CSE2100 DS & Algorithms 1
CSE 2100
Dynamic Programming
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Dynamic Programming and the Knapsack Problem: Overcoming Redundancy, Study notes of Computer Science
An overview of dynamic programming and its application to the knapsack problem. It discusses the motivation behind using dynamic programming, the issue of redundancy in recursion, and how to overcome it through bottom-up computation and memoization. Examples using fibonacci numbers and the knapsack problem.
Typology: Study notes
Pre 2010
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CSE 2100
Dynamic Programming
Overview
Motivation
One-dimension example
Recurrence - Recursion
The Knapsack
Problem
Recurrence
Solution method
Tracing a solution
Redundancy
Recursion speed?
Slow because many sub-problems can be redundant
Sub Problems & Redudancy
Sub-Problems can even by identical!
One Dimension Example
Remember the Fibonacci Numbers?
Fib(0) = 0
Fib(1) = 1
Fib(k) = Fib(k-1)+Fib(k-2)
public class Fibonacci { public Fibonacci() {} public int fib(int k) { if (k==0) return 0; else if (k==1) return 1; else return fib(k-1)+fib(k-2); } public static void main(String[] args) { System.out.println("Fib(12) = " + new Fibonacci().fib(12)); } }
Recursion Tree
Fib(6)
Fib(5) Fib(4)
Fib(4) Fib(3) Fib(3) Fib(2)
Fib(3) Fib(2) Fib(2) Fib(1) Fib(2) Fib(1) Fib(2) Fib(1)
Fib(2) Fib(1) Fib(1) Fib(0) Fib(1) Fib(0)
Fib(1) Fib(0)
How To Achieve It?
Two ways to achieve it
Top-Down computation
Memoization
Bottom-Up computation
Dynamic Programming
Bottom Up!
Fib(6)
Fib(5)
Fib(4)
Fib(3) Fib(2)
Fib(4)
Fib(3) Fib(2)
Fib(2) Fib(1) Fib(2) Fib(1)
Fib(2) Fib(1) Fib(1) Fib(0)
Fib(3)
Fib(2) Fib(1)
Fib(1) Fib(0)
Fib(1) Fib(0)
The Code
public class Fibonacci { public Fibonacci() {} public int fib(int k) { int tab[] = new int[k+1]; tab[0] = 0; tab[1] = 1; for(int i = 2;i<=k;i++) tab[i] = tab[i-1] + tab[i-2]; return tab[i]; } public static void main(String[] args) { System.out.println("Fib(12) = " + new Fibonacci().fib(12)); } }
The Code... Even Better....
public class Fibonacci { private int[] _tab; public Fibonacci() { _tab = new int[2];_tab[0]=0;_tab[1]=1;} public int fib(int k) { if (k >= _tab.length) { int[] nt = new int[k+1]; for(int k=0;k<_tab.length;k++) nt[k] = _tab[k]; for(int i = _tab.length;i<=k;i++) nt[i] = nt[i-1] + nt[i-2]; _tab = nt; } return _tab[i]; } public static void main(String[] args) { System.out.println("Fib(12) = " + new Fibonacci().fib(12)); } }