Message-passing multicomputers have been expected as the most promising way to construct massively parallel computers. The overall performance of these machines critically depends on the performance of the communication network to achieve the expected speedup and scalability. The communication networks of recently developed multicomputers have employed the oblivious wormhole routing strategy in which the cut-through switching technique makes the message latency insensitive to the distance traveled by a message, and the blocking flow control avoids using storage bandwidth in intermediate nodes where messages are routed. The combination of cut-through switching technique and the blocking flow control is called wormhole routing, and the network that adopts the wormhole routing technique is called wormhole-routed network. An important drawback of oblivious wormhole routing strategy is that the performance is seriously degraded due to the message contention that is caused by the deterministic routing algorithm when message injection rate is high and the traffic pattern is realistically non-uniform. And it is fault-intolerant to the node and channel failures. The adaptive routing has been expected as one of the best approaches to improve the network performance overcoming the message contention problem by utilizing available network bandwidth. In adaptive routing, multiple paths are provided for the messages and the selections are adaptively determined by local message congestion. Many adaptive routing strategies for wormholerouted networks have been developed. These strategies can be characterized by the number of additional virtual channels required to prevent deadlock, the degree of adaptiveness provided. All of the previous adaptive routing approaches restrict the adaptiveness to prevent the communication deadlock, and only a part of the shortest paths are permissible for the messages. This results in low utilization of the channels, hence low performance. In thi...