－動態規劃. 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.

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Presentation transcript:

－動態規劃

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)

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?

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

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

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

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 ?

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 ?

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 ?

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.

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 ?

Related Exercises on Uva OJ Fibonacci Freeze Walking on the Safe Side Let Me Count The Ways Longest Common Subsequence History Grading The jackpot The Skyline Problem

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

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