A lgorithms & D ata Structures for Games

1. Minor Games Programming Algorithms & Data Structuresfor Games Lecture 3

2. Next AD college

3. Algorithms and Data Structures • Feedback previous lectures • Theory: Dynamic Programming • Theory: Randomized Algorithms • Theory: Backtracking Jan Verhoeven j.verhoeven@windesheim.nl

4. Theory: Dynamic Programming • 10.3 Dynamic Programming • 10.3.1 Using a Table instead of Recursion

5. 10.3 Dynamic Programming • Rewrite the recursive algorithm as a non recursive algorithm that systematically records the answers to the subproblems in a table.

6. 10.3.1 Using a Table instead of Recursion The natural recursive program to compute the Fibonacci numbers is very inefficient. (Running time is exponential).

7. 10.3.1 Using a “Table” instead of Recursion a linearsolution

8. Fibonacci

9. Another very nice example Solve the recurrence: with C(0) = 1 And what about C(2) and C(3)?

10. Figure 10.43 Recursive solution(what is the running time?)

11. Figure 10.45 solution with a table(what is the running time?)

12. Running time

13. Theory: Randomized Algorithms • 10.4 Randomized Algorithms • 10.4.2 Skip Lists • And extra topic: How about Primes….

14. 10.4 Randomized Algorithms • At least once during the algorithm, a random number is used to make a decision. The running time depends on the particular input, but also on the random numbers that occur. • A sample application is: • 10.4.2 Skip Lists

15. 10.4.2 Skip Lists • A data structure that supports both searching and insertion in O(log N) expected time. • Also see: http://en.wikipedia.org/wiki/Skip_list

16. Linked lists with extra links to cells ahead

17. Searching for the value of 8

18. A skip list

19. Before and after an insertion in a skip list

20. Beautiful Applets !! Please run: http://people.ksp.sk/~kuko/bak/big/ And you might take a look at: http://iamwww.unibe.ch/~wenger/DA/SkipList/

21. Primes Do you know a method to generate prime numbers? See: Sieve_of_Eratosthenes

22. Primes boolisPrime (const long aNumber) { // Do you know a method to test // whether a number is prime? } And what about the running time ?

23. Primes Long nextPrime(const long aPrime) { // Do you know a method to generate // the next prime following aPrime? } And what about the running time ?

24. Well Known study: A Random Walk • Imagine now a drunkard walking randomly in a city. • The city is infinite and arranged in a square grid, and at every intersection, the drunkard chooses one of the four possible routes (including the one he came from) with equal probability. • Formally, this is a random walk on the set of all points in the plane with integer coordinates.

25. Random Walk (2) • Will the drunkard ever get back to his home from the bar? • It turns out that he will (almost surely).

26. Random Walk (3)

27. Algorithms and Data Structures for games

28. Backtracking is a form of recursion. • The usual scenario is that you are faced with a number of options, and you must choose one of these. • After you make your choice you will get a new set of options; just what set of options you get depends on what choice you made. • This procedure is repeated over and over until you reach a final state. • If you made a good sequence of choices, your final state is a goal state; if you didn't, it isn't.

29. An example …

30. The steps: • Starting at Root, your options are A and B. You choose A. • At A, your options are C and D. You choose C. • C is bad. Go back to A. • At A, you have already tried C, and it failed. Try D. • D is bad. Go back to A. • At A, you have no options left to try. Go back to Root. • At Root, you have already tried A. Try B. • At B, your options are E and F. Try E. • E is good. Congratulations!

31. In pseudo code booleansolve(Node n){ if n is a leaf node { if the leaf is a goal node, return true else return false } else { for each child c of n { if solve(c) succeeds, return true } return false } }

32. Cindy’s puzzle, the goal is to reverse the positions of the marbles: • The black marbles can only move to the right, and the white marbles can only move to the left (no backing up). At each move, a marble can either: • Move one space ahead, if that space is clear, or • Jump ahead over exactly one marble of the opposite color, if the space just beyond that marble is clear.

33. Cindy’s puzzle SEE: ..\..\Projects\BacktrackingCindyCS\BacktrackingCindyCS.sln

34. A little exercise • Can you describe an backtracking algorithm to solve the next problem? • The problem is to write an integer as the sum of 4 squares. This is always possible ! • So: 70 = 64 + 4 + 1 + 1 • And: 12345 = 11664 + 676 + 4 + 1 • See: ..\..\Projects\Backtracking4SquaresCS\Backtracking4SquaresCS.sln

35. boolsolve(intvalue, int num) { if (value == 0)return true; // Are we done? // we have no more numbers to work with if (num == 0) return false; // Start at 1 and work up for (int i = 1; i * i <= value; i++) { int sq = i * i; // Place guess if (solve(value - sq, num - 1)) return true; } return false; // Nothing worked:Backtrack }

36. Another example: SUDOKU • See: • ..\..\Projects\SudokuCS\SudokuCS.sln

37. The famous 8 queen’s puzzlehttp://en.wikipedia.org/wiki/Eight_queens_puzzle

38. Homework and Practice • Study the slides and the corresponding text in your study book. • Try to program one of the backtracking examples (Cindy’s puzzle, Sudoku, eight queens) • The practice is about a backtracking problem to be solved (using C#)