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Let A be a sequence of n real numbers, L1 and L2 be two integers such that L1 ≤ L2 , and R1 and R2 be two real numbers such that R1 ≤ R2. An interval of A is feasible if its length is between L1 and L2 and its average is between R1 and R2. In this paper, we study the following problems: finding all feasible intervals of A, counting all feasible intervals of A, finding a maximum cardinality set of non-overlapping feasible intervals of A, locating a longest feasible interval of A, and locating a shortest feasible interval of A. The problems are motivated from the problem of locating CpG islands in biomolecular sequences. In this paper, we firstly show that all the problems have an Ω(n log n)-time lower bound in the comparison model. Then, we use geometric approaches to design optimal algorithms for the problems. All the presented algorithms run in an on-line manner and use O(n) space.
algorithms, data structures, analysis of algorithms, geometrical problems and computations
Biing-Feng Wang, Chih-Chiang Yu, Yong-Hsiang Hsieh, "Optimal Algorithms for the Interval Location Problem with Range Constraints on Length and Average", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 5, no. , pp. 281-290, April-June 2008, doi:10.1109/TCBB.2007.70217
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