  The subset sum problem, also referred as SSP, is a NP-Hard computational
problem. SSP has its applications in broad domains like cryptography, number
theory, operation research and complexity theory. The most famous algorithm for
solving SSP is Backtracking Algorithm which has exponential time complexity.
Therefore, our goal is to design and develop better alternate enumeration
techniques for faster generation of SSP solutions. Given the set of first n
natural numbers which is denoted by Xn and a target sum S, we propose various
alternate enumeration techniques which find all the subsets of Xn that add up
to sum S.
  In this paper, we present the mathematics behind this exponential problem. We
analyze the distribution of power set of Xn and present formulas which show
definite patterns and relations among these subsets. We introduce three major
distributions for power set of Xn: Sum Distribution, Length-Sum Distribution
and Element Distribution. These distributions are prepossessing procedures for
various alternate enumeration techniques for solving SSP. We propose novel
algorithms: Subset Generation using Sum Distribution, Subset Generation using
Length-Sum Distribution, Basic Bucket Algorithm, Maximum and Minimum Frequency
Driven Bucket Algorithms and Local Search using Maximal and Minimal Subsets for
enumerating SSP.
  We compare the performance of these approaches against the traditional
backtracking algorithm. The efficiency and effectiveness of these algorithms
are presented with the help of these experimental results. Furthermore, we
studied the over solution set of subsets generated by various algorithms to get
the complete solution for subset sum problem. Finally, we present a conjecture
about upper bound on the number of subsets that has to be enumerated to get all
solutions for Subset Sum Problem.
