John Augustine
ABSTRACT
This paper examines two variants of strip packing: when the rectangles to be placed have precedence constraints and when the rectangles have release times. Strip packing can be used to model scheduling problems in which tasks require a contiguous subset of identical resources that are arranged in a linear topology. The particular variants studied here are motivated by scheduling tasks for dynamically reconfigurable Field-Programmable Gate Arrays (FPGAs) comprised of an array of computing columns. Each column is a computing resource and the array of columns forms the linear topology of resources. We assume that the given FPGA has K columns, where K is a fixed positive integer, and each task occupies a contiguous subset of these columns. For the case in which tasks have precedence constraints, we give an O(log n) approximation, where n is the number of tasks. We then consider the special case in which all the rectangles have uniform height and reduce it to the resource constrained scheduling studied by Garey, Graham, Johnson and Yao, thereby extending their asymptotic results to our special case problem. We also give an absolute 3-approximation for this special case problem. For strip packing with release times, we provide an asymptotic polynomial time (1 + ε)- approximation scheme. We make the standard assumption that the rectangles have height at most 1. In addition, we also require widths to be in [1<over>K, 1], i.e., the rectangles are at least as wide as a column in the FPGA. Our running time is polynomial in n and 1/ε, but exponential in K.
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Index Terms
- Strip packing with precedence constraints and strip packing with release times
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