Online Square Packing

TL;DR

This paper studies the online problem of packing squares into a strip with gravity constraints, introducing new algorithms with improved competitive ratios over previous methods.

Contribution

It presents a geometric analysis and new algorithms achieving better competitive ratios for online square packing with gravity constraints.

Findings

Achieved a 3.5-competitive ratio with the bottom-left heuristic.

Developed a 2.615-competitive algorithm for the problem.

Extended previous work by incorporating gravity into the analysis.

Abstract

We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4-competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares "hang in the air") based on ideas of shelf-packing: Squares are assigned to different horizontal levels, allowing an analysis that is reminiscent of some bin-packing arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottom-left heuristic and present a 34/13=2.615...-competitive algorithm.