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Algorithms Homework 19

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Homework 19

1. The Subset Sum Decision problem (SSD) accepts an array of integers
data and target t and returns whether data contains some subset that
sums up to t. Prove that SSD ∈ NP.
2. The Subset Sum problem (SS) accepts the same input as SSD, but it
returns a subset that sums to t rather than true or false. The following
algorithm describes a reduction from SS to SSD:
Input: data: array of integers
Input: n: size of data
Input: t: target value
Output: S: subset of data that sums to t, or ∅ if no such set exists
1 Algorithm: SSReduction
2 if ¬SSD(data, t) then
3 return ∅
4 end
5 sub = 0
6 top = n
7 while sub < top do
8 if ¬SSD(data[1..(top − 1)], t) then
/* Add data[top] to subset */
9 sub = sub + 1
10 Swap data[top] and data[sub]
11 else
/* Delete data[top] from array */
12 top = top − 1
13 end
14 end
15 return data[1..sub]
What do we know about the worst-case complexity of SS if the worst-case
complexity of SSD is O(S(n)) and Ω(s(n))?
1

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