10 research outputs found
Gerechte Designs with Rectangular Regions
A \emph{gerechte framework} is a partition of an array into
regions of cells each. A \emph{realization} of a gerechte framework is a
latin square of order with the property that when its cells are partitioned
by the framework, each region contains exactly one copy of each symbol. A
\emph{gerechte design} is a gerechte framework together with a realization.
We investigate gerechte frameworks where each region is a rectangle. It seems
plausible that all such frameworks have realizations, and we present some
progress towards answering this question. In particular, we show that for all
positive integers and , any gerechte framework where each region is
either an rectangle or a rectangle is realizable.Comment: 14 pages, 12 figure
An analogue of Ryser's Theorem for partial Sudoku squares
In 1956 Ryser gave a necessary and sufficient condition for a partial latin
rectangle to be completable to a latin square. In 1990 Hilton and Johnson
showed that Ryser's condition could be reformulated in terms of Hall's
Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as
saying that any partial latin rectangle can be completed if and only if
satisfies Hall's Condition for partial latin squares.
We define Hall's Condition for partial Sudoku squares and show that Hall's
Condition for partial Sudoku squares gives a criterion for the completion of
partial Sudoku rectangles that is both necessary and sufficient. In the
particular case where , , , the result is especially simple, as
we show that any partial -Sudoku rectangle can be completed
(no further condition being necessary).Comment: 19 pages, 10 figure
Testing of random matrices
Let be a positive integer and be an
\linebreak \noindent sized matrix of independent random variables
having joint uniform distribution \hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k
\leq n} = \frac{1}{n} \quad (1 \leq i, j \leq n) \koz. A realization
of is called \textit{good}, if its each row and
each column contains a permutation of the numbers . We present and
analyse four typical algorithms which decide whether a given realization is
good
Testing of sequences by simulation
Let be a random integer vector, having uniform distribution
A realization of is called
\textit{good}, if its elements are different. We present algorithms
\textsc{Linear}, \textsc{Backward}, \textsc{Forward}, \textsc{Tree},
\textsc{Garbage}, \textsc{Bucket} which decide whether a given realization is
good. We analyse the number of comparisons and running time of these algorithms
using simulation gathering data on all possible inputs for small values of
and generating random inputs for large values of
Quick Testing of Random Sequences
Abstract Let ξ be a random integer sequence, having uniform distribution A realization (i1, i2, . . . , in) of ξ is called good, if its elements are different. We present seven algorithms which decide whether a given realization is good. The investigated problem is connected with design of experiment