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Based on a realistic, yet simple cost model, we compute the switch radix that minimizes the cost of a fat tree network to support a given number of end nodes. The cost model comprises two parameters indicating the relative cost of a crosspoint vs. a link, and the crosspoint-independent base cost of a switch. These parameters can be adapted to represent a given technology used to implement links and switches. Based on these inputs, the resulting model allows a quick evaluation of the switch radix that minimizes the overall cost of the network. We demonstrate that the optimum radix depends most strongly on the relative cost of a link, and turns out to be largely independent of the network size. Using a first-order cost bounds analysis based on current CMOS and link technology, our model indicates that the optimum switch radix for large fat trees is driven almost entirely by link cost and as a result lies in the range of hundreds of ports, rather than the tens of ports being offered today by most commercial switch products today.