In this paper we consider several variations of the following basic tiling problem: given a sequence of real numbers with two size bound parameters, we want to find a set of tiles such that they satisfy the size bounds and the total weight of the tiles is maximized. This solution to this problem is important to a number of computational biology applications, such as selecting genomic DNA fragments for amplicon microarrays, or performing homology searches with long sequence queries. Our goal is to design efficient algorithms with linear or near-linear time and space in the normal range of parameter values for these problems. For this purpose, we discuss the solution of a basic online interval maximum problem via a sliding window approach and show how to use this solution in a non-trivial manner for many of our tiling problems. We also discuss NP-hardness and approximation algorithms for generalization of our basic tiling problem to higher dimensions.
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