Chapter 8: Dynamic Programming
Those who do not remember the past are con-
demned to repeat it. (George Santayana)
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Example: Computing Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Computing Binomial Coefficients 4
Definition of Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Table for Computing Binomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Binomial Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Comments on Binomial Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Warshall’s Algorithm 8
Transitive Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Bad Algorithm for Transitive Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Idea Behind Warshall’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Warshall’s Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Example of Warshall’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Example Continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Other Dynamic Programming Problems 14
Optimal Binary Search Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
The Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Example for the Knapsack Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1
Introduction
Dynamic programming is a technique for solving problems with the following
properties:
An instance is solved using the solutions for smaller instances.
The solution for a smaller instance might be needed multiple times.
The solutions to smaller instances are stored in a table, so that each smaller
instance is solved only once.
Additional space is used to save time.
CS 3343 Analysis of Algorithms Chapter 8: Slide – 2
Example: Computing Fibonacci Numbers
Compare recursive solution (exponential):
algorithm F(n)
if n= 0 or n= 1 then return n
else return F(n−1) + F(n−2)
To dynamic programming solution (linear):
algorithm F(n)
A[0] ←0;A[1] ←1
for i←2to ndo
A[i]←A[i−1] + A[i−2]
return A[n]
CS 3343 Analysis of Algorithms Chapter 8: Slide – 3
2
