build

en/docs/chapter_divide_and_conquer/binary_search_recur.md

@@ -7,7 +7,7 @@ comments: true
 We have already learned that search algorithms are divided into two major categories.
 
 - **Brute-force search**: Implemented by traversing the data structure, with a time complexity of $O(n)$.
-- **Adaptive search**: Utilizes unique data organization forms or prior information, with time complexity reaching $O(\log n)$ or even $O(1)$.
+- **Adaptive search**: Leverages specific data organization or prior information, with time complexity reaching $O(\log n)$ or even $O(1)$.
 
 In fact, **search algorithms with time complexity of $O(\log n)$ are typically implemented based on the divide and conquer strategy**, such as binary search and trees.
 
@@ -28,7 +28,7 @@ In previous sections, binary search was implemented based on iteration. Now we i
 
 !!! question
 
-    Given a sorted array `nums` of length $n$, where all elements are unique, find the element `target`.
+    Given a sorted array `nums` of length $n$, where all elements are unique, find `target`.
 
 From a divide and conquer perspective, we denote the subproblem corresponding to the search interval $[i, j]$ as $f(i, j)$.
 
@@ -36,7 +36,7 @@ Starting from the original problem $f(0, n-1)$, perform binary search through th
 
 1. Calculate the midpoint $m$ of the search interval $[i, j]$, and use it to eliminate half of the search interval.
 2. Recursively solve the subproblem reduced by half in size, which could be $f(i, m-1)$ or $f(m+1, j)$.
-3. Repeat steps `1.` and `2.` until `target` is found or the interval is empty and return.
+3. Repeat steps `1.` and `2.` until `target` is found, or return when the interval is empty.
 
 Figure 12-4 shows the divide and conquer process of binary search for element $6$ in an array.
 

en/docs/chapter_divide_and_conquer/build_binary_tree_problem.md

@@ -41,7 +41,7 @@ Using the data from the figure above as an example, we can obtain the division r
 
 ### 3.   Describing Subtree Intervals Based on Variables
 
-Based on the above division method, **we have obtained the index intervals of the root node, left subtree, and right subtree in `preorder` and `inorder`**. To describe these index intervals, we need to use several pointer variables.
+Based on the above division method, **we have obtained the index intervals of the root node, left subtree, and right subtree in `preorder` and `inorder`**. To describe these index intervals, we need to use several index variables.
 
 - Denote the index of the current tree's root node in `preorder` as $i$.
 - Denote the index of the current tree's root node in `inorder` as $m$.

en/docs/chapter_divide_and_conquer/divide_and_conquer.md

@@ -4,7 +4,7 @@ comments: true
 
 # 12.1   Divide and Conquer Algorithms
 
-<u>Divide and conquer</u> is a very important and common algorithm strategy. Divide and conquer is typically implemented based on recursion, consisting of two steps: "divide" and "conquer".
+<u>Divide and conquer</u> is a very important and common algorithmic strategy. Divide and conquer is typically implemented based on recursion, consisting of two steps: "divide" and "conquer".
 
 1. **Divide (partition phase)**: Recursively divide the original problem into two or more subproblems until the smallest subproblem is reached.
 2. **Conquer (merge phase)**: Starting from the smallest subproblems with known solutions, merge the solutions of subproblems from bottom to top to construct the solution to the original problem.
@@ -34,13 +34,13 @@ Clearly, merge sort satisfies these three criteria.
 
 ## 12.1.2 &nbsp; Improving Efficiency Through Divide and Conquer
 
-**Divide and conquer can not only effectively solve algorithmic problems but often also improve algorithm efficiency**. In sorting algorithms, quick sort, merge sort, and heap sort are faster than selection, bubble, and insertion sort because they apply the divide and conquer strategy.
+**Divide and conquer can not only effectively solve algorithmic problems, but can often also improve algorithmic efficiency**. In sorting algorithms, quick sort, merge sort, and heap sort are faster than selection, bubble, and insertion sort because they apply the divide and conquer strategy.
 
 This raises the question: **Why can divide and conquer improve algorithm efficiency, and what is the underlying logic**? In other words, why is dividing a large problem into multiple subproblems, solving the subproblems, and merging their solutions more efficient than directly solving the original problem? This question can be discussed from two aspects: operation count and parallel computation.
 
 ### 1. &nbsp; Operation Count Optimization
 
-Taking "bubble sort" as an example, processing an array of length $n$ requires $O(n^2)$ time. Suppose we divide the array into two subarrays from the midpoint as shown in Figure 12-2, the division requires $O(n)$ time, sorting each subarray requires $O((n / 2)^2)$ time, and merging the two subarrays requires $O(n)$ time, resulting in an overall time complexity of:
+Taking "bubble sort" as an example, processing an array of length $n$ requires $O(n^2)$ time. Suppose we divide the array at the midpoint into two subarrays, as shown in Figure 12-2. The division requires $O(n)$ time, sorting each subarray requires $O((n / 2)^2)$ time, and merging the two subarrays requires $O(n)$ time, resulting in an overall time complexity of:
 
 $$
 O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n)
@@ -68,7 +68,7 @@ Thinking further, **what if we set multiple division points** and evenly divide
 
 ### 2. &nbsp; Parallel Computation Optimization
 
-We know that the subproblems generated by divide and conquer are independent of each other, **so they can typically be solved in parallel**. This means divide and conquer can not only reduce the time complexity of algorithms, **but also benefits from parallel optimization by operating systems**.
+We know that the subproblems generated by divide and conquer are independent of each other, **so they can typically be solved in parallel**. This means divide and conquer can not only reduce the time complexity of algorithms, **but is also amenable to parallel optimization by the operating system**.
 
 Parallel optimization is particularly effective in multi-core or multi-processor environments, as the system can simultaneously handle multiple subproblems, making fuller use of computing resources and significantly reducing overall runtime.
 
@@ -80,7 +80,7 @@ For example, in the "bucket sort" shown in Figure 12-3, we evenly distribute mas
 
 ## 12.1.3 &nbsp; Common Applications of Divide and Conquer
 
-On one hand, divide and conquer can be used to solve many classic algorithmic problems.
+On the one hand, divide and conquer can be used to solve many classic algorithmic problems.
 
 - **Finding the closest pair of points**: This algorithm first divides the point set into two parts, then finds the closest pair of points in each part separately, and finally finds the closest pair of points that spans both parts.
 - **Large integer multiplication**: For example, the Karatsuba algorithm, which decomposes large integer multiplication into several smaller integer multiplications and additions.
@@ -90,12 +90,12 @@ On one hand, divide and conquer can be used to solve many classic algorithmic pr
 
 On the other hand, divide and conquer is widely applied in the design of algorithms and data structures.
 
-- **Binary search**: Binary search divides a sorted array into two parts from the midpoint index, then decides which half to eliminate based on the comparison result between the target value and the middle element value, and performs the same binary operation on the remaining interval.
+- **Binary search**: Binary search divides a sorted array into two parts from the midpoint index, then decides which half to eliminate based on the comparison result between the target value and the middle element value, and performs the same binary-search step on the remaining interval.
 - **Merge sort**: Already introduced at the beginning of this section, no further elaboration needed.
 - **Quick sort**: Quick sort selects a pivot value, then divides the array into two subarrays, one with elements smaller than the pivot and the other with elements larger than the pivot, then performs the same division operation on these two parts until the subarrays have only one element.
 - **Bucket sort**: The basic idea of bucket sort is to scatter data into multiple buckets, then sort the elements within each bucket, and finally extract the elements from each bucket in sequence to obtain a sorted array.
 - **Trees**: For example, binary search trees, AVL trees, red-black trees, B-trees, B+ trees, etc. Their search, insertion, and deletion operations can all be viewed as applications of the divide and conquer strategy.
 - **Heaps**: A heap is a special complete binary tree, and its various operations, such as insertion, deletion, and heapify, actually imply the divide and conquer idea.
-- **Hash tables**: Although hash tables do not directly apply divide and conquer, some hash collision resolution solutions indirectly apply the divide and conquer strategy. For example, long linked lists in chaining may be converted to red-black trees to improve query efficiency.
+- **Hash tables**: Although hash tables do not directly apply divide and conquer, some methods for resolving hash collisions indirectly apply the divide and conquer strategy. For example, long linked lists in chaining may be converted to red-black trees to improve lookup efficiency.
 
-It can be seen that **divide and conquer is a "subtly pervasive" algorithmic idea**, embedded in various algorithms and data structures.
+It can be seen that **divide and conquer is a "quietly pervasive" algorithmic idea**, embedded in various algorithms and data structures.

en/docs/chapter_divide_and_conquer/hanota_problem.md

@@ -552,4 +552,4 @@ As shown in Figure 12-15, the hanota problem forms a recursion tree of height $n
 
     The hanota problem originates from an ancient legend. In a temple in ancient India, monks had three tall diamond pillars and $64$ golden discs of different sizes. The monks continuously moved the discs, believing that when the last disc was correctly placed, the world would come to an end.
 
-    However, even if the monks moved one disc per second, it would take approximately $2^{64} \approx 1.84×10^{19}$ seconds, which is about $5850$ billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.
+    However, even if the monks moved one disc per second, it would take approximately $2^{64} \approx 1.84×10^{19}$ seconds, which is about $585$ billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.

en/docs/chapter_divide_and_conquer/index.md

@@ -11,12 +11,12 @@ icon: material/set-split
 
     Difficult problems are decomposed layer by layer, with each decomposition making them simpler.
 
-    Divide and conquer reveals an important truth: start with simplicity, and nothing remains complex.
+    Divide and conquer reveals an important truth: start with what is simple, and nothing remains complex.
 
 ## Chapter contents
 
 - [12.1   Divide and Conquer Algorithms](divide_and_conquer.md)
 - [12.2   Divide and Conquer Search Strategy](binary_search_recur.md)
 - [12.3   Building a Binary Tree Problem](build_binary_tree_problem.md)
-- [12.4   Hanoi Tower Problem](hanota_problem.md)
+- [12.4   Hanota Problem](hanota_problem.md)
 - [12.5   Summary](summary.md)

en/docs/chapter_divide_and_conquer/summary.md

@@ -6,11 +6,11 @@ comments: true
 
 ### 1.   Key Review
 
-- Divide and conquer is a common algorithm design strategy, consisting of two phases: divide (partition) and conquer (merge), typically implemented based on recursion.
+- Divide and conquer is a common algorithm design strategy consisting of two phases, divide (partition) and conquer (merge), and is typically implemented recursively.
 - The criteria for determining whether a problem is a divide and conquer problem include: whether the problem can be decomposed, whether subproblems are independent, and whether subproblems can be merged.
 - Merge sort is a typical application of the divide and conquer strategy. It recursively divides an array into two equal-length subarrays until only one element remains, then merges them layer by layer to complete the sorting.
-- Introducing the divide and conquer strategy can often improve algorithm efficiency. On one hand, the divide and conquer strategy reduces the number of operations; on the other hand, it facilitates parallel optimization of the system after division.
-- Divide and conquer can both solve many algorithmic problems and is widely applied in data structure and algorithm design, appearing everywhere.
+- Introducing the divide and conquer strategy can often improve algorithm efficiency. On one hand, it reduces the number of operations; on the other hand, it makes parallel optimization by the system easier.
+- Divide and conquer can solve many algorithmic problems and is also widely used in data structures and algorithm design, making it ubiquitous.
 - Compared to brute-force search, adaptive search is more efficient. Search algorithms with time complexity of $O(\log n)$ are typically implemented based on the divide and conquer strategy.
 - Binary search is another typical application of divide and conquer. It does not include the step of merging solutions of subproblems. We can implement binary search through recursive divide and conquer.
 - In the problem of building a binary tree, building the tree (original problem) can be divided into building the left subtree and right subtree (subproblems), which can be achieved by dividing the index intervals of the preorder and inorder traversals.