COMP3010: Algorithm Theory and Design
Daniel Sutantyo, Department of Computing, Macquarie University
4.2 - Developing a DP Algorithm
4.2 - Developing a DP Algorithm
Developing a DP Algorithm
- Questions:
-
- Does the problem have the optimal substructure property?
- Does the problem have overlapping subproblems?
- If the answer is yes to both of these questions, then we can use dynamic programming to solve the problem
4.2 - Developing a DP Algorithm
Developing a DP Algorithm
- In developing a dynamic-programming solution, we follow a sequence of four steps (from CLRS, page 359)
- Characterise the structure of an optimal solution
- Recursively define the value of an optimal solution
- Compute the value of an optimal solution
- (optional) Construct an optimal solution from computed information
4.2 - Developing a DP Algorithm
Developing a DP Algorithm
- ... or informally:
- Show that there is an optimal substructure
- Show the recursive relation that gives optimal solution
- Find the VALUE of the optimal solution
- e.g. find the length of the LCSS
- (optional) Find the COMBINATION that gives the optimal solution
- e.g. find the LCSS itself (not just the length)
4.2 - Developing a DP Algorithm
Rod Cutting
- We have one piece of rod that we can cut into smaller pieces. Suppose that cutting a rod is free and we can sell a rod of length \(i\) for \(p_i\). Given a rod of length \(n\), how should we cut it up to maximise our revenue?
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
$9
$1
$8
$8
$1
$5
$5
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$5
$5
$5
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
$20
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
$9
$9
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
$9
$9
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
$9
$9
$17
$5
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
4.2 - Developing a DP Algorithm
Rod Cutting
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
4.2 - Developing a DP Algorithm
Rod Cutting
4.2 - Developing a DP Algorithm
Rod Cutting
4.2 - Developing a DP Algorithm
Rod Cutting
4.2 - Developing a DP Algorithm
Rod Cutting
4.2 - Developing a DP Algorithm
Rod Cutting
4.2 - Developing a DP Algorithm
Rod Cutting
1
4
6
4
1
4.2 - Developing a DP Algorithm
Rod Cutting
- no cut: \(\binom{4}{0}\) = 1 way
- cut in 1 place: \(4 \choose 1\) = 4 ways
- cut in 2 places: \(4 \choose 2\) = 6 ways
- cut in 3 places: \(4 \choose 3\) = 4 ways
- cut in 4 places: \(4 \choose 4\) = 1 way
4.2 - Developing a DP Algorithm
Rod Cutting
- If n = 5, you have 4 possible places to cut
- In total, there are \(2^{n-1}\) possible combinations
- hence the brute-force approach is \(O(2^n)\)
4.2 - Developing a DP Algorithm
Rod Cutting
- Do you see the problems and subproblems?
- Do you see the overlapping subproblems?
4.2 - Developing a DP Algorithm
Rod Cutting
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
- New rule:
- After each cut, don't cut the left most segment any further
- Why? Symmetry
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
so where are the overlapping subproblems?
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
- so our game plan is to find the optimal price when we have a rod of length 1, length 2, length 3, etc
- this is NOT the same with the price of a rod of length 1, length 2, length 3, etc
Rod Cutting - Overlapping Subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Overlapping Subproblems
\(i\)
\(p_i\)
1 2 3 4 5 6 7 8 9 10
1 5 8 9 10 17 17 20 24 30
\(i\)
optimal
1 2 3 4 5 6 7 8 9 10
1 5 8 10 13 17 ?? ?? ?? ??
4.2 - Developing a DP Algorithm
Rod Cutting - Optimal Substructure
- Does the problem exhibit optimal substructure?
- You have to show that the optimal solution to the problem contains the optimal solutions to the subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Optimal Substructure
- To solve the problem, you have to make a choice
Rod Cutting - Optimal Substructure
- The choice you make results in more subproblems
- You have to show that the optimal solution to your problem MUST contain the optimal solution to your subproblems
4.2 - Developing a DP Algorithm
Rod Cutting - Optimal Substructure
- Imagine that you already have the optimal solution to the problem
- i.e. you already know which path to follow
- we can show that we are using the optimal solution to the subproblem as well
- Suppose I have the optimal solution to the problem of cutting a rod of length \(n\)
- the solution requires that I cut the rod to length \(r\) and \((n-r)\) (the next subproblem)
- the solution to cutting \((n-r)\) must be optimal
4.2 - Developing a DP Algorithm
Rod Cutting - Optimal Substructure
- Suppose that the solution to the subproblem of cutting a rod of length \((n-r)\) gives you \(y\) dollars (reminder, this is NOT the price of a rod of length \((n-r)\))
- Suppose that the optimal solution to the problem of cutting a rod of length \(n\) gives you \(\$(x+y)\)
\(\$x\)
\(\$y\)
\(\$y\)
- then \(\$y\) must be the best that you can do, i.e. it is optimal
4.2 - Developing a DP Algorithm
\(<\$y\)
\(<\$y\)
\(<\$y\)
Rod Cutting - Optimal Substructure
- Why? Because if the subproblem with \((n-r)\) has a better solution, then you WILL be able to find a better solution for the problem
- e.g. if \(z > y\), then the optimal solution should be \(\$(x+z)\), not \(\$(x+y)\)
- so if the solution to the problem is \(\$(x+y)\), then it must be the case that \(y > z\)
\(\$x\)
\(\$y\)
\(\$y\)
\(<\$y\)
4.2 - Developing a DP Algorithm
\(<\$y\)
\(\$z>\)
\(<\$y\)
Rod Cutting - Optimal Substructure
- Make sure you understand the logic, this is a proof by contradiction (we're doing something similar next week with greedy algorithm)
- Summary:
- FACT: we have the optimal solution
- don't worry about how you get it, all it matters is that you have it
- PROPOSITION: we must have the optimal solution to the subproblem as well
- because if we don't, then there is a better solution than the one we are using, which means, our FACT is wrong, that's a contradiction
- FACT: we have the optimal solution
4.2 - Developing a DP Algorithm
Rod Cutting - Optimal Substructure
- Another way to think about this:
- I have the most expensive computer in the world
- So, every component must also be the most expensive in the world
- if my monitor is not the most expensive in monitor in the world, then I can find another monitor which is more expensive, and so my computer wouldn't be the most expensive
- that's a contradiction
4.2 - Developing a DP Algorithm
Optimal Substructure
4.2 - Developing a DP Algorithm
- In general, to show the existence of optimal substructure, you always follow the same steps
- Show that the optimal solution requires you to solve one or more subproblems and assume that you have the optimal solution
- You argue that the solution to the subproblems must also be optimal
- otherwise, you can construct a better solution for the problem using cut-and-paste technique (cut the better solution, and paste it)
- but this is a contradiction: we assumed we have the optimal solution!
- See page 379 of CLRS if you want a more detailed breakdown
Rod Cutting - Optimal Substructure
- One more time, just to be clear:
- Show that the rod cutting problem exhibits optimal structure:
- Suppose that we have the solution to the problem of cutting a rod of length \(n\)
- I would also have the solution to the subproblems of cutting the rods of length \((n-r)\), \(0 < r < n\), and my solution to the problem will include one of these (whichever gives me the most value)
- The solution to the subproblem that I use, say \(y\), must also be optimal, because if it is not, then I can find a better solution to the subproblem, say \(z > y\), and this results in the better solution than what we claim to be the optimal solution
4.2 - Developing a DP Algorithm
4.2 - Developing a DP Algorithm
Developing a DP Algorithm
- Remember that there are 4 steps:
- Show that there is an optimal substructure
- Show the recursive relation that gives optimal solution
- Find the VALUE of the optimal solution
- (optional) Find the COMBINATION that gives the optimal solution
- We have only shown the first one, but we are going slowly with this, so don't get too worried
- The rest should be simpler since optimal substructure is usually the part where students have problems
COMP3010 - 4.2 - Developing a DP Algorithm - Part 1 - Optimal Substructure
By Daniel Sutantyo
COMP3010 - 4.2 - Developing a DP Algorithm - Part 1 - Optimal Substructure
Identifying overlapping subproblems, optimal substructure, rod-cutting example
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