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The tightest lower bound on the number of comparisons, in the worst case, for comparison-based sorting is 

The tightest upper bound on the number of comparisons, in the worst case, for comparison-based sorting is

The tightest lower bound on the number of comparisons, in the best case, for comparison-based sorting is

The tightest upper bound on the number of comparisons, in the best case, for comparison-based sorting is

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