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#### Unbiased Matrix Rounding

##### MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44338

Doerr,  Benjamin
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44447

Friedrich,  Tobias
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44793

Klein,  Christian
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45146

Osbild,  Ralf
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Zitation

Doerr, B., Friedrich, T., Klein, C., & Osbild, R. (2006). Unbiased Matrix Rounding. In Algorithm theory - SWAT 2006: 10th Scandinavian Workshop on Algorithm Theory (pp. 102-112). Berlin, Germany: Springer.

We show several ways to round a real matrix to an integer one in such a way that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular, statistics. We improve earlier solutions of different authors in two ways. For rounding $m \times n$ matrices, we reduce the runtime from $O( (m n)^2 )$ to $O(m n \log(m n))$. Second, our roundings also have a rounding error of less than one in all initial intervals of rows and columns. Consequently, arbitrary intervals have an error of at most two. This is particularly useful in the statistics application of controlled rounding. The same result can be obtained via (dependent) randomized rounding. This has the additional advantage that the rounding is unbiased, that is, for all entries $y_{ij}$ of our rounding, we have $E(y_{ij}) = x_{ij}$, where $x_{ij}$ is the corresponding entry of the input matrix.