Linear-time constant-space 1/2-approximation algorithm for the maximum subset sum problem

Question

The following problem statement is given: Let $S = \{s_1, s_2, \cdots, s_n\}$ be a sequence of unique positive integers and $K$ a positive integer, where $K \ge s_i$ for every $i$ between $1$ and $n$. The goal is to find a subset of $S$ whose sum is maximum, subject to the constraint that the sum must be $\le K$. Write an approximation algorithm which computes a sum which is at least half of the optimal solution on any given input. This algorithm must run in $O(n)$ time-complexity and $O(1)$ space-complexity.

I've been searching for some simple algorithm to satisfy these conditions and the closest I've come to is this, but since it makes use of sorting, the time-complexity is brought to $O(n\log n)$.

This is taken from a set of exercises given by my uni professor as preparation for the final exam. To be honest, I am not sure whether it is his original work or not, so I don't have any relevant reference to provide.

Answer 1

The idea is to set $K/2$ as the target.

• If there is any given number that is at least $K/2$, just return it.
• Otherwise, all given numbers are $<K/2$.
  • If the sum of all given numbers is $\le K$, return the sum, which is the optimal solution.
  • Otherwise the sum of all given numbers is $>K$.
    Let $prefix\_sum$ be $0$ initially. We will try adding all given numbers one by one to $prefix\_sum$ until it is $\ge K/2$, at which moment it cannot be $>K$ -- otherwise the last number added must be $>K/2$. Return it.

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Here is an algorithm that implements the idea concisely. It runs in $O(n)$ time and $O(1)$ space.

1. Let $prefix\_sum = 0$.
2. For $i$ from $1$ to $n$:
   1. If $s_i\ge K/2$, return $s_i$. Else add $s_i$ to $prefix\_sum$.
   2. If $prefix\_sum\ge K/2$, return $prefix\_sum$.
3. Return $prefix\_sum$.