Implementation and analysis of search algorithms in single player connect four game
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The document discusses the implementation and analysis of search algorithms in a single-player Connect Four game. It outlines the game rules and previous work analyzing strategies. It then describes the problem statement, algorithms implemented including minimax and alpha-beta pruning, heuristics to evaluate board positions, and a comparative analysis of the algorithms. Exponential heuristics were found to explore more nodes than linear heuristics but require less than 1 second to search to a depth of 10. Alpha-beta pruning reduced the number of nodes explored by 10 to 100 times compared to not using pruning.
![Mathematical Analysis
● 4,531,985,219,092 positions[1] for all games populated with 0-42 pieces
● Perfect Information
● Adversarial
● Zero-Sum Game
● Final outcomes of game:
○ Win/Lose
○ Tie
Image Source: https://dl.mandiant.com/EE/library/MIRcon2014/MIRcon_2014_Mgmt_Track_Application_Game_Theory.pdf](https://image.slidesharecdn.com/implementationandanalysisofsearchalgorithmsinsingle-playerconnectfourgame-160424050339/85/Implementation-and-analysis-of-search-algorithms-in-single-player-connect-four-game-6-320.jpg)
![Previous Work
● 1998 - James D. Allen & Victor Allis individually found the following:
○ First player has a winning strategy to the game
● Victor gave Knowledge Based Approach[2]
● James described Winning Strategies[3]
● John Trump - Brute force approach [4]
○ 8 ply database - 40,000 hours of computation
○ Any position in tree can be evaluated in seconds
● Work by James & Victor proved - First player win game
● Demonstrated first player win in Perfect Play](https://image.slidesharecdn.com/implementationandanalysisofsearchalgorithmsinsingle-playerconnectfourgame-160424050339/85/Implementation-and-analysis-of-search-algorithms-in-single-player-connect-four-game-7-320.jpg)
![Heuristics
● Explore all possibilities
○ Rows - 4 possible space -> 24 combinations
■ [1,2,3,4], [2,3,4,5], [3,4,5,6], [4,5,6,7] for one row
○ Columns - 3 possible space -> 21 combinations
■ [1,8,15,22], [8,15,22,29], [15,22,29,36] for one
column
○ Diagonals -> 24 combinations
■ 12 for major diagonal
■ 12 for minor diagonal
● Total possible spaces = 69
36 37 38 39 40 41 42
29 30 31 32 33 34 35
22 23 24 25 26 27 28
15 16 17 18 19 20 21
8 9 10 11 12 13 14
1 2 3 4 5 6 7](https://image.slidesharecdn.com/implementationandanalysisofsearchalgorithmsinsingle-playerconnectfourgame-160424050339/85/Implementation-and-analysis-of-search-algorithms-in-single-player-connect-four-game-11-320.jpg)
Implementation and analysis of search algorithms in single player connect four game
- 1. Implementation and Analysis of Search Algorithms in Single-Player Connect Four Game Anmol Rajpurohit Harsh Bahua Krish Furia Fall 2015 University of California, Irvine
- 2. Outline ● Motivation ● Game & Rules ● Analysis & Previous Work ● Problem Statement & Algorithms ● Heuristics & Evaluation ● Comparative Analysis & Conclusion Image Source: http://www.business.mcmaster.ca/mktg/nbontis/shape011.jpg
- 3. Motivation ● Implementation enhances understanding of algorithms ● Thorough analysis of algorithms ● Understanding the role of Heuristics and its variations ● Role & Impact of optimization ● Comparative study ● Love for Games Image Source: http://hunterphoenixcoaching.com/wp-content/uploads/2013/05/motivation-300x300.jpg
- 4. Connect Four ● Centuries Old Game ● Vertical Grid/Board ● Most Common - 6 Rows & 7 Columns ● Initially Board - Empty ● 42 discs - 21 of each color ● Players choose color of disc ● Players take turn to drop their color discs in columns ○ Takes lowest unoccupied row of that column Image Source: https://upload.wikimedia.org/wikipedia/commons/a/ad/Connect_Four.gif
- 5. Rules ● Simple ● Place disc in any 1 of 7 columns - Takes lowest row available ● Once the column is full no player can choose to drop disc in that column ● Winner: Player who connects four of one's own discs(same color) next to each other first in any of following manner: ○ Vertically ○ Horizontally ○ Diagonally ● Game Tie possible Image Source: http://synergyexpert.com/wp-content/uploads/2015/06/Rules5.jpg
- 6. Mathematical Analysis ● 4,531,985,219,092 positions[1] for all games populated with 0-42 pieces ● Perfect Information ● Adversarial ● Zero-Sum Game ● Final outcomes of game: ○ Win/Lose ○ Tie Image Source: https://dl.mandiant.com/EE/library/MIRcon2014/MIRcon_2014_Mgmt_Track_Application_Game_Theory.pdf
- 7. Previous Work ● 1998 - James D. Allen & Victor Allis individually found the following: ○ First player has a winning strategy to the game ● Victor gave Knowledge Based Approach[2] ● James described Winning Strategies[3] ● John Trump - Brute force approach [4] ○ 8 ply database - 40,000 hours of computation ○ Any position in tree can be evaluated in seconds ● Work by James & Victor proved - First player win game ● Demonstrated first player win in Perfect Play
- 8. Problem Statement ● Develop AI component which ○ makes best move on computer’s behalf ● State Description ○ 6 x 7 two-dimensional matrix/board ○ Each location has disc of certain color or its empty ● Initial State ○ Empty board ● Goal State ○ Board with first four connecting pieces – vertically, horizontally, or diagonally ○ Board is full and no one is able to connect, then Tie ● Search Space ○ Large search space ● Operation ○ Place a disc Image Source: http://www.dnaevolution.com.au/images/what_we_do1.jpg
- 9. Algorithms Implemented / Analyzed ● Minimax Algorithm ○ Tries to minimize whatever value the opponent is trying to maximize ○ Opponent tries to maximize its value ● Alpha Beta Pruning ○ Optimization of Minimax algorithm ○ Avoids searching subtrees of moves, which won't be selected ○ α is maximum lower bound of possible solutions ○ β is minimum upper bound of possible solutions ○ Sub-trees/nodes are pruned if the current estimate of node(based on heuristics) does not lie between α and β ● Cut-off ○ Limit depth of game tree to get next move quickly Image Source: http://connect4me102b.weebly.com/about.html The above diagram is just for illustration purpose. It does not represent actual numbers as per our implementation.
- 10. Approach ● Depth First Search ● Algorithm works in following order: a. Offensive b. Defensive c. Apply Minimax ● Five difficulty levels (depth cutoff) ● For each computer move our algorithm computes the best move ● Worst Branching Factor: 7 ● Implemented 6 x 7 Board but is scalable. Difficulty Level Depth of Game Tree explored 1 2 2 4 3 6 4 8 5 10
- 11. Heuristics ● Explore all possibilities ○ Rows - 4 possible space -> 24 combinations ■ [1,2,3,4], [2,3,4,5], [3,4,5,6], [4,5,6,7] for one row ○ Columns - 3 possible space -> 21 combinations ■ [1,8,15,22], [8,15,22,29], [15,22,29,36] for one column ○ Diagonals -> 24 combinations ■ 12 for major diagonal ■ 12 for minor diagonal ● Total possible spaces = 69 36 37 38 39 40 41 42 29 30 31 32 33 34 35 22 23 24 25 26 27 28 15 16 17 18 19 20 21 8 9 10 11 12 13 14 1 2 3 4 5 6 7
- 12. Heuristics ● For a particular space the heuristic is calculated in following manner ○ If count of pieces by C and P are equal ■ If computer’s turn then 1 ■ If player’s turn then -1 ○ If count of pieces by C > P ■ If computer’s turn then (C’s count + 1) – P’s count ■ If player’s turn then C’s count – P’s count ○ If count of pieces by computer is less than player ■ If computer’s turn then C’s count– P’s count ■ If player’s turn then C’s count – (P’s count +1) ● Count is determined using ○ Linear equation ( y = 2x) ○ Exponential ( y = 2x ) ○ Score is zero for count = 0 Image Source: https://citelighter-cards.s3.amazonaws.com/p16nqvblds4c4kcfmpuc1c19h40_79793.jpg
- 13. Evaluation: Number of Board Analyzed Depth Linear w/ AlphaBeta Exponential w/ AlphaBeta No AlphaBeta 2 58 58 73 4 1,002 1,112 2,817 6 13,000 17,000 137,000 8 152,000 300,000 6,500,000 10 1,900,000 3,900,000 305 million
- 14. Evaluation: Execution Time Depth Linear w/ AlphaBeta Exponential w/ AlphaBeta No AlphaBeta 2 0 0 0 4 0.005 0.004 0.007 6 0.015 0.017 0.08 8 0.088 0.097 0.89 10 0.329 0.6 34.7 12 4.033 6.38 >300
- 15. Evaluation: 10 Plays Difficulty Level Win Loss Draw Beginner 8 1 1 Intermediate 6 4 0 Advanced 4 5 1 Expert 4 6 0 Ultra 2 7 1
- 16. Evaluation: 10 Plays Difficulty Level Win Loss Draw Beginner 6 2 2 Intermediate 6 4 0 Advanced 5 3 2 Expert 1 7 2 Ultra 1 8 1
- 17. Evaluation
- 18. Conclusion ● Exponential heuristics works better than Linear heuristics. ● Exponential heuristics explores more number of nodes hence increasing computation time. ● Computation time of exponential heuristics is more but time required is less than 1 sec for depth of up to 10. ● Using Alpha-Beta pruning can save explore 10 to 100 times less nodes as compared to not using Alpha-Beta pruning.
- 19. Future Work ● Improve UI ● Multi-Threaded Programming ● Leverage Transposition Table ● More heuristics ● Analysis with sufficiently large board size
- 20. Thank You! Image Source: http://usercontent2.hubimg.com/6800731_f520.jpg