Abstract
We consider the problem: Given a set of n vectors in the d-dimensional Euclidean space, find a subsetmaximizing the length of the sum vector.We propose an algorithm that finds an optimal solution to this problem in time O(nd−1(d + logn)). In particular, if the input vectors lie in a plane then the problem is solvable in almost linear time.
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References
A. E. Baburin, E. Kh. Gimadi, N. I. Glebov, and A. V. Pyatkin, “The Problem of Finding a Subset of Vectors with theMaximum TotalWeight,” Diskretn. Anal. Issled.Oper. Ser. 2, 14 (1), 32–42 (2007) [J. Appl. Indust. Math. 2 (1), 32–38 (2008)].
A. E. Baburin and A. V. Pyatkin, “Polynomial Algorithms for Solving the Vector Sum Problem,” Diskretn. Anal. Issled. Oper. Ser. 1, 13 (2), 3–10 (2006) [J. Appl. Indust. Math. 1 (3), 268–272 (2007)].
E. Kh. Gimadi, Yu. V. Glazkov, and I. A. Rykov, “On Two Problems of Choosing Some Subset of Vectors with Integer Coordinates That HasMaximum Norm of the Sum ofElements in EuclideanSpace,” Diskretn. Anal. Issled. Oper. 15 (4), 30–43 (2008) [J. Appl. Indust. Math. 3 (3), 343–352 (2009)].
E. Kh. Gimadi, A. V. Kel’manov, M. A. Kel’manova, and S. A. Khamidullin, “A Posteriori Detection of a Quasiperiodic Fragment with a Given Number of Repetitions in a Numerical Sequence,” Sibir. Zh. Industr. Mat. 9 (1), 55–74 (2006) [Pattern Recognit. Image Anal. 18 (1), 30–42 (2008)].
E. Kh. Gimadi, A. V. Pyatkin, and I. A. Rykov, “On Polynomial Solvability of Some Problems of a Vector Subset Choice in a Euclidean Space of Fixed Dimension,” Diskretn. Anal. Issled.Oper. 15 (6), 11–19 (2008) [J. Appl. Indust. Math. 4 (1), 48–53 (2010)].
A. V. Pyatkin, “On Complexity of a Choice Problem of the Vector Subset with the Maximum Sum Length,” Diskretn. Anal. Issled. Oper. 16 (6), 68–73 (2009) [J. Appl. Indust.Math. 4 (4), 549–552 (2010)].
V. V. Shenmaier, “Solving Some Vector Subset Problems by Voronoi Diagrams,” Diskretn. Anal. Issled. Oper. 23 (4), 102–115 (2016) [J. Appl. Indust.Math. 10 (4), 560–566 (2016)].
R. C. Buck, “Partition of Space,” Amer.Math.Monthly 50 (9), 541–544 (1943).
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms (MIT Press and McGraw-Hill, Cambridge,MA, 2001).
H. Edelsbrunner, J. O’Rourke, and R. Seidel, “Constructing Arrangements of Lines and Hyperplanes with Applications,” SIAM J. Comput. 15 (2), 341–363 (1986).
F. K. Hwang, S. Onn, and U. G. Rothblum, “A Polynomial Time Algorithm for Shaped Partition Problems,” SIAM J. Optim. 10 (1), 70–81 (1999).
D. S. Johnson and F. P. Preparata, “The Densest Hemisphere Problem,” Theoret. Comput. Sci. 6 (1), 93–107 (1978).
S. Onn and L. J. Schulman, “The Vector Partition Problem for Convex Objective Functions,” Math. Oper. Res. 26 (3), 583–590 (2001).
S. Onn (private communication, Nov. 2016).
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Original Russian Text © V.V. Shenmaier, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 4, pp. 111–129.
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Shenmaier, V.V. An exact algorithm for finding a vector subset with the longest sum. J. Appl. Ind. Math. 11, 584–593 (2017). https://doi.org/10.1134/S1990478917040160
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DOI: https://doi.org/10.1134/S1990478917040160