Abstract
In an array of n numbers each of the \(\binom{n}{2}+n\) contiguous subarrays define a sum. In this paper we focus on algorithms for selecting and reporting maximal sums from an array of numbers. First, we consider the problem of reporting k subarrays inducing the k largest sums among all subarrays of length at least l and at most u. For this problem we design an optimal O(n + k) time algorithm. Secondly, we consider the problem of selecting a subarray storing the k’th largest sum. For this problem we prove a time bound of Θ(n · max {1,log(k/n)}) by describing an algorithm with this running time and by proving a matching lower bound. Finally, we combine the ideas and obtain an O(n· max {1,log(k/n)}) time algorithm that selects a subarray storing the k’th largest sum among all subarrays of length at least l and at most u.
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References
Allison, L.: Longest biased interval and longest non-negative sum interval. Bioinformatics 19(10), 1294–1295 (2003)
Altschul, S.F., Gish, W., Miller, W., Myers, E.W., Lipman, D.J.: Basic local alignment search tool. Journal of Molecular Biology 215(3), 403–410 (1990)
Arge, L., Vitter, J.S.: Optimal external memory interval management. SIAM Journal on Computing 32(6), 1488–1508 (2003)
Bae, S.E., Takaoka, T.: Algorithms for the problem of k maximum sums and a vlsi algorithm for the k maximum subarrays problem. In: Proc. 7th International Symposium on Parallel Architectures, Algorithms, and Networks, pp. 247–253. IEEE Computer Society, Los Alamitos (2004)
Bentley, J.: Programming pearls: algorithm design techniques. Commun. ACM 27(9), 865–873 (1984)
Blum, M., Floyd, R.W., Pratt, V.R., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. J. Comput. Syst. Sci. 7(4), 448–461 (1973)
Brodal, G.S., Jørgensen, A.G.: A linear time algorithm for the k maximal sums problem. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 442–453. Springer, Heidelberg (2007)
Driscoll, J.R., Sarnak, N., Sleator, D.D., Tarjan, R.E.: Making data structures persistent. Journal of Computer and System Sciences 38(1), 86–124 (1989)
Fan, T.-H., Lee, S., Lu, H.-I., Tsou, T.-S., Wang, T.-C., Yao, A.: An optimal algorithm for maximum-sum segment and its application in bioinformatics. In: Proc. 8th International Conference on Implementation and Application of Automata. LNCS, pp. 46–66. Springer, Heidelberg (2003)
Frederickson, G.N.: An optimal algorithm for selection in a min-heap. Inf. Comput. 104(2), 197–214 (1993)
Frederickson, G.N., Johnson, D.B.: The complexity of selection and ranking in X+Y and matrices with sorted columns. J. Comput. Syst. Sci. 24(2), 197–208 (1982)
Fukuda, T., Morimoto, Y., Morishita, S., Tokuyama, T.: Data mining with optimized two-dimensional association rules. ACM Trans. Database Syst. 26(2), 179–213 (2001)
Fukuda, T., Morimoto, Y., Morishta, S., Tokuyama, T.: Interval finding and its application to data mining. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E80-A(4), 620–626 (1997)
Hannenhalli, S., Levy, S.: Promoter prediction in the human genome. Bioinformatics 17, S90–S96 (2001)
Huang, X.: An algorithm for identifying regions of a DNA sequence that satisfy a content requirement. Computer Applications in the Biosciences 10(3), 219–225 (1994)
Lin, T.-C., Lee, D.T.: Efficient algorithms for the sum selection problem and k maximum sums problem. In: The 17th International Symposium on Algorithms and Computation. LNCS, pp. 460–473. Springer, Heidelberg (2006)
Lin, T.-C., Lee, D.T.: Randomized algorithm for the sum selection problem. Theor. Comput. Sci. 377(1-3), 151–156 (2007)
Lin, Y.-L., Jiang, T., Chao, K.-M.: Efficient algorithms for locating the length-constrained heaviest segments, with applications to biomolecular sequence analysis. In: Proc. 27th International Symposium of Mathematical Foundations of Computer Science 2002. LNCS, pp. 459–470. Springer, Heidelberg (2002)
Nievergelt, J., Reingold, E.M.: Binary search trees of bounded balance. In: STOC 1972: Proceedings of the fourth annual ACM symposium on Theory of computing, pp. 137–142. ACM, New York (1972)
Okasaki, C.: Purely functional random-access lists. In: Functional Programming Languages and Computer Architecture, pp. 86–95 (1995)
Takaoka, T.: Efficient algorithms for the maximum subarray problem by distance matrix multiplication. Electr. Notes Theor. Comput. Sci. 61 (2002)
Tamaki, H., Tokuyama, T.: Algorithms for the maximum subarray problem based on matrix multiplication. In: Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms, pp. 446–452. Society for Industrial and Applied Mathematics, Philadelphia (1998)
Walder, R.Y., Garrett, M.R., McClain, A.M., Beck, G.E., Brennan, T.M., Kramer, N.A., Kanis, A.B., Mark, A.L., Rapp, J.P., Sheffield, V.C.: Short tandem repeat polymorphic markers for the rat genome from marker-selected libraries. Mammalian Genome 9(12), 1013–1021 (1998)
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Brodal, G.S., Jørgensen, A.G. (2008). Selecting Sums in Arrays. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_12
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DOI: https://doi.org/10.1007/978-3-540-92182-0_12
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