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en/docs/chapter_divide_and_conquer/binary_search_recur.md

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 # Divide and conquer search strategy
 
-We have learned that search algorithms fall into two main categories.
+We have learned from the [chapter 10.5](../chapter_searching/searching_algorithm_revisited.md) that search algorithms fall into two main categories.
 
-- **Brute-force search**: It is implemented by traversing the data structure, with a time complexity of $O(n)$.
-- **Adaptive search**: It utilizes a unique data organization form or prior information, and its time complexity can reach $O(\log n)$ or even $O(1)$.
+- **Brute-force search**: It is implemented by traversing every element of a data structure, with a time complexity of $O(n)$.
+- **Adaptive search**: It utilizes a unique data structure or prior information, with a time complexity of $O(\log n)$ or even $O(1)$.
 
-In fact, **search algorithms with a time complexity of $O(\log n)$ are usually based on the divide-and-conquer strategy**, such as binary search and trees.
+In fact, **search algorithms with a time complexity of $O(\log n)$ are typically based on the divide-and-conquer strategy**, such as binary search and tree.
 
-- Each step of binary search divides the problem (searching for a target element in an array) into a smaller problem (searching for the target element in half of the array), continuing until the array is empty or the target element is found.
-- Trees represent the divide-and-conquer idea, where in data structures like binary search trees, AVL trees, and heaps, the time complexity of various operations is $O(\log n)$.
+- Binary search divides the original problem (searching for a target element in an array) into a smaller subproblem (searching for a target element in half of the array) at each step. This process continues until the array is empty or the target element is found.
+- Trees are a classic example of the divide-and-conquer paradigm. In data structures such as binary search trees, AVL trees, and heaps, the time complexity of operations is $O(\log n)$.
 
-The divide-and-conquer strategy of binary search is as follows.
+The divide-and-conquer strategy of binary search is characterized as follows.
 
-- **The problem can be divided**: Binary search recursively divides the original problem (searching in an array) into subproblems (searching in half of the array), achieved by comparing the middle element with the target element.
-- **Subproblems are independent**: In binary search, each round handles one subproblem, unaffected by other subproblems.
-- **The solutions of subproblems do not need to be merged**: Binary search aims to find a specific element, so there is no need to merge the solutions of subproblems. When a subproblem is solved, the original problem is also solved.
+- **Dividing the problem**: Binary search recursively divides the original problem (searching in the full array) into subproblems (searching in half of the array) by comparing the middle element with the target element.
+- **Independence of subproblems**: Each round of binary search handle only one subproblem, unaffected by other subproblems.
+- **No subproblem solution merging**: Binary search aims to find a specific element, so there is no need to merge the solutions of subproblems. Solving the subproblem inherently solves the original problem.
 
-Divide-and-conquer can enhance search efficiency because brute-force search can only eliminate one option per round, **whereas divide-and-conquer can eliminate half of the options**.
+Divide-and-conquer improves search efficiency. This is because, in brute-force search, only one option can be eliminated per round. In contrast, **divide-and-conquer search eliminates half of the options in each round**.
 
 ### Implementing binary search based on divide-and-conquer
 
-In previous chapters, binary search was implemented based on iteration. Now, we implement it based on divide-and-conquer (recursion).
+In previous chapters, binary search was implemented iteratively. Here, we will implement it using a divide-and-conquer (recursive) approach.
 
 !!! question
 
-    Given an ordered array `nums` of length $n$, where all elements are unique, please find the element `target`.
+    Given a sorted array `nums` of length $n$ with unique elements, find the index of element `target`.
 
-From a divide-and-conquer perspective, we denote the subproblem corresponding to the search interval $[i, j]$ as $f(i, j)$.
+From a divide-and-conquer perspective, we define $f(i, j)$ as the subproblem of the search interval $[i, j]$.
 
-Starting from the original problem $f(0, n-1)$, perform the binary search through the following steps.
+Starting from the original problem $f(0, n-1)$, the binary search proceeds as follows.
 
-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 returns.
+1. Calculate the midpoint $m$ of the search interval $[i, j]$ to eliminate half of the search space.
+2. Recursively solve the subproblem on the reduced interval, either $f(i, m-1)$ or $f(m+1, j)$.
+3. Repeat steps `1.` and `2.`, until `target` is found or the interval becomes empty.
 
-The figure below shows the divide-and-conquer process of binary search for element $6$ in an array.
+The figure below shows the divide-and-conquer process of performing binary search for element $6$ in a sorted array.
 
 ![The divide-and-conquer process of binary search](binary_search_recur.assets/binary_search_recur.png)