Subset-Sum Problem

Definition

Algorithm:

• Variation
• Return largest sum $\leq$ t of elements in S

Exact-Subset-Sum(S, t) $\;\;\;\;\;$ $\;\;\;\;\;$; S = {x₁, .., x_n}
$\;\;\;\;\;$n = $\mid$S$\mid$
$\;\;\;\;\;$L₀ = $\langle$0$\rangle$
$\;\;\;\;\;$for i = 1 to n
$\;\;\;\;\;$ $\;\;\;\;\;$L_i = Merge-Lists( )
$\;\;\;\;\;$ $\;\;\;\;\;$remove from L_i elements > t
$\;\;\;\;\;$return largest element in L_n

Example

S = {1, 2, 3}
L₀ = $\langle$0$\rangle$
L₁ = $\langle$0, 1$\rangle$
L₂ = $\langle$0, 1, 2, 3$\rangle$
L₃ = $\langle$0, 1, 2, 3, 4, 5, 6$\rangle$

Analysis

• L could double in size after each iteration
• Final merge could take 2ⁿ steps
• Exponential running time