Sum of subset reduce to
Partition
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林季謙
Sum of Subset Problem
• The Sum of Subset Problem (部份集合的和問題):
▫ 給予一組正整數的集合S={a1, a2, … , an}及一個常數c，問: 集合S中
是否存在一組子集合S’，此子集合S’的數字總合為c。
▫ Ex: 假設有一個集合 S = {12, 9, 33, 42, 7, 10, 5} 與常數c = 24.
▫ 問 S’ = {9, 10, 5}的總和會不會等於 c
▫ 在此 C=24 所以 會回答Yes
▫ If C=6 會回答No
Partition Problem
▫ 給予一組正整數的集合S={a1, a2, … , an}，問: 是否可以將其分割成
兩個子集合S1與S2，而此兩個子集合的個別總和相等。
▫ Ex: 設有一集合 S = {13, 2, 17, 20, 8}.
▫ S1 = {13, 17} and S2 = {2, 20, 8}.
Proof Sum of subset reduce to
Partition
• Sum of Subsets : A = { a1, a2, …, an }, M
• Partition : B = { b1, b2, …,bn,bn+1,bn+2 },
and bi = ai, 1<i<n
• 設bn+1=M+1
• bn+2=( )+1-M
• Consider a sum of subset problem S with
{a1, a2, … , an} and B.
• Let SUM(ai)=M for i= 1, 2, … , n.
• We find an integer Q s.t. B+Q= ½(M+Q).
(or Q=M-2B)
• Let the corresponding partition problem
instance P be {Q,a1,a2,…,an}.
• We can partition P if and only if there is
some subset of S that sums to B.
• Ex. S={5, 7, 3, 8} and B=11. Set Q=M2B=23-22=1. Then. P={1, 5, 7, 3, 8} has a
partition {1, 3, 8}//{5, 7}.The former
contains a subset with sum B=11.
Sum of Subsets : A = { a1, a2, …, an }, M
Partition : B = { b1, b2, …,bn,bn+1,bn+2 },
and bi = ai, 1<i<n
設bn+1=M+1
bn+2=(
)+1-M
假設有一個Sum of subset 的解 S 即
那麼
假設有一個partition的解，然而
(
)則存在S
需要不一樣且