Abstract
The master theorem provides a solution to a well-known divide-and-conquer recurrence, called here the master recurrence. This paper proves two cook-book style generalizations of this master theorem. The first extends the treated class of driving functions to the natural class of exponential-logarithmic (EL) functions. The second extends the result to the multiterm master recurrence. The power and simplicity of our approach comes from re-interpreting integer recurrences as real recurrences, with emphasis on elementary techniques and real induction.
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This work is supported by an National Science Foundation Grants #CCF-0728977 and #CCF-0917093, and also with KIAS support.
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References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms. Addison-Wesley, Reading (1983)
Akra, M., Bazzi, L.: On the solution of linear recurrences. Computational Optimizations and Applications 10(2), 195–210 (1998)
Bentley, J.L., Haken, D., Saxe, J.B.: A general method for solving divide-and-conquer recurrences. ACM SIGACT News 12(3), 36–44 (1980)
Brassard, G., Bratley, P.: Fundamentals of Algorithms. Prentice-Hall, Englewood Cliffs (1996)
Corman, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press & McGraw-Hill Book Co. (2001)
Edelsbrunner, H., Welzl, E.: Halfplanar range search in linear space and O(n 0.695) query time. Info. Processing Letters 23, 289–293 (1986)
Escardó, M.H., Streicher, T.: Induction and recursion on the partial real line with applications to Real PCF. Theor. Computer Sci. 210(1), 121–157 (1999)
Gonnet, G.H.: Handbook of Algorithms and Data Structures. Addison-Wesley Pub. Co., London (1984)
Goursat, É.: A Course in Mathematical Analysis, vol. 1. Ginn & Co., Boston (1904); Trans. by Earle Raymod Hedrick. Available from Google books
Greene, D.H., Knuth, D.E.: Mathematics for the Analysis of Algorithms, 2nd edn. Birkhäuser, Basel (1982)
Kao, M.: Multiple-size divide-and-conquer recurrences. SIGACT News 28(2), 67–69 (1997); also Proc. 1996 Intl. Conf. on Algorithms, Natl. Sun Yat-Sen U., Taiwan, pp. 159–161
Karp, R.M.: Probabilistic recurrence relations. J. ACM 41(6), 1136–1150 (1994)
Knuth, D.E.: The Art of Computer Programming: Sorting and Searching, vol. 3. Addison-Wesley, Boston (1972)
Leighton, T.: Notes on better master theorems for divide-and-conquer recurrences (1996) (class notes)
Mahony, B.P., Hayes, I.J.: Using continuous real functions to model timed histories. In: 6th Australian Software Eng. Conf (ASWEC), pp. 257–270 (1991)
Paul Walton Purdom, J., Brown, C.A.: The Analysis of Algorithms. Holt, Rinehart and Winston, New York (1985)
Roura, S.: An improved master theorem for divide-and-conquer recurrences. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 449–459. Springer, Heidelberg (1997)
Roura, S.: Improved master theorems for divide-and-conquer recurrences. J. ACM 48(2), 170–205 (2001)
Verma, R.M.: A general method and a master theorem for divide-and-conquer recurrences with applications. J. Algorithms 16, 67–79 (1994)
Wang, X., Fu, Q.: A frame for general divide-and-conquer recurrences. Info. Processing Letters 59, 45–51 (1996)
Yap, C.K.: Theory of real computation according to EGC. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds.) Real Number Algorithms. LNCS, vol. 5045, pp. 193–237. Springer, Heidelberg (2008)
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Yap, C. (2011). A Real Elementary Approach to the Master Recurrence and Generalizations. In: Ogihara, M., Tarui, J. (eds) Theory and Applications of Models of Computation. TAMC 2011. Lecture Notes in Computer Science, vol 6648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20877-5_3
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DOI: https://doi.org/10.1007/978-3-642-20877-5_3
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