Sorting Balls and Water: Equivalence and Computational Complexity

TL;DR

This paper analyzes the computational complexity of two popular sorting puzzles involving balls and water, proving they are equivalent, NP-complete, and providing polynomial algorithms for special cases.

Contribution

It establishes the equivalence of ball and water puzzles, proves NP-completeness, and offers bounds and polynomial solutions for specific scenarios.

Findings

Puzzles are equivalent in solvability

NP-complete for general cases

Polynomial algorithms for special cases

Abstract

Various forms of sorting problems have been studied over the years. Recently, two kinds of sorting puzzle apps are popularized. In these puzzles, we are given a set of bins filled with colored units, balls or water, and some empty bins. These puzzles allow us to move colored units from a bin to another when the colors involved match in some way or the target bin is empty. The goal of these puzzles is to sort all the color units in order. We investigate computational complexities of these puzzles. We first show that these two puzzles are essentially the same from the viewpoint of solvability. That is, an instance is sortable by ball-moves if and only if it is sortable by water-moves. We also show that every yes-instance has a solution of polynomial length, which implies that these puzzles belong to in NP. We then show that these puzzles are NP-complete. For some special cases, we give polynomial-time algorithms. We finally consider the number of empty bins sufficient for making all instances solvable and give non-trivial upper and lower bounds in terms of the number of filled bins and the capacity of bins.