The cell probe complexity of dynamic data structures
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Lower bounds for data structure problems on RAMs (extended abstract)
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Journal of the ACM (JACM)
Surpassing the information theoretic bound with fusion trees
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
On data structures and asymmetric communication complexity
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
On dynamic algorithms for algebraic problems
Journal of Algorithms
The Complexity of Maintaining an Array and Computing Its Partial Sums
Journal of the ACM (JACM)
Sorting and Searching on the Word RAM
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
Tight bounds for depth-two superconcentrators
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Optimal bi-weighted binary trees and the complexity of maintaining partial sums
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Subquadratic algorithm for dynamic shortest distances
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
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We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, ..., xn) = (y1, ..., ym) is an algebraic problem, we consider serving on-line requests of the form "change input xi to value v" or "what is the value of output yi?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Ω(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an Ω(√n) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and Tate's O(√n log n) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Ω(√n) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an Ω((log n)2/log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields. Due to the space constraints imposed by these proceedings, in this version of the paper we only present the third technique, proving the lower bound for dynamic discrete Fourier transform and refer to the full version of the paper which is currently available as a BRICS technical report, for the rest of the proofs.