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en/docs/chapter_sorting/bucket_sort.md

@@ -4,7 +4,7 @@ comments: true
 
 # 11.8   Bucket Sort
 
-The sorting algorithms discussed earlier are all comparison-based sorting algorithms, which sort by comparing the relative order of elements. The time complexity of such algorithms cannot beat $O(n \log n)$. Next, we will explore several non-comparison sorting algorithms, whose time complexity can be linear.
+The sorting algorithms discussed earlier are all comparison-based sorting algorithms, which sort by comparing the relative order of elements. The worst-case time complexity of such algorithms has a lower bound of $\Omega(n \log n)$. Next, we will explore several non-comparison sorting algorithms, whose time complexity can be linear.
 
 <u>Bucket sort</u> is a typical application of the divide-and-conquer strategy. It works by creating a sequence of ordered buckets, each corresponding to a data range, and distributing the data evenly among them. The elements within each bucket are then sorted separately. Finally, all buckets are merged in order.
 

en/docs/chapter_sorting/selection_sort.md

@@ -326,7 +326,7 @@ In the code, we use $k$ to track the smallest element within the unsorted interv
 
 ## 11.2.1   Algorithm Characteristics
 
-- **Time complexity $O(n^2)$, non-adaptive sorting**: The outer loop has $n - 1$ rounds in total. The length of the unsorted interval in the first round is $n$, and the length of the unsorted interval in the last round is $2$. That is, the rounds of the outer loop contain inner loops with $n$, $n - 1$, $\dots$, $3$, and $2$ iterations, summing to $\frac{(n - 1)(n + 2)}{2}$.
+- **Time complexity $O(n^2)$, non-adaptive sorting**: The outer loop has $n - 1$ rounds in total. The inner loop runs $n - 1$ times in the first round and $1$ time in the last round. Thus, it runs $n - 1$, $n - 2$, $\dots$, $2$, and $1$ times across the rounds, summing to $\frac{n(n - 1)}{2}$.
 - **Space complexity $O(1)$, in-place sorting**: Pointers $i$ and $j$ use a constant amount of extra space.
 - **Unstable sorting**: As shown in Figure 11-3, element `nums[i]` may be swapped to the right of an element equal to it, causing a change in their relative order.
 

en/docs/chapter_sorting/sorting_algorithm.md

@@ -43,7 +43,7 @@ Stable sorting is a necessary condition for multi-level sorting scenarios. Suppo
 
 **Adaptability**: <u>Adaptive sorting</u> can utilize the existing order information in the input data to reduce the amount of computation, achieving better time efficiency. The best-case time complexity of adaptive sorting algorithms is typically better than the average time complexity.
 
-**Comparison-based or non-comparison**: <u>Comparison-based sorting</u> relies on comparison operators ($<$, $=$, $>$) to determine the relative order of elements, thereby sorting the entire array, with a theoretical optimal time complexity of $O(n \log n)$. <u>Non-comparison sorting</u> does not use comparison operators and can achieve a time complexity of $O(n)$, but its versatility is relatively limited.
+**Comparison-based or non-comparison**: <u>Comparison-based sorting</u> relies on comparison operators ($<$, $=$, $>$) to determine the relative order of elements, thereby sorting the entire array. Its worst-case time complexity has a lower bound of $\Omega(n \log n)$. <u>Non-comparison sorting</u> does not use comparison operators and can achieve a time complexity of $O(n)$, but its versatility is relatively limited.
 
 ## 11.1.2 &nbsp; Ideal Sorting Algorithm