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### Dynamic Programming

Day 3: Examples

Longest Common Subsequence

- Given two sequences of characters A, B
- Find the longest sequence C such that C is a subsequence of A as well as of B
- Example:
- ABCDGH
- ZBAEDH
- Length of LCS = 3

Longest Common SubsequenceIterative DP

Longest Common Subsequence Length (UVA 10405) – Iterative DP

int a[1010][1010];

int main() {

...

memset(a, 0, sizeof(a));

for(int i = 0; i < str1.size(); i++) {

for(int j = 0; j < str2.size(); j++) {

if (str1[i] == str2[j])

a[i+1][j+1] >?= a[i][j] + 1;

a[i+1][j+1] >?= a[i+1][j];

a[i+1][j+1] >?= a[i][j+1];

}

}

cout << a[str1.size()][str2.size()] << endl;

}

Longest Common Subsequence Length (UVA 10405) - Memoization DP

int lcs(int a, int b) {

if (cache[a][b] >= 0) return cache[a][b];

if (a == str1.size() || b == str2.size())

return 0;

int ret = 0;

if (str1[a] == str2[b])

ret = 1 + lcs(a+1, b+1);

return cache[a][b] = max(ret,

lcs(a, b + 1),

lcs(a + 1, b));

}

Longest Common Subsequence DPPath Extraction (UVA 531)

- It is wonderful that we know how to compute the length of the longest common subsequence, BUT:
- How do we find out what the subsequence really is?
- In a more general case, we often need to find out not only the “quality” of the optimal solution, but also what it is
- How to do this? Let’s look at our DP table again.

Longest Common Subsequence DPPath Extraction (UVA 531)

Longest Common Subsequence DPPath Extraction (UVA 531)

- We can trace our way back through the DP table:
vector<string> path;

int i = str1.size() - 1, j = str2.size() - 1;

while (i >= 0 && j >= 0) {

if (a[i+1][j+1] == a[i][j+1]) i--;

else if (a[i+1][j+1] == a[i+1][j]) j--;

else {

path.push_back(str1[i]);

i--; j--;

}

}

reverse(path.begin(), path.end());

Critical Mass DP(UVA 580)

- 3 types of boxes: lead and uranium
- Stacking boxes on top of each other
- Stack is dangerous if 3 or more uranium boxes on top of each other
- How many dangerous stacks of height H are there?
- For example, 3 dangerous stacks if H = 4:
- LUUU
- UUUL
- UUUU

Step 1: Identify the sub-problem DP

- Modify the definition of a dangerous stack
- Stack is dangerous if:
- It contains 3 adjacent uranium boxes
OR

- It contains K adjacent uranium boxes at the bottom (K <= 3)

- It contains 3 adjacent uranium boxes

Step 2: Form a Recursive Solution DP

int doit(int height, int bottom)

{

if (bottom == 0) return 1 << height;

if (height == 0) return 0;

return doit(height - 1, 3)

+ doit(height - 1, bottom - 1);

}

Step 3: Recursion -> DP DP

int doit(int height, int bottom) {

if (bottom == 0) return 1 << height;

if (height == 0) return 0;

if (cache[height][bottom] >= 0)

return cache[height][bottom];

return cache[height][bottom] =

doit(height - 1, 3)

+ doit(height - 1, bottom - 1);

}

Headmaster’s Headache DP(UVA 10817)

Headmaster’s Headache DP(UVA 10817)

- S subjects, 1 <= S <= 8
- M teachers you have, each teaches some of the subjects, 1 <= M <= 20
- N applicants from whom you can hire, each teaches some of the subjects 1 <= N <= 100
- Each teacher and each applicant has a salary they require
- What is the minimum budget the headmaster needs so that each subject is taught by at least 2 teachers?

Headmaster’s Headache DP(UVA 10817)

Questions to Think About DP

- What are different possible representation of the sub-problem? What are the advantages and disadvantages of each?
- Would you prefer to solve this problem using memoization or iterative DP? What are the advantages and disadvantages of each?

Mailbox Manufacturer’s Problem (UVA 882) DP

- You have 1 <= k <= 10 mailboxes
- At most 100 crackers fit into a mailbox
- If a mailbox can withstand x fire-crackers, it can also withstand x - 1 fire-crackers
- After explosion, a mailbox is either totally destroyed or unharmed
- How many fire-crackers do you need to guarantee you will find out the smallest number of crackers that blow up a mailbox?

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