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<p><b>Abstract</b>—Mesh is one of the most commonly used interconnection networks and, therefore, embedding between different meshes becomes a basic embedding problem. Not only does an efficient embedding between meshes allow one mesh-connected computing system to efficiently simulate another, but it also provides a useful tool for solving other embedding problems. In this paper, we study how to embed an <it>s</it><sub>1</sub>×<it>t</it><sub>1</sub> mesh into an <it>s</it><sub>2</sub>×<it>t</it><sub>2</sub> mesh, where <it>s</it><sub><it>i</it></sub>≤<it>t</it><sub><it>i</it></sub>(<it>i</it> = 1, 2), <it>s</it><sub>1</sub><it>t</it><sub>1</sub> = <it>s</it><sub>2</sub><it>t</it><sub>2</sub>, such that the minimum dilation and congestion can be achieved. First, we present a lower bound on the dilations and congestions of such embeddings for different cases. Then, we propose an embedding with dilation <tmath>$\lfloor s_1/s_2 \rfloor + 2$</tmath> and congestion <tmath>$\lfloor s_1/s_2 \rfloor + 4$</tmath> for the case <it>s</it><sub>1</sub>≥<it>s</it><sub>2</sub>, both of which almost match the lower bound <tmath>$\lceil s_1/s_2 \rceil.$</tmath> Finally, for the case <it>s</it><sub>1</sub> < <it>s</it><sub>2</sub>, we present an embedding which has a dilation less than or equal to <tmath>$2\sqrt {s_1}.$</tmath></p>
Dilation, embedding, mesh, parallel processing, vertex partition.

Q. Hu, W. Liang and X. Shen, "On Embedding Between 2D Meshes of the Same Size," in IEEE Transactions on Computers, vol. 46, no. , pp. 880-889, 1997.
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