Example 1: Critical Mass (UVA 580)

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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 Subsequence
Iterative DP
--
A
B
C
D
G
H
--
0
0
0
0
0
0
0
Z
0
0
B
0
0
A
0
1
E
0
1
D
0
1
A
0
1
Longest Common Subsequence
Iterative DP
--
A
B
C
D
G
H
--
0
0
0
0
0
0
0
Z
0
0
0
0
0
0
0
B
0
0
1
1
1
1
1
A
0
1
1
1
1
1
1
E
0
1
1
1
1
1
1
D
0
1
1
1
2
2
2
H
0
1
1
1
2
2
3
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
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
Path 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
Path Extraction (UVA 531)
--
A
B
C
D
G
H
--
0
0
0
0
0
0
0
Z
0
0
0
0
0
0
0
B
0
0
1
1
1
1
1
A
0
1
1
1
1
1
1
E
0
1
1
1
1
1
1
D
0
1
1
1
2
2
2
H
0
1
1
1
2
2
3
Longest Common Subsequence
Path 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
(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
• 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)
Step 2: Form a Recursive Solution
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
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
(UVA 10817)
Headmaster’s Headache
(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
(UVA 10817)
Name
Anna
Must Salary
Hire?
YES 55,000
Bob
YES
50,000
Carl
YES
60,000
Math?
Phys?
Chem?
YES
YES
YES
Donald
75,000
YES
Emil
90,000
YES
Fred
50,000
YES
YES
YES
Questions to Think About
• 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)
• 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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