Open Access
Fast evaluation and interpolation
Hsiang-Tsung Kung
- 01 Jan 1973
41
TL;DR: It is proved that the evaluation of an nth degree polynomial at n+1 arbitrary points can be done in 0(n log^ n) arithmetic operations, and consequently, its dual problem, interpolation of annth degreePolynomial from 2 n-1 arbitrary Points can be performed in 0("n log n") arithmetic operations.
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Abstract: A method for dividing a polynomial of degree (2n-l) by a precomputed nth degree polynomial in 0(n log n) arithmetic operations is given. This is used to prove that the evaluation of an nth degree polynomial at n+1 arbitrary points can be done in 0(n log^ n) arithmetic operations, and consequently, its dual problem, interpolation of an nth degree polynomial from 2 n+1 arbitrary points can be performed in 0(n log n) arithmetic operations. The best previously known algorithms required 0(n log^ n) arithmetic operations .
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Citations
“Short-Dot”: Computing Large Linear Transforms Distributedly Using Coded Short Dot Products
TL;DR: The key novelty in this work is that in the particular regime where the number of available processing nodes is greater than the total number of dot products, Short-Dot has lower expected computation time under straggling under an exponential model compared to existing strategies.
On the Optimal Recovery Threshold of Coded Matrix Multiplication
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TL;DR: Novel coded computation strategies for distributed matrix–matrix products that outperform the recent “Polynomial code” constructions in recovery threshold, i.e., the required number of successful workers are provided.
302
Gradient Coding From Cyclic MDS Codes and Expander Graphs
TL;DR: In this article, an approximate variant of the gradient coding problem is introduced, in which they settle for approximate gradient computation instead of the exact one, which enables graceful degradation, i.e., the approximation error of the approximate gradient is a decreasing function of the number of straggglers.
•Posted Content
"Short-Dot": Computing Large Linear Transforms Distributedly Using Coded Short Dot Products
TL;DR: In this paper, the authors propose a technique called Short-Dot to reduce the number of redundant computations in a coding theory inspired fashion for computing linear transforms of long vectors.
202
Coded convolution for parallel and distributed computing within a deadline
Sanghamitra Dutta,Viveck R. Cadambe,Pulkit Grover +2 more
- 01 Jun 2017
TL;DR: In this paper, the authors consider the problem of computing the convolution of two long vectors using parallel processors in the presence of stragglers and demonstrate that coding can dramatically improve the probability of finishing the computation within a target deadline.
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