Abstract
The problem of designing an electrical circuit with the smallest number of make and break switches which will allow a current to flow when m (or more)out of n voters close their switches (vote yes) is discussed. An upper bound j(n,m) for this number is obtained as the maximal solution of a generalized subadditive inequality which is then shown to satisfy the recurrence relation of the title. It is shown how to find explicit solutions of this equation and asymptotic solutions are also found. A second, direct construction gives another upper bound and the two upper bounds are compared, j(n,m) being the better for m small in comparison with n. Problems involving other voting procedures are also considered.
Keywords
- Recurrence Relation
- Electrical Circuit
- Asymptotic Solution
- Maximal Solution
- Vote Procedure
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References
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Hammersley, J.M., and Grimmett, G.R. Maximal Solutions of the Generalized Subadditive Inequality. In Stochastic Geometry, ed. Kendall, D.G. and Harding, E.F., J. Wiley, London (1974).
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Mendelson, E. Introduction to Mathematical Logic. D. van Nostrand, Princeton (1964).
Morris, R. Some Theorems on Sorting. SIAM J. Appl. Math. 17(1969), 1–6.
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© 1977 Springer-Verlag
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Pelling, M.J., Rogers, D.G. (1977). A problem in the design of electrical circuits, a generalized subadditive inequality and the recurrence relation j(n,m)=j([n/2],m)+j([n+1/2],m)+j(n,m–1). In: Little, C.H.C. (eds) Combinatorial Mathematics V. Lecture Notes in Mathematics, vol 622. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069190
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DOI: https://doi.org/10.1007/BFb0069190
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08524-9
Online ISBN: 978-3-540-37020-8
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