diff --git a/en/docs/chapter_searching/binary_search_edge.md b/en/docs/chapter_searching/binary_search_edge.md index 3af95f050..2ccafc2cb 100644 --- a/en/docs/chapter_searching/binary_search_edge.md +++ b/en/docs/chapter_searching/binary_search_edge.md @@ -6,9 +6,9 @@ Given a sorted array `nums` of length $n$, which may contain duplicate elements, return the index of the leftmost element `target`. If the element is not present in the array, return $-1$. -Recall the method of binary search for an insertion point, after the search is completed, $i$ points to the leftmost `target`, **thus searching for the insertion point is essentially searching for the index of the leftmost `target`**. +Recall the method of binary search for an insertion point, after the search is completed, the index $i$ will point to the leftmost occurrence of `target`. Therefore, **searching for the insertion point is essentially the same as finding the index of the leftmost `target`**. -Consider implementing the search for the left boundary using the function for finding an insertion point. Note that the array might not contain `target`, which could lead to the following two results: +We can use the function for finding an insertion point to find the left boundary of `target`. Note that the array might not contain `target`, which could lead to the following two results: - The index $i$ of the insertion point is out of bounds. - The element `nums[i]` is not equal to `target`. @@ -21,27 +21,27 @@ In these cases, simply return $-1$. The code is as follows: ## Find the right boundary -So how do we find the rightmost `target`? The most straightforward way is to modify the code, replacing the pointer contraction operation in the case of `nums[m] == target`. The code is omitted here, but interested readers can implement it on their own. +How do we find the rightmost occurrence of `target`? The most straightforward way is to modify the traditional binary search logic by changing how we adjust the search boundaries in the case of `nums[m] == target`. The code is omitted here. If you are interested, try to implement the code on your own. -Below we introduce two more cunning methods. +Below we are going to introduce two more ingenious methods. -### Reusing the search for the left boundary +### Reuse the left boundary search -In fact, we can use the function for finding the leftmost element to find the rightmost element, specifically by **transforming the search for the rightmost `target` into a search for the leftmost `target + 1`**. +To find the rightmost occurrence of `target`, we can reuse the logic for finding the leftmost occurrence of `target`. Specifically, we can find the leftmost `target`, and then adjust the result to point to the rightmost `target` by simply adding 1 to the index of the leftmost `target`. -As shown in the figure below, after the search is completed, the pointer $i$ points to the leftmost `target + 1` (if it exists), while $j$ points to the rightmost `target`, **thus returning $j$ is sufficient**. +As shown in the figure below, after the search is complete, pointer $i$ will point to the position just after the leftmost `target` (i.e., `target + 1`), and pointer $j$ will point to the rightmost `target`. Therefore, returning $j$ will give us the right boundary. ![Transforming the search for the right boundary into the search for the left boundary](binary_search_edge.assets/binary_search_right_edge_by_left_edge.png) -Please note, the insertion point returned is $i$, therefore, it should be subtracted by $1$ to obtain $j$: +Note that the insertion point returned is $i$, therefore, it should be subtracted by $1$ to obtain $j$: ```src [file]{binary_search_edge}-[class]{}-[func]{binary_search_right_edge} ``` -### Transforming into an element search +### Transform into an element search -We know that when the array does not contain `target`, $i$ and $j$ will eventually point to the first element greater and smaller than `target` respectively. +When the array does not contain `target`, $i$ and $j$ will eventually point to the first element greater and smaller than `target` respectively. Thus, as shown in the figure below, we can construct an element that does not exist in the array, to search for the left and right boundaries. @@ -50,7 +50,7 @@ Thus, as shown in the figure below, we can construct an element that does not ex ![Transforming the search for boundaries into the search for an element](binary_search_edge.assets/binary_search_edge_by_element.png) -The code is omitted here, but two points are worth noting. +The code is omitted here, but here are two important points to note about this approach. -- The given array does not contain decimals, meaning we do not need to worry about how to handle equal situations. -- Since this method introduces decimals, the variable `target` in the function needs to be changed to a floating point type (no change needed in Python). +- The given array `nums` does not contain decimal, so handling equal cases is not a concern. +- However, the introduction of decimal in this approach requires modifying the `target` variable to a floating-point type (no change needed in Python). diff --git a/en/docs/chapter_searching/binary_search_insertion.md b/en/docs/chapter_searching/binary_search_insertion.md index 8eb4a06b0..dc7be697e 100644 --- a/en/docs/chapter_searching/binary_search_insertion.md +++ b/en/docs/chapter_searching/binary_search_insertion.md @@ -6,21 +6,21 @@ Binary search is not only used to search for target elements but also to solve m !!! question - Given an ordered array `nums` of length $n$ and an element `target`, where the array has no duplicate elements. Now insert `target` into the array `nums` while maintaining its order. If the element `target` already exists in the array, insert it to its left side. Please return the index of `target` in the array after insertion. See the example shown in the figure below. + Given a sorted array `nums` of length $n$ with unique elements and an element `target`, insert `target` into `nums` while maintaining its sorted order. If `target` already exists in the array, insert it to the left of the existing element. Return the index of `target` in the array after insertion. See the example shown in the figure below. ![Example data for binary search insertion point](binary_search_insertion.assets/binary_search_insertion_example.png) If you want to reuse the binary search code from the previous section, you need to answer the following two questions. -**Question one**: When the array contains `target`, is the insertion point index the index of that element? +**Question one**: If the array already contains `target`, would the insertion point be the index of existing element? -The requirement to insert `target` to the left of equal elements means that the newly inserted `target` replaces the original `target` position. Thus, **when the array contains `target`, the insertion point index is the index of that `target`**. +The requirement to insert `target` to the left of equal elements means that the newly inserted `target` will replace the original `target` position. In other words, **when the array contains `target`, the insertion point is indeed the index of that `target`**. -**Question two**: When the array does not contain `target`, what is the index of the insertion point? +**Question two**: When the array does not contain `target`, at which index would it be inserted? -Further consider the binary search process: when `nums[m] < target`, pointer $i$ moves, meaning that pointer $i$ is approaching an element greater than or equal to `target`. Similarly, pointer $j$ is always approaching an element less than or equal to `target`. +Let's further consider the binary search process: when `nums[m] < target`, pointer $i$ moves, meaning that pointer $i$ is approaching an element greater than or equal to `target`. Similarly, pointer $j$ is always approaching an element less than or equal to `target`. -Therefore, at the end of the binary, it is certain that: $i$ points to the first element greater than `target`, and $j$ points to the first element less than `target`. **It is easy to see that when the array does not contain `target`, the insertion index is $i$**. The code is as follows: +Therefore, at the end of the binary, it is certain that: $i$ points to the first element greater than `target`, and $j$ points to the first element less than `target`. **It is easy to see that when the array does not contain `target`, the insertion point is $i$**. The code is as follows: ```src [file]{binary_search_insertion}-[class]{}-[func]{binary_search_insertion_simple} @@ -32,21 +32,21 @@ Therefore, at the end of the binary, it is certain that: $i$ points to the first Based on the previous question, assume the array may contain duplicate elements, all else remains the same. -Suppose there are multiple `target`s in the array, ordinary binary search can only return the index of one of the `target`s, **and it cannot determine how many `target`s are to the left and right of that element**. +When there are multiple occurrences of `target` in the array, a regular binary search can only return the index of one occurrence of `target`, **and it cannot determine how many occurrences of `target` are to the left and right of that position**. -The task requires inserting the target element to the very left, **so we need to find the index of the leftmost `target` in the array**. Initially consider implementing this through the steps shown in the figure below. +The problem requires inserting the target element to the very left, **so we need to find the index of the leftmost `target` in the array**. Initially consider implementing this through the steps shown in the figure below. -1. Perform a binary search, get an arbitrary index of `target`, denoted as $k$. -2. Start from index $k$, and perform a linear search to the left until the leftmost `target` is found and return. +1. Perform a binary search to find any index of `target`, say $k$. +2. Starting from index $k$, conduct a linear traversal to the left until the leftmost occurrence of `target` is found, then return this index. ![Linear search for the insertion point of duplicate elements](binary_search_insertion.assets/binary_search_insertion_naive.png) Although this method is feasible, it includes linear search, so its time complexity is $O(n)$. This method is inefficient when the array contains many duplicate `target`s. -Now consider extending the binary search code. As shown in the figure below, the overall process remains the same, each round first calculates the midpoint index $m$, then judges the size relationship between `target` and `nums[m]`, divided into the following cases. +Now consider extending the binary search code. As shown in the figure below, the overall process remains the same. In each round, we first calculate the middle index $m$, then compare the value of `target` with `nums[m]`, which results in the following cases. -- When `nums[m] < target` or `nums[m] > target`, it means `target` has not been found yet, thus use the normal binary search interval reduction operation, **thus making pointers $i$ and $j$ approach `target`**. -- When `nums[m] == target`, it indicates that the elements less than `target` are in the interval $[i, m - 1]$, therefore use $j = m - 1$ to narrow the interval, **thus making pointer $j$ approach elements less than `target`**. +- When `nums[m] < target` or `nums[m] > target`, it means `target` has not been found yet, thus use the normal binary search to narrow the search range, **bring pointers $i$ and $j$ closer to `target`**. +- When `nums[m] == target`, it indicates that the elements less than `target` are in the range $[i, m - 1]$, therefore use $j = m - 1$ to narrow the range, **thus bring pointer $j$ closer to the elements less than `target`**. After the loop, $i$ points to the leftmost `target`, and $j$ points to the first element less than `target`, **therefore index $i$ is the insertion point**. @@ -74,9 +74,9 @@ After the loop, $i$ points to the leftmost `target`, and $j$ points to the first === "<8>" ![binary_search_insertion_step8](binary_search_insertion.assets/binary_search_insertion_step8.png) -Observe the code, the operations of the branch `nums[m] > target` and `nums[m] == target` are the same, so the two can be combined. +Observe the following code. The operations in the branches `nums[m] > target` and `nums[m] == target` are the same, so these two branches can be merged. -Even so, we can still keep the conditions expanded, as their logic is clearer and more readable. +Even so, we can still keep the conditions expanded, as it makes the logic clearer and improves readability. ```src [file]{binary_search_insertion}-[class]{}-[func]{binary_search_insertion} @@ -84,8 +84,8 @@ Even so, we can still keep the conditions expanded, as their logic is clearer an !!! tip - The code in this section uses "closed intervals". Readers interested can implement the "left-closed right-open" method themselves. + The code in this section uses "closed interval". If you are interested in "left-closed,right-open", try to implement the code on your own. -In summary, binary search is merely about setting search targets for pointers $i$ and $j$, which might be a specific element (like `target`) or a range of elements (like elements less than `target`). +In summary, binary search essentially involves setting search targets for pointers $i$ and $j$, which might be a specific element (like `target`) or a range of elements (like elements less than `target`). In the continuous loop of binary search, pointers $i$ and $j$ gradually approach the predefined target. Ultimately, they either find the answer or stop after crossing the boundary. diff --git a/en/mkdocs.yml b/en/mkdocs.yml index c98eeabb9..291cc385d 100644 --- a/en/mkdocs.yml +++ b/en/mkdocs.yml @@ -117,7 +117,7 @@ nav: # [icon: material/text-search] - chapter_searching/index.md - 10.1 Binary search: chapter_searching/binary_search.md - - 10.2 Binary search insertion: chapter_searching/binary_search_insertion.md + - 10.2 Binary search for insertion point: chapter_searching/binary_search_insertion.md - 10.3 Binary search boundaries: chapter_searching/binary_search_edge.md - 10.4 Hashing optimization strategies: chapter_searching/replace_linear_by_hashing.md - 10.5 Search algorithms revisited: chapter_searching/searching_algorithm_revisited.md