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#### Routing and sorting on circular arrays

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MPI-I-93-138.pdf

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##### Citation

Sibeyn, J. F.(1993). *Routing and sorting on circular arrays*
(MPI-I-93-138). Saarbrücken: Max-Planck-Institut für Informatik.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0014-B428-E

##### Abstract

We analyze routing and sorting problems on circular processor arrays
with bidirectional connections. We assume that initially and finally
each PU holds $k \geq 1$ packets. On linear processor arrays the
routing and sorting problem can easily be solved for any $k$, but
for the circular array it is not obvious how to exploit the
wrap-around connection.
We show that on an array with $n$ PUs $k$-$k$ routing, $k \geq 4$,
can be performed optimally in $k \cdot n / 4 + \sqrt{n}$ steps by a
deterministical algorithm. For $k = 1$, the routing problem is
trivial. For $k = 2$ and $k = 3$, we prove lower-bounds and show
that these (almost) can be matched. A very simple algorithm has
good performance for dynamic routing problems.
For the $k$-$k$ sorting problem we use a powerful algorithm which
also can be used for sorting on higher-dimensional tori and meshes.
For the ring the routing time is $\max\{n, k \cdot n / 4\} + {\cal
O}((k \cdot n)^{2/3})$ steps. For large $k$ we take the computation
time into account and show that for $n = o(\log k)$ optimal speed-up
can be achieved. For $k < 4$, we give specific results, which
come close to the routing times.