Center Robin Lopez and forward Hakim Warrick went from the Suns to the New Orleans Hornets, while the contract of retiring center Brad Miller and guard Jerome Dyson were sent from the Hornets to the Suns.
Phoenix gets a future first-round pick from Minnesota and the Timberwolves obtained two second-round picks (2013 and 2016) from New Orleans.
The key player in the transaction was Johnson -- the fourth overall pick of the 2010 NBA Draft. He averaged 7.7 points and 2.9 rebounds in 144 games for Minnesota.
The T-wolves also announced Friday they have signed free agent forward Andrei Kirilenko with the salary cap space freed up by Johnson's departure.
Terms of the contract were not disclosed, but the (Minneapolis) Star-Tribune reports it is for two years at $20 million.
Kirilenko, 31, who played 10 years for the Utah Jazz, last season averaged 14.1 points and 7.5 rebounds for CSKA Moscow of the Euroleague.
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