Selective algorithm processing of subset sum distributions

TL;DR

This paper introduces a system that adaptively selects the most efficient subset sum algorithm for different input subsets, significantly improving computational efficiency through dynamic partitioning and modular arithmetic.

Contribution

It presents a novel adaptive approach that maps subset sums into bins to optimize algorithm selection for the subset sum problem.

Findings

Achieves efficiency of O(max(T, n^2)) with space complexity O(max(T, n))

Effectively handles cases like all even values using modular arithmetic

Demonstrates experimentally validated improvements in subset sum computations

Abstract

The efficiency of exact subset sum problem algorithms which compute individual subset sums is defined as $e=min(T/z, 1)$, where $z$ is the number of subset sums computed. $e$ is related to these algorithms' computational complexity. This system maps the sums into $kn$ bins to select its most efficient algorithm for each bin for each input value. These algorithms include additive, subtractive and repeated value dynamic programming. Cases which would otherwise be processed inefficiently (eg: all even values) are handled by modular arithmetic and by dynamically partioning the input values. The system's experimentally validated efficiency corresponds to O(max($T$, $n^2$)) with space complexity O(max($T$, $n$)), for $k=2$.