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.
strip packing problem!Tetris constraint Tetris-like online packing has been considered before. Most notably, Azar and Epstein @cite_12 consider online packing of rectangles into a strip; just like in Tetris, they consider the situation with or without rotation of objects. For the case without rotation, they show that no constant competitive ratio is possible, unless there is a fixed-size lower bound of @math on the side length of the objects, in which case there is an upper bound of @math on the competitive ratio.
@cite_12
2752916745 2098666887 2012437618 1586863324
In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receive an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares, isosceles right triangles of leg length one, and unit cubes, respectively. For Online Circle Packing in Squares, we improve the previous best-known competitive ratio for the bounded space version, when at most a constant number of bins can be open at any given time, from 2.439 to 2.3536. For Online Circle Packing in Isosceles Right Triangles and Online Sphere Packing in Cubes we show bounded space algorithms of asymptotic competitive ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and 2.7707 on the competitive ratio of any online bounded space algorithm for these two problems. We also considered the online unbounded space variant of these three problems which admits a small reorganization of the items inside the bin after their packing, and we present algorithms of competitive ratios 2.3105, 2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes, respectively. We study strip packing, which is one of the most classical two-dimensional packing problems: given a collection of rectangles, the problem is to find a feasible orthogonal packing without rotations into a strip of width 1 and minimum height. In this paper we present an approximation algorithm for the strip packing problem with absolute approximation ratio of 5 3+@e for any @e>0. This result significantly narrows the gap between the best known upper bound and the lower bound of 3 2; previously, the best upper bound was 1.9396 due to Harren and van Stee. 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 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 (0.31382733 ... )n under any on-line packing algorithm. An on-line algorithm is presented that achieves an asymptotic expected height of (0.36976421 ... )n. 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-like constraint. We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ( @math ) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided @math .