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We study oblivious routing in fat-tree-based system area networks with deterministic routing under the assumption that the traffic demand is uncertain. The performance of a routing algorithm under uncertain traffic demands is characterized by the oblivious performance ratio that bounds the relative performance of the routing algorithm with respect to the optimal algorithm for any given traffic demand. We consider both single-path routing, where only one path is used to carry the traffic between each source-destination pair, and multipath routing, where multiple paths are allowed. For single-path routing, we derive lower bounds of the oblivious performance ratio for different fat-trees and develop routing schemes that achieve the optimal oblivious performance ratios for commonly used topologies. Our evaluation results indicate that the proposed oblivious routing schemes not only provide the optimal worst-case performance guarantees but also outperform existing schemes in average cases. For multipath routing, we show that it is possible to obtain an optimal scheme for all traffic demands (an oblivious performance ratio of 1). These results quantitatively demonstrate the performance difference between single-path routing and multipath routing in fat-trees.