Ruler Wrapping

TL;DR

This paper proves that the ruler wrapping problem, a variation of a known NP-complete problem, can be solved in linear time, and also presents a linear-time algorithm for partitioning sequences into non-decreasing sum substrings.

Contribution

The paper introduces a linear-time solution for O'Rourke's ruler wrapping variation and a linear-time partitioning algorithm for sequences based on sum properties.

Findings

Ruler wrapping problem is solvable in linear time.

Partitioning sequences into maximum non-decreasing sum substrings is achievable in linear time.

Provides efficient algorithms for problems previously considered complex.

Abstract

In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter's ruler with segments of given positive lengths can be folded into a line of at most a given length, such that the folded hinges alternate between 180 degrees clockwise and 180 degrees counter-clockwise. At the open-problem session of 33rd Canadian Conference on Computational Geometry (CCCG '21), O'Rourke proposed a natural variation of this problem called {\em ruler wrapping}, in which all folded hinges must be folded the same way. In this paper we show O'Rourke's variation has an linear-time solution. We also show how, given a sequence of positive numbers, in linear time we can partition it into the maximum number of substrings whose totals are non-decreasing.