Cut a Numeric String into Required Pieces

TL;DR

The paper investigates algorithms for partitioning a numeric string into segments with minimum sum constraints, presenting efficient greedy, dynamic programming, and fixed-parameter tractable algorithms.

Contribution

It introduces multiple algorithms with different complexities for the string partitioning problem, including a novel FPT algorithm for specific cases.

Findings

Greedy algorithm runs in O(n) time.

Dynamic programming solution operates in O(kn log n) time.

FPT algorithm achieves O(n + k log n + 2^k k) time complexity.

Abstract

We study the problem of cutting a length-$n$ string of positive real numbers into $k$ pieces so that every piece has sum at least $b$. The problem can also be phrased as transforming such a string into a new one by merging adjacent numbers. We discuss connections with other problems and present several algorithms in connection with the problem: an $O(n)$-time greedy algorithm, an $O(kn\log n)$-time dynamic programming algorithm, and an $O(n+ k \log n + 2^kk)$-time FPT algorithm for $n-k$ pieces.