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Increasing integrated-circuit pin bandwidth has motivateda corresponding increase in the degree or radix of interconnection networksand their routers. This paper introduces the flattened butterfly, a cost-efficient topology for high-radix networks. On benign (load-balanced) traffic, the flattened butterfly approaches the cost/performance of a butterfly network and has roughly half the cost of a comparable performance Clos network.The advantage over the Clos is achieved by eliminating redundant hopswhen they are not needed for load balance. On adversarial traffic, the flattened butterfly matches the cost/performance of a folded-Clos network and provides an order of magnitude better performance than a conventional butterfly.In this case, global adaptive routing is used to switchthe flattened butterfly from minimal to non-minimal routing - usingredundant hops only when they are needed. Minimal and non-minimal, oblivious and adaptive routing algorithms are evaluated on the flattened butterfly.We show that load-balancing adversarial traffic requires non-minimalglobally-adaptive routing and show that sequential allocators are required to avoid transient load imbalance when using adaptive routing algorithms.We also compare the cost of the flattened butterfly to folded-Clos, hypercube,and butterfly networks with identical capacityand show that the flattened butterfly is more cost-efficient thanfolded-Clos and hypercube topologies.