# Deterministic 1 -*k* routing on meshes with applications to worm-hole routing

## Abstract

In 1-*k* routing each of the *n*^{2} processing units of an *n* x *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 has great practical importance in itself and by its implications for hot-potato worm-hole routing.

We present a near-optimal deterministic algorithm running in \(\sqrt k \cdot {n \mathord{\left/{\vphantom {n 2}} \right.\kern-\nulldelimiterspace} 2} + \mathcal{O}\left( n \right)\) steps, and an algorithm with slightly worse routing time but working queue size three. Non-trivial extensions are given to *l-k* routing, and for routing on higher dimensional meshes. We show that under a natural condition 1-*k* routing can be performed in \(\mathcal{O}\left( n \right)\) steps. Finally we show that *k-k* routing can be performed in \(\mathcal{O}\left( {k \cdot n} \right)\) steps with working queue size four. Hereby hot-potato worm-hole routing can be performed in \(\mathcal{O}\left( {k^{{3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}} \cdot n} \right)\) steps.

## Keywords

theory of parallel and distributed computation meshes packet routing hot-potato worm-hole routing## Preview

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