Packing Rectangles in a Strip

Rectangles with dimensions independently chosen from a uniform distribution on [0,1] are packed on-line into a unit-width strip under a constraint like that of the Tetris[TM] game: rectangles arrive from the top and must be moved inside the strip to reach their place; once placed, they cannot be moved again. Cargo loading applications impose similar constraints. This paper assumes that rectangles must be moved without rotation. For n rectangles, the resulting packing height is shown to have an asymptotic expected value of at least 0.31382733n under any on-line packing algorithm. An on-line algorithm is presented that achieves an asymptotic expected height of 0.36976421n. This algorithm improves the bound achieved in Next Fit Level (NFL) packing, by compressing the items packed on two successive levels of an NFL packing via on-line movement admissible under the Tetris constraint.