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On Dynamic Algorithms for Algebraic Problems

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Stephen R. Tate, Professor and Department Head (Creator)
The University of North Carolina at Greensboro (UNCG )
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Abstract: In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, if f(x1, x2, …, xn) = (y1, y2, …, ym) is an algebraic problem, we consider answering on-line requests of the form "change input xi to value v" or "what is the value of output yj?" We first present lower bounds for some simply stated algebraic problems such as multipoint polynomial evaluation, polynomial reciprocal, and extended polynomial GCD, proving an Ω(n). lower bound for the incremental evaluation of these functions. In addition, we prove two time-space trade-off theorems that apply to incremental algorithms for almost all algebraic functions. We then derive several general-purpose algorithm design techniques and apply them to several fundamental algebraic problems. For example, we give an O( n  ) time per request algorithm for incremental DFT. We also present a design technique for serving incremental requests using a parallel machine, giving a choice of either optimal work with respect to the sequential incremental algorithm or superfast algorithms with O(log log n) time per request with a sublinear number of processors.

Additional Information

Journal of Algorithms, Vol. 22, No. 2, 1997, pp. 347–371.
Language: English
Date: 1997
Algorithms, Incremental, Mathematical models, Algebra