1. Dynamic Programming 1 Neil Tang 4/20/2010
CS223 Advanced Data Structures and Algorithms

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

3. CS223 Advanced Data Structures and Algorithms
Class Overview
Basic Idea
Fibonacci numbers
Recursive equation evaluation
All-pairs shortest paths
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
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

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

6. CS223 Advanced Data Structures and Algorithms
Fibonacci Numbers
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Fibonacci Numbers
A dynamic programming based algorithm
Time Complexity: O(n)
CS223 Advanced Data Structures and Algorithms

8. Recursive Equation Evaluation
CS223 Advanced Data Structures and Algorithms

9. Recursive Equation Evaluation
CS223 Advanced Data Structures and Algorithms

10. Recursive Equation Evaluation
A dynamic programming based algorithm
Time Complexity: O(n2)
CS223 Advanced Data Structures and Algorithms

11. 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

12. 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

13. All-Pairs Shortest Path
Time Complexity: O(|V|3)
CS223 Advanced Data Structures and Algorithms