Employee Benefit Plans - Narrative (Details) $ in Millions	3 Months Ended		9 Months Ended
	May 28, 2022 USD ($) RetirementPlan	May 29, 2021 USD ($)	May 28, 2022 USD ($) RetirementPlan	May 29, 2021 USD ($)
Postemployment Benefits [Abstract]
Contributions charged to expense under the plan	$ 4.7	$ 5.1	$ 14.2	$ 15.2
Number of frozen non-contributory defined benefit pension plans | RetirementPlan	1		1
Amounts charged to expense related to the plans	$ 0.6	$ 0.8	$ 1.9	$ 2.3
Deferred compensation plan, employee or employer contribution charged to expense	$ 0.0		$ 0.0