The cell probe complexity of dynamic data structures
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
On rank properties of Toeplitz matrices over finite fields
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Fast on-line integer multiplication
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
Tight bounds for the partial-sums problem
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Perceptrons: An Introduction to Computational Geometry
Perceptrons: An Introduction to Computational Geometry
Logarithmic Lower Bounds in the Cell-Probe Model
SIAM Journal on Computing
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
A Black Box for Online Approximate Pattern Matching
CPM '08 Proceedings of the 19th annual symposium on Combinatorial Pattern Matching
Lower bound techniques for data structures
Lower bound techniques for data structures
Pattern matching in pseudo real-time
Journal of Discrete Algorithms
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We show time lower bounds for both online integer multiplication and convolution in the cell-probe model with word size w. For the multiplication problem, one pair of digits, each from one of two n digit numbers that are to be multiplied, is given as input at step i. The online algorithm outputs a single new digit from the product of the numbers before step i + 1. We give a lower bound of Ω(δ/w log n) time on average per output digit for this problem where 2δ is the maximum value of a digit. In the convolution problem, we are given a fixed vector V of length n and we consider a stream in which numbers arrive one at a time. We output the inner product of V and the vector that consists of the last n numbers of the stream. We show an Ω(δ/w log n) lower bound for the time required per new number in the stream. All the bounds presented hold under randomisation and amortisation. Multiplication and convolution are central problems in the study of algorithms which also have the widest range of practical applications.