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Efficient Tree Coding Algorithms

Wang X., Tian J.

Przegląd Elektrotechniczny

2012

R. 88, nr 7b

350--352

This paper studies the algorithms for coding and decoding second Neville’s codes of a labeled tree. The algorithms for coding and decoding second Neville’s codes of a labeled tree in the literatures require O(n log n) time usually. As stated in [1][2], no linear time algorithms for the second Neville’s codes. In this paper we consider the second Neville’s code problem in a different angle and a more direct manner. We start from a naïve algorithm, then improved it gradually and finally we obtain a very practical linear time algorithm. The techniques we used in this paper are interesting themselves.

W artykule rozważano problem kodu Neville drugiego rzędu stosowanego do etykietowania elementów struktury typu drzewo.

Randomized and quantum algorithms for solving initial-value problems in ordinary differential equations of order k

Goćwin M., Szczęsny M.

Opuscula Mathematica

2008

Vol. 28, no. 3

247-277

The complexity of initial-value problems is well studied for systems of equations of first order. In this paper, we study the ε-complexity for initial-value problems for scalar equations of higher order. We consider two models of computation, the randomized model and the quantum model. We construct almost optimal algorithms adjusted to scalar equations of higher order, without passing to systems of first order equations. The analysis of these algorithms allows us to establish upper complexity bounds. We also show (almost) matching lower complexity bounds. The ε-complexity in the randomized and quantum setting depends on the regularity of the right-hand side function, but is independent of the order of equation. Comparing the obtained bounds with results known in the deterministic case, we see that randomized algorithms give us a speed-up by 1/2, and quantum algorithms by 1 in the exponent. Hence, the speed-up does not depend on the order of equation, and is the same as for the systems of equations of first order. We also include results of some numerical experiments which confirm theoretical results.