Nick Dawes | Searcharxiv

arXiv2025

An O(nlogn) approximate knapsack algorithm

A modified dynamic programming algorithm rapidly and accurately solves large 0/1 knapsack problems. It has computational O(nlogn), space O(nlogn) and predictable maximum error. Experimentally it's accuracy increases faster than linearly with the solution size k. Problems with k=1e3 are solved with an average maximum fractional error of 1e-4 and problems with k=1e5 with an average maximum fractional error of 1e-7. The algorithm runs in constant time for all problems with a given n. On a common desktop computer the algorithm processes n=1e3 problems in 1e-3 seconds and n=1e6 problems in 2 seconds.

cs.DS↗

arXiv2024

Selective algorithm processing of subset sum distributions

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 $k=2$.

cs.DS↗