<p><it>Abstract—</it>In this paper, we give two algorithms for the 1-1 routing problems on a mesh-connected computer. The first algorithm, with queue size 28, solves the 1-1 routing problem on an $<tmath>n\times n</tmath>$ mesh-connected computer in $<tmath>2n+O(1)</tmath>$ steps. This improves the previous queue size of $<tmath>75</tmath>$. The second algorithm solves the 1-1 routing problem in $<tmath>2n-2</tmath>$ steps with queue size $<tmath>12t_s/s</tmath>$ where $<tmath>t_s</tmath>$ is the time for sorting an $<tmath>s\times s</tmath>$ mesh into a row major order for all $<tmath>s\geq 1</tmath>$. This result improves the previous queue size $<tmath>18.67t_s/s</tmath>$.</p><p><it>Index Terms—</it>Parallel algorithms, mesh-connected computer, packet routing, queue size, time complexity.</p>