# Algorithm Analysis Neil Tang 01222008

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```					               Dynamic Programming 1

Neil Tang
4/20/2010

CS223 Advanced Data Structures and Algorithms   1
Course Survey
Please complete the course survey at:

http://www.cs.montana.edu/survey/

CS223 Advanced Data Structures and Algorithms         2
Class Overview
 Basic Idea

 Fibonacci numbers

 Recursive equation evaluation

 All-pairs shortest paths

CS223 Advanced Data Structures and Algorithms   3
Basic Idea
 Mathematically express the problem in the recursive form.

 Solve it by a non-recursive algorithm that systematically
records the answers to the subproblems in a table.

CS223 Advanced Data Structures and Algorithms                 4
Fibonacci Numbers
 fib(N) = fib(N-1) + fib(N-2)

CS223 Advanced Data Structures and Algorithms   5
Fibonacci Numbers

CS223 Advanced Data Structures and Algorithms   6
Fibonacci Numbers
 A dynamic programming based algorithm

Time Complexity: O(n)
CS223 Advanced Data Structures and Algorithms   7
Recursive Equation Evaluation
N 1
2
C(N ) 
N
 C (i)  N
i 0

CS223 Advanced Data Structures and Algorithms   8
Recursive Equation Evaluation

CS223 Advanced Data Structures and Algorithms   9
Recursive Equation Evaluation
 A dynamic programming based algorithm

Time Complexity: O(n2)
CS223 Advanced Data Structures and Algorithms   10
All-pairs Shortest Path
 The all-pairs shortest path problem: Given a weighted
graph G, find the shortest (minimum cost) path between
every pair of nodes in G.

 A simple solution: run Dijsktra’s algorithms |V| times, which
leads to a time complexity of O(|E||V|log|V|).

CS223 Advanced Data Structures and Algorithms                 11
All-Pairs Shortest Path
 Recursive expression for any node pair (i, j)

Dk,i,j = min{Dk-1,i,j, Dk-1,i,k+Dk-1,k,j}

CS223 Advanced Data Structures and Algorithms     12
All-Pairs Shortest Path

Time Complexity: O(|V|3)

CS223 Advanced Data Structures and Algorithms                          13

```
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