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**1 - 3**of**3**### Approximation of a Straight Line in a Bounded Lattice

"... this paper, we focus on a particular problem inspired from voice encoding in the GSM norm. The encoding process involves several steps, more precisely, ..."

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this paper, we focus on a particular problem inspired from voice encoding in the GSM norm. The encoding process involves several steps, more precisely,

### Incremental Construction Algorithms For Lattices Generated By Many Lattice Points

, 2000

"... . We have adopted the incremental construction algorithm paradigm which is a standard technique in computational geometry for lattice problems. The considered lattice problems are: lattice basis, successive minima, orthogonal decomposition. For these problems we give algorithms which are reasonable ..."

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. We have adopted the incremental construction algorithm paradigm which is a standard technique in computational geometry for lattice problems. The considered lattice problems are: lattice basis, successive minima, orthogonal decomposition. For these problems we give algorithms which are reasonable fast. Using a classical theorem of MINKOWSKI we prove a worst-case upper bound for the number of update steps for every insertion order. It seems to be obvious that randomization yields a much lower expected upper bound for the number of update steps. We leave this problem as a major open question of this paper. Key words. lattice basis problem, successive minima, orthogonal decomposition of lattices and positive definite quadratic forms, randomized incremental algorithms AMS subject classifications. 11Y16 Algorithms, 68Q25 Analysis of algorithms and problem complexity, 11H06 Lattices and convex bodies (number theoretic results), 11H55 Quadratic forms 1. Introduction. Many problems in com...