build

en/docs/chapter_computational_complexity/summary.md

@@ -10,16 +10,16 @@ comments: true
 
 - Time efficiency and space efficiency are the two primary evaluation metrics for measuring algorithm performance.
 - We can evaluate algorithm efficiency through actual testing, but it is difficult to eliminate the influence of the testing environment, and it consumes substantial computational resources.
-- Complexity analysis can eliminate the drawbacks of actual testing, with results applicable to all running platforms, and it can reveal algorithm efficiency under different data scales.
+- Complexity analysis can overcome the limitations of actual testing. Its results apply across running platforms, and it can reveal algorithm efficiency under different data scales.
 
 **Time Complexity**
 
-- Time complexity is used to measure the trend of algorithm runtime as data volume increases. It can effectively evaluate algorithm efficiency, but may fail in certain situations, such as when the input data volume is small or when time complexities are identical, making it impossible to precisely compare algorithm efficiency.
+- Time complexity is used to measure the trend of algorithm runtime as data volume increases. It can effectively evaluate algorithm efficiency, but it may be less informative in certain situations, such as when the input data volume is small or when time complexities are identical, making it impossible to precisely compare algorithm efficiency.
 - Worst-case time complexity is represented using Big $O$ notation, corresponding to the asymptotic upper bound of a function, reflecting the growth level of the number of operations $T(n)$ as $n$ approaches positive infinity.
 - Deriving time complexity involves two steps: first, counting the number of operations, then determining the asymptotic upper bound.
 - Common time complexities arranged from low to high include $O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$, $O(2^n)$, and $O(n!)$.
 - The time complexity of some algorithms is not fixed, but rather depends on the distribution of input data. Time complexity is divided into worst-case, best-case, and average-case time complexity. Best-case time complexity is rarely used because input data generally needs to satisfy strict conditions to achieve the best case.
-- Average time complexity reflects the algorithm's runtime efficiency under random data input, and is closest to the algorithm's performance in practical applications. Calculating average time complexity requires statistical analysis of input data distribution and the combined mathematical expectation.
+- Average time complexity reflects the algorithm's runtime efficiency under random data input, and is closest to the algorithm's performance in practical applications. Calculating average time complexity requires analyzing the input data distribution and the resulting mathematical expectation.
 
 **Space Complexity**
 
@@ -36,7 +36,7 @@ Theoretically, the space complexity of tail recursive functions can be optimized
 
 **Q**: What is the difference between the terms function and method?
 
-A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object, is implicitly passed to the object that invokes it, and can operate on data contained in class instances.
+A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object, is implicitly bound to the object that invokes it, and can operate on data contained in class instances.
 
 The following examples use several common programming languages for illustration.
 
@@ -50,7 +50,7 @@ No, the diagram shows space complexity, which reflects growth trends rather than
 
 Assuming $n = 8$, you might find that the values of each curve do not correspond to the functions. This is because each curve contains a constant term used to compress the value range into a visually comfortable range.
 
-In practice, because we generally do not know what the "constant term" complexity of each method is, we usually cannot select the optimal solution for $n = 8$ based on complexity alone. But for $n = 8^5$, the choice is straightforward, as the growth trend already dominates.
+In practice, because we generally do not know the "constant-term" cost of each method, we usually cannot choose the optimal solution for cases like $n = 8$ based on complexity alone. But for $n = 8^5$, the choice is straightforward, because the growth trend already dominates.
 
 **Q**: Are there situations where algorithms are designed to sacrifice time (or space) based on actual use cases?