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Abstract:
In $1$-$k$ routing each of the $n^2$ processing units of an $n
\times n$ mesh connected computer initially holds $1$ packet which
must be routed such that any processor is the destination of at most
$k$ packets. This problem reflects practical desire for routing
better than the popular routing of permutations. $1$-$k$ routing
also has implications for hot-potato worm-hole routing, which is of
great importance for real world systems.
We present a near-optimal deterministic algorithm running in
$\sqrt{k} \cdot n / 2 + \go{n}$ steps. We give a second
algorithm with slightly worse routing time but working queue size
three. Applying this algorithm considerably reduces the routing
time of hot-potato worm-hole routing.
Non-trivial extensions are given to the general $l$-$k$ routing
problem and for routing on higher dimensional meshes. Finally we
show that $k$-$k$ routing can be performed in $\go{k \cdot n}$ steps
with working queue size four. Hereby the hot-potato worm-hole routing
problem can be solved in $\go{k^{3/2} \cdot n}$ steps.
