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The Collaborative International Dictionary of English v.0.48:

Rout \Rout\, v. t. [imp. & p. p. Routed; p. pr. & vb. n. Routing.] To break the ranks of, as troops, and put them to flight in disorder; to put to rout. [1913 Webster] That party . . . that charged the Scots, so totally routed and defeated their whole army, that they fied. --Clarendon. [1913 Webster] Syn: To defeat; discomfit; overpower; overthrow. [1913 Webster]
The Free On-line Dictionary of Computing (30 December 2018):

routing /row'ting/ Using a kind of rotating cutting tool called a router, pronounced /row't*/. In the USA a router, pronounced /row't*/, is also a network device that performs "routing". In the UK, the network device is pronounced /roo't*/ and what it does is spelled "routeing". (2002-07-31)