Solving the Subset Sum Problem with Heap-Ordered Subset Trees

TL;DR

This paper introduces a novel tree-based data structure called subset tree, inspired by binomial heaps, to solve the subset sum problem more efficiently than brute-force methods.

Contribution

It presents the subset tree data structure that extends min-heap solutions to handle all integers, improving the computational approach to the subset sum problem.

Findings

Achieves a solution time of O(N^3k log k) for the subset sum problem.

Extends min-heap methods to all integers, including negatives.

Provides a new data structure for subset sum problem solving.

Abstract

In the field of algorithmic analysis, one of the more well-known exercises is the subset sum problem. That is, given a set of integers, determine whether one or more integers in the set can sum to a target value. Aside from the brute-force approach of verifying all combinations of integers, several solutions have been found, ranging from clever uses of various data structures to computationally-efficient approximation solutions. In this paper, a unique approach is discussed which builds upon the existing min-heap solution for positive integers, introducing a tree-based data structure influenced by the binomial heap. Termed the subset tree, this data structure solves the subset sum problem for all integers in time $O(N^3k\log k)$, where $N$ is the length of the set and $k$ is the index of the list of subsets that is being searched.