  The subset sum problem (SSP) can be briefly stated as: given a target integer
$E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$
summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the
ratio of $n$ to $m$, where $m$ is the logarithm of the largest integer within
$A$. Based on the structural and statistical properties of subset sums, we
present an improved enumeration scheme for SSP, and implement it as a complete
and exact algorithm (EnumPlus). The algorithm always equivalently reduces an
instance to be low-density, and then solve it by enumeration. Through this
approach, we show the possibility to design a sole algorithm that can
efficiently solve arbitrary density instance in a uniform way. Furthermore, our
algorithm has considerable performance advantage over previous algorithms.
Firstly, it extends the density scope, in which SSP can be solved in expected
polynomial time. Specifically, It solves SSP in expected $O(n\log{n})$ time
when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best
density scope is $d \geq c\cdot n/(\log{n})^{2}$. In addition, the overall
expected time and space requirement in the average case are proven to be
$O(n^5\log n)$ and $O(n^5)$ respectively. Secondly, in the worst case, it
slightly improves the previously best time complexity of exact algorithms for
SSP. Specifically, the worst-case time complexity of our algorithm is proved to
be $O((n-6)2^{n/2}+n)$, while the previously best result is $O(n2^{n/2})$.
