1 Algorithms & Data Structures for Games Lecture 2A Minor Games Programming

2 Algorithms and Data Structures  Feedback previous lectures  Theory: Greedy Algorithms Jan Verhoeven

3 Theory: Greedy Algorithms  Algorithm Design Techniques  10.1 Greedy Algorithms  A Simple Scheduling Problem  Hufman Codes  Approximate Bin Packing  Finally: some exercises

4 Greedy Algorithms  Definition: An algorithm that always takes the best immediate, or local, solution while finding an answer.  Greedy algorithms find the overall, or globally, optimal solution for some optimization problems, but may find less-than-optimal solutions for some instances of other problems. 

Coin-changing problem  How to minimize the number of bills and coins, when paying an amount in US currency Solution:  Greedy algorithm: repeatedly dispense the largest denomination. 5

6 Example

7  Okay, the algorithm usually works. In fact, for the U.S. coin system {1, 5, 10, 25}, it always yields the optimal set. However, for other sets, like {1, 5, 8, 10} and a goal of 13, this greedy algorithm would take one 10, and then three 1's, for a total of four coins, when the two coin solution {5, 8} also exists.

A small exercise  I buy a hamburger in Amsterdam and I pay in coins the amount of 3.98 euro’s.  What is the minimal numbers of coins I need to pay for the burger?  And when I buy 100 burgers?  The euro coins are 1, 2, 5, 10, 20, 50, 100 and 250 cents. 8

9 Problem: A Simple Scheduling Problem  What is the best way to schedule jobs in order to minimize the average completion time?  Greedy Solution: arrange by shortest job first

10 Possible schedules (single processor) Average completion time:Top: 25, Bottom: 17,75 (optimal) Optimal Schedule: arranged by shortest job first.

11 Multiprocessor case

12 Possible Schedules (3 processors) case I

13 Possible Schedules (3 processors) case II

14 Minimizing the final completion time (much harder to solve …)

A small exercise  How do I schedule on my 2 processor computer the following jobs and get the lowest possible average completion time?  J1(5), J2(2), J3(8), J4(1), J5(6), J6(4) with the restriction J4 can only run when J2 has been finished.  What is the average completion time? 15

16 Huffman Codes  File compression strategy: Allow the code length to vary from character to character and ensure that the frequently occurring characters have short codes.

17 Using a standard coding scheme

Representation of the original code in a tree 18

19 Optimal prefix code

20 Optimal prefix code: TREE

A small exercise  Please decode the following (hexadecimal) bit string in normal characters, using the optimal prefix code tree of previous slide: 1E06 D

22 Huffman’s Algorithm  Greedy,  To produce the optimal prefix code  See example in figure’s thru 10.19

A small exercise  Construct the Huffman code of the following text:  COFFEE TIME FOR YOU!!!!!!!!!!  What is the compression rate? 23

24 Approximate Bin Packing  We are given N items of sizes s 1, s 2,.., s n. The problem is to pack these items in the fewest number of bins.

25 Approximate Bin Packing  Greedy algorithms to solve the bin packing problem.  On-line bin packing:  Next fit  First fit  Best fit  Off-line bin packing:  First fit decreasing  Best fit decreasing

On-line and Off-line  On-line:  Each item must be placed in a bin before the next item can be processed.  Off-line:  We do not need to do anything until all the input has been read. 26

Next fit for 0.2, 0.5, 0.4, 0.7, 0.1, 0.3,

First fit for 0.2, 0.5, 0.4, 0.7, 0.1, 0.3,

Best fit for 0.2, 0.5, 0.4, 0.7, 0.1, 0.3,

Off-line  First sort the items in decreasing order.  Then execute first-fit or best-fit 30

31 Optimal packing for: 0.2, 0.5, 0.4, 0.7, 0.1, 0.3, 0.8

32 A small exercise Show the results of all the bin-packing strategies on the input: 0.1, 0.3, 0.5, 0.2, 0.8, 0.4, 0.7

33 Homework  Homework  Chapter 10.1  Exercises (should not be handed in!)  10.3  10.7 (program)  10.10