1. －動態規劃

2. Getting started : Fibonacci Sequence Given initial value F(1) = 1, F(2) = 1. Recursive relation F(n) = F(n-1) + F(n-2) Given any number k, ask F(k)

3. Fibonacci Sequence – Normal Solution Use recursive Continuity call the previous one Return while it reach known value – F(1), F(2) +intuitive -Time-consuming and memory-consuming →How many counts needed?

4. Fibonacci Sequence – Another Way Start from smaller one. Continuous add the last two to get a new one. Do until reach the value we want. +Faster -Not so easy to think at first glimpse

5. DP Elements Table Element Definition Recursive Formula Computing Sequence – Recursion – Mathematical Induction – Divide-and-Conquer ※ This part adapt from Pang-Feng’s PPT.

6. Relative Topics LCS / LIS /.. BFS / DFS Back Tracking Searching - More than once

7. Recall – LCS/LIS Given two or more sequences. Want to find the maximum length of common subsequence. If add increasing restriction? How about if we need to print the result out ?

8. Recall – BFS/DFS Given a plane, may refine region or not. Given starting point and ending point. How many ways to reach with least steps? (Means how many different shortest paths?) Visualize ? http://www.neopets.com/games/play.phtml?game_id=356

9. An example : Spilt coin We have some values of coins now. $1, $5, … Unlimited amount for each kind. Given specific amount, find ways to combine. How if there are many test cases ?

10. Direct thought : Top down For example : $100, compose with $5 & $10 We may take $5 or $10 from it. → So ways(100) = ways(100-5) + ways(100-10) → ways(n) = ways(n-5) + ways(n-10) Use recursion until reach the known ones.

11. DP method : Bottom up Consider the value we can form from lower one. For example, if we have $5 and $10, we can add value by 5 or 10 through add one coin. How to use it ?

12. Related Exercises on Uva OJ 495 - Fibonacci Freeze 825 - Walking on the Safe Side 357 - Let Me Count The Ways 10405 - Longest Common Subsequence 111 - History Grading 10684 - The jackpot 105 - The Skyline Problem

13. References Pangfeng Liu, ACP-DP, 2004, NTU UVa Online Judge

14. Further Reading BFS,DFS and backtracking, by yugi340238, 2009 -- Visualized illustration about search. DP slider, by muming, 2009 -- Tiling problem, DP method of LIS. DP slider, by geniusdog, 2008 -- Tiling problem, formal definition and properties about dynamic programming.