Accumulated Other Comprehensive Loss - Components of Accumulated Other Comprehensive Loss, Net of Tax (Details) - USD ($) $ in Thousands		3 Months Ended		9 Months Ended
		May 27, 2023	May 28, 2022	May 27, 2023	May 28, 2022
AOCI Attributable to Parent, Net of Tax [Roll Forward]
Balance at beginning of period		$ 1,955,088	$ 1,898,975	$ 1,915,871	$ 1,872,952
Other comprehensive income (loss)		768	(996)	(2,372)	(2,653)
Balance at end of period		1,976,851	1,904,011	1,976,851	1,904,011
Foreign Currency Translation
AOCI Attributable to Parent, Net of Tax [Roll Forward]
Balance at beginning of period		(28,054)	(19,451)	(24,803)	(17,801)
Other comprehensive (loss) income before reclassification		762	(982)	(2,489)	(2,632)
Amounts reclassified from accumulated other comprehensive loss		0	0	0	0
Other comprehensive income (loss)		762	(982)	(2,489)	(2,632)
Balance at end of period		(27,292)	(20,433)	(27,292)	(20,433)
Pension-related
AOCI Attributable to Parent, Net of Tax [Roll Forward]
Balance at beginning of period	[1]	452	(7,066)	452	(7,066)
Other comprehensive (loss) income before reclassification	[1]	0	0	0	0
Amounts reclassified from accumulated other comprehensive loss	[1]	0	0	0	0
Other comprehensive income (loss)	[1]	0	0	0	0
Balance at end of period	[1]	452	(7,066)	452	(7,066)
Derivative Financial Instruments
AOCI Attributable to Parent, Net of Tax [Roll Forward]
Balance at beginning of period	[1]	172	23	61	31
Other comprehensive (loss) income before reclassification	[1]	31	(10)	186	35
Amounts reclassified from accumulated other comprehensive loss	[1]	(25)	(4)	(69)	(57)
Other comprehensive income (loss)	[1]	6	(14)	117	(22)
Balance at end of period	[1]	178	9	178	9
Accumulated Other Comprehensive Loss
AOCI Attributable to Parent, Net of Tax [Roll Forward]
Balance at beginning of period		(27,430)	(26,494)	(24,290)	(24,836)
Other comprehensive (loss) income before reclassification		793	(992)	(2,303)	(2,597)
Amounts reclassified from accumulated other comprehensive loss		(25)	(4)	(69)	(57)
Other comprehensive income (loss)		768	(996)	(2,372)	(2,654)
Balance at end of period		$ (26,662)	$ (27,490)	$ (26,662)	$ (27,490)

[1] Amounts are shown net of tax