Slideshow:
A simple (and naive) way to find the optimal solution
A general approach to finding the best solution:
A better way to find the optimal solution
A general approach to finding the best solution:
A better way to find the optimal solution
A general approach to finding the best solution:
Pre-requsite of applying Dynamic Programming explained in graphic way
Problem statement:
Pre-requsite of applying Dynamic Programming explained in graphic way
Given:
Pre-requsite of applying Dynamic Programming explained in graphic way
Requirement to use dynamic programming:
Example problem solved using Dynamic Programming
Example problem solved using Dynamic Programming
Example on how to solve MakeChangeFor(15) using smaller (optimal) soultions:
Example problem solved using Dynamic Programming
Example on how to solve MakeChangeFor(15) using smaller (optimal) soultions:
/home/cs554001/demo/dyn-prog/coins.java
- General approach to find the
best solution:
- Brute force search.....
- Compute the cost
of every solution
- Find the solution with lowest cost
- Compute the cost
of every solution
- Brute force search.....
- Dynamic Programming:
- Dynamic Programming =
a smarter way to
find the optimal solution through
the brute force search method
- Dynamic programming uses the
optimal solution of
some smaller problem
to find the
optimal solution of
a larger problem
- Dynamic Programming can only be applied if the problem is recursive
- Dynamic Programming =
a smarter way to
find the optimal solution through
the brute force search method
- Pre-requisite to
apply the
dynamic programming
problem solving technique:
- The problem can be solved by using the solution of one or more smaller problems of identical nature
Graphically:
- We need to
solve this
problem N:
- Suppose we
have the
solutions for
the smaller problems:
- Pre-requisite to
using
the dynamic programming
problem solving method:
- Problem description:
- There are 4 different denominations of
coins in use with values:
- 1, 4, 7, 12
- Problem:
- Find the minimum # of coins that you need to use to make the change for $N
- There are 4 different denominations of
coins in use with values:
- Example:
Denominations: 1, 4, 7, 12 Change amount | Min. # coins | How to do it ----------------+--------------+--------------- 1 1 1 2 2 1+1 3 3 1+1+1 4 1 4 5 2 4+1 6 3 4+1+1 7 1 7 8 2 7+1 9 3 7+1+1 10 4 7+1+1+1 11 2 7+4 12 1 12 13 2 12+1 14 2 7+7 15 3 7+7+1 16 2 12+4 and so on...
- We can find the
solution for
$15
using the
smaller solutions for
$3,
$8, 11
and $14 as follows:
Possible denominations: 1, 4, 7, 12
We can make the value 15 using any one of the above denomincations: change(15) = 1 + change(14) change(15) = 4 + change(11) change(15) = 7 + change(8) change(15) = 12 + change(3) ==> One of these must be the optimum solution !!!!
If we know the optimum solution for change(14), change(11), change(8), change(3), we can solve the problem as follows: 15 = 1 + 14 | 4 + 11 | 7 + 8 | 12 + 3 | | | | USE the solution for smaller prob ! V V V V 7+7 7+4 7+1 1+1+1 -------------------------------------------------- 1+7+7 4+7+4 7+7+1 12+1+1+1 Best solution for 15: 1 + 7 + 7 (chosen arbitrarily among the best solutions)
- Example Program:
(Demo above code)
- /home/cs554001/demo/dyn-prog/coins.java
- If you like to know more about dynamic programming: click here