On Dynamic Algorithms for Algebraic Problems
In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, iff(x1,x2,…,xn)=(y1,y2,…,ym) is an algebraic problem, we consider answering on-line requests of the form “change inputxito valuev” or “what is the value of outputyj?” We first present lower bounds f...
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| Vydané v: | Journal of algorithms Ročník 22; číslo 2; s. 347 - 371 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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San Diego, CA
Elsevier Inc
01.02.1997
Elsevier |
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| ISSN: | 0196-6774, 1090-2678 |
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| Abstract | In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, iff(x1,x2,…,xn)=(y1,y2,…,ym) is an algebraic problem, we consider answering on-line requests of the form “change inputxito valuev” or “what is the value of outputyj?” 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 anO(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 withO(loglogn) time per request with a sublinear number of processors. |
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| AbstractList | In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, iff(x1,x2,…,xn)=(y1,y2,…,ym) is an algebraic problem, we consider answering on-line requests of the form “change inputxito valuev” or “what is the value of outputyj?” 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 anO(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 withO(loglogn) time per request with a sublinear number of processors. |
| Author | Tate, Stephen R. Reif, John H. |
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ACM doi: 10.1145/358645.358660 – ident: 10.1006/jagm.1995.0807_AL950807RF1 doi: 10.1109/SFCS.1992.267816 – ident: 10.1006/jagm.1995.0807_AL950807RF11 doi: 10.1109/SFCS.1991.185429 – ident: 10.1006/jagm.1995.0807_AL950807RF6 – volume: 61 start-page: 163 year: 1955 ident: 10.1006/jagm.1995.0807_AL950807RF17 article-title: Evaluation of polynomials and evaluation of rational functions publication-title: Bull. Amer. Math. Soc. – year: 1984 ident: 10.1006/jagm.1995.0807_AL950807RF23 – volume: 20, No start-page: 740 year: 1978 ident: 10.1006/jagm.1995.0807_AL950807RF14 article-title: A survey of parallel algorithms in numerical linear algebra publication-title: SIAM Rev. doi: 10.1137/1020096 – year: 1975 ident: 10.1006/jagm.1995.0807_AL950807RF5 – volume: 26 start-page: 362 year: 1983 ident: 10.1006/jagm.1995.0807_AL950807RF20 article-title: A data structure for dynamic trees publication-title: J. Comput. 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SSSR – volume: 29 start-page: 250 year: 1982 ident: 10.1006/jagm.1995.0807_AL950807RF10 article-title: The complexity of maintaining an array and computing its partial sums publication-title: J. Assoc. Comput. Mech. doi: 10.1145/322290.322305 – ident: 10.1006/jagm.1995.0807_AL950807RF21 – year: 1994 ident: 10.1006/jagm.1995.0807_AL950807RF4 article-title: Polynomial and Matrix Computations – volume: 8 start-page: 189 year: 1979 ident: 10.1006/jagm.1995.0807_AL950807RF24 article-title: The complexity of computing the permanent publication-title: Theoret. Comput. Sci. doi: 10.1016/0304-3975(79)90044-6 – ident: 10.1006/jagm.1995.0807_AL950807RF12 – volume: 40 start-page: 2921 year: 1992 ident: 10.1006/jagm.1995.0807_AL950807RF13 article-title: Recursive fast algorithms and the role of the tensor product publication-title: IEEE Trans. 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| SubjectTerms | Algorithmics. Computability. Computer arithmetics Applied sciences Computer science; control theory; systems Exact sciences and technology Theoretical computing |
| Title | On Dynamic Algorithms for Algebraic Problems |
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