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Share-based Compensation - Narrative (Details) - USD ($)
$ / shares in Units, $ in Millions
12 Months Ended
Oct. 27, 2014
Aug. 26, 2023
Aug. 27, 2022
Aug. 28, 2021
Oct. 30, 2010
Stock Appreciation Rights (SARs)          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Percentage vested after each anniversary of the grant date   20.00%      
Expiration period   10 years      
Weighted average fair values of Share-Based Awards granted (in dollars per share)   $ 60.38 $ 51.25 $ 42.63  
Stock Appreciation Rights (SARs) | Cliff Vesting          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Period awards become fully vested or exercisable   5 years      
Share-Based Awards          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Total compensation cost not yet recognized related to non-vested awards   $ 15.5      
Period for recognition   1 year 4 months 9 days      
Time-Based Restricted Stock Units          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Percentage vested after each anniversary of the grant date   20.00%      
Period awards become fully vested or exercisable   5 years      
Non-Employee Directors | Unrestricted Stock          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Shares granted (in shares)   1,575 1,025 0  
The 2010 Plan          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Number of shares reserved for issuance (in shares)         600,000
Number of additional shares reserved for issuance (in shares) 750,000        
Shares available for future grants (in shares)   98,769      
The 2010 Plan | Non-Employee Directors | Stock Appreciation Rights (SARs)          
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]          
Shares granted (in shares)   6,864 9,750 5,000