Dynamic Programming

1
Dynamic Programming

• Day 3 Examples

2
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

3
Longest Common Subsequence Iterative DP
-- A B C D G H
-- 0 0 0 0 0 0 0
Z 0
B 0
A 0
E 0
D 0
A 0
0
0
1
1
1
1
4
Longest Common SubsequenceIterative 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
5
Longest Common Subsequence Length (UVA 10405)
Iterative DP

• int a10101010
• int main()
• ...
• memset(a, 0, sizeof(a))
• for(int i 0 i lt str1.size() i)
• for(int j 0 j lt str2.size() j)
• if (str1i str2j)
• ai1j1 gt? aij 1
• ai1j1 gt? ai1j
• ai1j1 gt? aij1
• cout ltlt astr1.size()str2.size() ltlt
  endl

6
Longest Common Subsequence Length (UVA 10405) -
Memoization

• int lcs(int a, int b)
• if (cacheab gt 0) return cacheab
• if (a str1.size() b str2.size())
• return 0
• int ret 0
• if (str1a str2b)
• ret 1 lcs(a1, b1)
• return cacheab max(ret,
• lcs(a, b 1),
• lcs(a 1, b))

7
Longest Common SubsequencePath 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? Lets look at our DP table again.

8
Longest Common SubsequencePath 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
9
Longest Common SubsequencePath Extraction (UVA
531)

• We can trace our way back through the DP table
• vectorltstringgt path
• int i str1.size() - 1, j str2.size() - 1
• while (i gt 0 j gt 0)
• if (ai1j1 aij1) i--
• else if (ai1j1 ai1j) j--
• else
• path.push_back(str1i)
• i-- j--
• reverse(path.begin(), path.end())

10
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

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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 lt 3)

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Step 2 Form a Recursive Solution

• int doit(int height, int bottom)
• if (bottom 0) return 1 ltlt height
• if (height 0) return 0
• return doit(height - 1, 3)
• doit(height - 1, bottom - 1)

13
Step 3 Recursion -gt DP

• int doit(int height, int bottom)
• if (bottom 0) return 1 ltlt height
• if (height 0) return 0
• if (cacheheightbottom gt 0)
• return cacheheightbottom
• return cacheheightbottom
• doit(height - 1, 3)
• doit(height - 1, bottom - 1)

14
Headmasters Headache(UVA 10817)

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Headmasters Headache(UVA 10817)

• S subjects, 1 lt S lt 8
• M teachers you have, each teaches some of the
  subjects, 1 lt M lt 20
• N applicants from whom you can hire, each teaches
  some of the subjects 1 lt N lt 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?

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Headmasters Headache(UVA 10817)
Name Must Hire? Salary Math? Phys? Chem?
Anna YES 55,000 YES
Bob YES 50,000 YES
Carl YES 60,000 YES
Donald 75,000 YES YES
Emil 90,000 YES YES
Fred 50,000 YES
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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?

18
Mailbox Manufacturers Problem (UVA 882)

• You have 1 lt k lt 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?