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

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    In 11-kk routing each of the n2n^2 processing units of an n×nn \times n mesh connected computer initially holds 11 packet which must be routed such that any processor is the destination of at most kk packets. This problem reflects practical desire for routing better than the popular routing of permutations. 11-kk 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 ll-kk routing problem and for routing on higher dimensional meshes. Finally we show that kk-kk 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