won.
================ Input Format ================
To enter pieces to be removed at game start, enter the x and y coordinates of the piece to be removed. To enter moves, type in first the x coordinate, then the y coordinate of the piece that is moving, then the x and y coordinate of where it should be going.
================ Bugs ================
All errors that we could think of were handled, and the game itself plays perfectly with correct input. One issue is that the computer is not able to decide the very first move by itself. If the computer is going first, the user must input its first move for it. We were going to have the computer choose a random move for its first move, but R5RS does not have any sort of random functionality.
================ Data Structures ================
For this project, we used Scheme lists and Scheme vectors (similar to arrays in C-based languages). The Scheme lists are what everything in Scheme is composed of, and we used them wherever car and cdr functionality would be helpful. The vectors made it easy to select various coordinates on the game board, because they are indexed by values.
================ Algorithms ================
We used a minimax algorithm, which tries to find the best possible move by comparing all of the possible outcomes 3 moves out. It uses a static evaluation function which returns a ratio of the number of computer player moveable pieces by the number of opponent's moveable pieces. Our research indicated that this evaluation function yielded the highest chance of success (see pages 13-16 of http://cs.brynmawr.edu/Theses/Thompson.pdf) Finally, the Alpha-Beta pruning algorithm determined whether it was possible for a branch's minimum node to be higher than an already-explored branch's minimum. If not, then the branch can be 'pruned' - no longer searched.
================ Depth and Alpha-Beta Pruning Results ================ Please see file Plots.pdf for visual graphs of this data
Our results were:
Minimax without Alpha-Beta Pruning
Depth 1: evaluations: 630, cuts: 0 Avg branch factor: 5.4 Depth 2: evaluations: 1439, cuts: 0, Avg. branch factor: 5.2 Depth 3: evaluations: 17108, cuts: 0, Avg. branch factor: 13.3 Depth 4: evaluations: 193060, cuts: 0, Avg. branch factor: 9.1 Depth 5: evaluations: 1597450, cuts: 0, Avg. branch factor: 13.4 Depth 6: evaluations: 21549265, cuts: 0, Avg. branch factor: 9.2
Minimax with Alpha-Beta Pruning
Depth 1: Number of evaluations: 176 Number of cuts: 2 Average branch factor: 13/5 ~= 2.6 Depth 2: Number of evaluations: 698 Number of cuts: 2 Average branch factor: 703/194 ~= 3.62 Depth 3: Number of evaluations: 5809 Number of cuts: 695 Average branch factor: 1730/317 ~= 5.46 Depth 4: Number of evaluations: 45155 Number of cuts: 8779 Average branch factor: 58934/13977 ~= 4.22 Depth 5: Number of evaluations: 864093 Number of cuts: 92718 Average branch factor: 1022379/158462 ~= 6.46 Depth 6: Number of evaluations: 2972575 Number of cuts: 917364 Average branch factor: 4178495/1206184 ~= 3.46
================ Analysis of Depth and Alpha-Beta Pruning ================
By looking at these results, it is very clear that adding Alpha-Beta pruning has a great impact on the game statistics. First of all, it drastically reduces the number of static evaluations performed. Next, it decreases the average branching factor. Finally, it increases the number of cuts, since the Alpha-Beta pruning is what does the actual cuts. One statistic not measured, but witnessed by our eyes, was that the game took considerably less time when using pruning.