Victor Y. Pan
ABSTRACT
We exploit various matrix structures to decrease the running time and memory space of the known practical deterministic schemes for erasure-resilient encoding/decoding. Polynomial interpolation and multipoint evaluation enable both encoding and decoding in nearly linear time but the overhead constants are large (particularly, for interpolation), and more straightforward quadratic time algorithms prevail in practice. We propose faster algorithms. At the encoding stage, we decrease the running time per information packet from C log2 r, for a large constant C, or from r (for practical encoding) to log r. For decoding, our improvement is by the factors C and N/log N, respectively, for the input of size N. Our computations do not involve polynomial interpolation. Multipoint polynomial evaluation is either also avoided or is confined to decoding.
- 1.A. Albanese, J. B16mer, J. Edmonds, M. Luby, M. Sudan, Priority Encoding Transmission, Prec. 35th Ann. Syrup. on Foundations of Computer Science (FOCS), 604-613, IEEE Computer Society Press, 1994.]]Google Scholar
- 2.N. Alon, J. Edmonds, M. Luby, Linear Time Erasure Codes with Nearly Optimal Recovery, Prec. 36th Ann. Syrup. on Foundations of Computer Science (FOCS), 512-519, IEEE Computer Society Press, 1995.]] Google ScholarDigital Library
- 3.D. Bini, V.Y. Pan, Polynomial and Matrix Computations, Volume 1: Fundamental Algorithms, BirkhSuser, Boston, 1994.]] Google ScholarDigital Library
- 4.D. Bini, V.Y. Pan, Polynomial and Matrix Computations, Volume 2: Fundamental and Practical Algorithms, BirkhSuser, Boston, 2000.]] Google ScholarDigital Library
- 5.J. B15mer, M. Kalfane, R. Karp, M. Karpinski, M. Luby, D. Zuckerman, An XOR-Based Erasure-Resilient Coding Scheme, Technical Report TR-95-48, International Computer Science Institute, Berkeley, Califormia, 1995.]]Google Scholar
- 6.T. Fink, G. Heinig, K. Rost, An Inversion Formula and Fast Algorithms for Cauchy-Vandermonde Matrices, Linear Algebra Appl., 183, 179-191, 1993.]]Google ScholarCross Ref
- 7.N. Gastinel, Inversion d'une Matrice Generalisant la Matrice de Hilbert, Chiffres, 3, 149-152, 1960.]]Google Scholar
- 8.A. Gerasoulis, A Fast Algorithm for the Multiplication of Generalized Hilbert Matrices with Vectors, Math. Cemp., 50~ 181, 179-188, 1987.]]Google Scholar
- 9.I. Gohberg, V. Olshevsky, Fast Algorithms with Preprocessing for Matrix-Vector Multiplication Problem, J. of Complexity, 10~4, 411-427, 1994.]] Google ScholarDigital Library
- 10.T. Kailath, A. Sayed, Fast Reliable Algorithms for Matrices with Structure, SIAM Publications, Philadelphia, 1999.]] Google ScholarDigital Library
- 11.M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, V. Stemann, Practical Loss-Resilient Codes, Prec. 29th Ann. Syrup. on Theory of Computing (STOC'97), ACM Press, New York, 1997.]] Google ScholarDigital Library
- 12.L. Mirsky, An Introduction to Linear Algebra, Dover, New York, 1982.]]Google Scholar
- 13.B. Mourrain, V. Y. Pan, Multivariate Polynomials, Duality and Structured Matrices, J. of Complexity, 16~ 1, 110-180, 2000.]] Google ScholarDigital Library
- 14.F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New York, 1977.]]Google Scholar
- 15.V. Olshevsky, V. Y. Pan, A Unified Superfast Algorithm for Boundary Rational Tangential Interpolation Problem and for Inversion and Factorization of Dense Structured Matrices, Prec. 39th Annual IEEE Symposium on Foundations of Computer Science, 192-201, IEEE Computer Society Press, 1998.]] Google ScholarDigital Library
- 16.V. Olshevsky, V. Y. Pan, Polynomial and Rational Interpolation and Multipoint Evaluation (with Structured Matrices), Prec. 26th Intern. Colloquium on Automata, Languages and Programming (ICALP'99), 1644, 585-594, Springer's LNCS, Berlin, 1999.]] Google ScholarDigital Library
- 17.V. Olshevsky, M. A. Shokrollahi, A Displacement Approach to Efficient Decoding of Algebraic-Geometric Codes, Prec. 31st Ann. Syrup. on Theory of Computing, 235-244, ACM Press, New York, May 1999.]] Google ScholarDigital Library
- 18.V. Y. Pan, On Computations with Dense Structured Matrices, Prec. A CM-SIGSAM Intern. Syrup. on Symbolic and Alg. Cemp., 34-42, ACM Press, New York, 1989, and Math. of Computation., 55~ 191, 179-190, 1990.]] Google ScholarDigital Library
- 19.V. Y. Pan, Complexity of Computations with Matrices and Polynomials, SIAM Review, 34~ 2, 225-262, 1992.]] Google ScholarDigital Library
- 20.V. Y. Pan, Nearly Optimal Computations with Structured Matrices, Prec. 11th Ann. A CM-SIAM Syrup. on Discrete Algorithms, SODA'2000), 953-962, ACM Press, New York, and SIAM Publications, Philadelphia, 2000.]] Google ScholarDigital Library
- 21.V. Y. Pan, M. AbuTabanjeh, Z. Chen, E. Landowne, A. Sadikou, New Transformations of Cauchy Matrices and Trummer's Problem, Computer and Math. (with Applics.), 35, 12, 1-5, 1998.]]Google Scholar
- 22.V. Y. Pan, Z. Chen, The Complexity of the Matrix Eigenproblem, Prec. 31st Annual A CM Syrup. on Theory of Computing, 507-516, ACM Press, New York, 1999.]] Google ScholarDigital Library
- 23.P. Penfield Jr., R. Spencer, S. Duinker, Tellegen's Theorem and Electrical Networks, MIT Press, Cambridge, Massachusetts, 1970.]]Google Scholar
- 24.M. O. Rabin, Efficient Dispersal of Information for Security, Load Balancing, and Fault Tolerance, J. ACM, 36~ 2, 335-348, 1989.]] Google ScholarDigital Library
- 25.J. F. Traub, Associated Polynomials and Uniform Methods for the Solution of Linear Problems, SIAM Review, 8, 277-301, 1966.]]Google ScholarCross Ref
- 26.S. B. Wicker, V. Bhargava, Reed-Solomon Codes and Their Applications, IEEE Press, New York, 1994.]] Google ScholarDigital Library
Index Terms
- Matrix structure, polynomial arithmetic, and erasure-resilient encoding/decoding
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