1. Algorithms & Data Structures for Games
Minor Games Programming
Algorithms & Data Structures for Games
Lecture 3

2. Next AD college
Monday

3. Algorithms and Data Structures
Feedback previous lectures
Theory: Dynamic Programming
Theory: Randomized Algorithms
Theory: Backtracking
Jan Verhoeven

4. Theory: Dynamic Programming
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 linear solution

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
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:
Skip Lists

15. Skip Lists
A data structure that supports both searching and insertion in O(log N) expected time.
Also see:

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: And you might take a look at:

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

22. bool isPrime (const long aNumber) { // Do you know a method to test
Primes
bool isPrime (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
Backtracking
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
boolean solve(Node n){ if n is a leaf node { if the leaf is a goal node, return true else return false } { for each child c of n { if solve(c) succeeds, return true } }

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 =
And: =
See: ..\..\Projects\Backtracking4SquaresCS\Backtrackin g4SquaresCS.sln

35. bool solve(int value, 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 puzzle http://en. wikipedia

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#)