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hello-algo/en/docs/chapter_searching/binary_search_edge.md

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---
comments: true
---
# 10.3   Binary search boundaries
## 10.3.1   Find the left boundary
!!! question
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`**.
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:
- The index $i$ of the insertion point is out of bounds.
- The element `nums[i]` is not equal to `target`.
In these cases, simply return $-1$. The code is as follows:
=== "Python"
```python title="binary_search_edge.py"
def binary_search_left_edge(nums: list[int], target: int) -> int:
"""二分查找最左一个 target"""
# 等价于查找 target 的插入点
i = binary_search_insertion(nums, target)
# 未找到 target ，返回 -1
if i == len(nums) or nums[i] != target:
return -1
# 找到 target ，返回索引 i
return i
```
=== "C++"
```cpp title="binary_search_edge.cpp"
/* 二分查找最左一个 target */
int binarySearchLeftEdge(vector<int> &nums, int target) {
// 等价于查找 target 的插入点
int i = binarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i == nums.size() || nums[i] != target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "Java"
```java title="binary_search_edge.java"
/* 二分查找最左一个 target */
int binarySearchLeftEdge(int[] nums, int target) {
// 等价于查找 target 的插入点
int i = binary_search_insertion.binarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i == nums.length || nums[i] != target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "C#"
```csharp title="binary_search_edge.cs"
/* 二分查找最左一个 target */
int BinarySearchLeftEdge(int[] nums, int target) {
// 等价于查找 target 的插入点
int i = binary_search_insertion.BinarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i == nums.Length || nums[i] != target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "Go"
```go title="binary_search_edge.go"
/* 二分查找最左一个 target */
func binarySearchLeftEdge(nums []int, target int) int {
// 等价于查找 target 的插入点
i := binarySearchInsertion(nums, target)
// 未找到 target ，返回 -1
if i == len(nums) || nums[i] != target {
return -1
}
// 找到 target ，返回索引 i
return i
}
```
=== "Swift"
```swift title="binary_search_edge.swift"
/* 二分查找最左一个 target */
func binarySearchLeftEdge(nums: [Int], target: Int) -> Int {
// 等价于查找 target 的插入点
let i = binarySearchInsertion(nums: nums, target: target)
// 未找到 target ，返回 -1
if i == nums.endIndex || nums[i] != target {
return -1
}
// 找到 target ，返回索引 i
return i
}
```
=== "JS"
```javascript title="binary_search_edge.js"
/* 二分查找最左一个 target */
function binarySearchLeftEdge(nums, target) {
// 等价于查找 target 的插入点
const i = binarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i === nums.length || nums[i] !== target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "TS"
```typescript title="binary_search_edge.ts"
/* 二分查找最左一个 target */
function binarySearchLeftEdge(nums: Array<number>, target: number): number {
// 等价于查找 target 的插入点
const i = binarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i === nums.length || nums[i] !== target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "Dart"
```dart title="binary_search_edge.dart"
/* 二分查找最左一个 target */
int binarySearchLeftEdge(List<int> nums, int target) {
// 等价于查找 target 的插入点
int i = binarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i == nums.length || nums[i] != target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "Rust"
```rust title="binary_search_edge.rs"
/* 二分查找最左一个 target */
fn binary_search_left_edge(nums: &[i32], target: i32) -> i32 {
// 等价于查找 target 的插入点
let i = binary_search_insertion(nums, target);
// 未找到 target ，返回 -1
if i == nums.len() as i32 || nums[i as usize] != target {
return -1;
}
// 找到 target ，返回索引 i
i
}
```
=== "C"
```c title="binary_search_edge.c"
/* 二分查找最左一个 target */
int binarySearchLeftEdge(int *nums, int numSize, int target) {
// 等价于查找 target 的插入点
int i = binarySearchInsertion(nums, numSize, target);
// 未找到 target ，返回 -1
if (i == numSize || nums[i] != target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
```
=== "Kotlin"
```kotlin title="binary_search_edge.kt"
/* 二分查找最左一个 target */
fun binarySearchLeftEdge(nums: IntArray, target: Int): Int {
// 等价于查找 target 的插入点
val i = binarySearchInsertion(nums, target)
// 未找到 target ，返回 -1
if (i == nums.size || nums[i] != target) {
return -1
}
// 找到 target ，返回索引 i
return i
}
```
=== "Ruby"
```ruby title="binary_search_edge.rb"
### 二分查找最左一个 target ###
def binary_search_left_edge(nums, target)
# 等价于查找 target 的插入点
i = binary_search_insertion(nums, target)
# 未找到 target ，返回 -1
return -1 if i == nums.length || nums[i] != target
i # 找到 target ，返回索引 i
end
```
=== "Zig"
```zig title="binary_search_edge.zig"
[class]{}-[func]{binarySearchLeftEdge}
```
??? pythontutor "Code Visualization"
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E6%8F%92%E5%85%A5%E7%82%B9%EF%BC%88%E5%AD%98%E5%9C%A8%E9%87%8D%E5%A4%8D%E5%85%83%E7%B4%A0%EF%BC%89%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E9%97%AD%E5%8C%BA%E9%97%B4%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%AD%E7%82%B9%E7%B4%A2%E5%BC%95%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bm%2B1,%20j%5D%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m-1%5D%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20%E9%A6%96%E4%B8%AA%E5%B0%8F%E4%BA%8E%20target%20%E7%9A%84%E5%85%83%E7%B4%A0%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m-1%5D%20%E4%B8%AD%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E6%8F%92%E5%85%A5%E7%82%B9%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_left_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E6%9C%80%E5%B7%A6%E4%B8%80%E4%B8%AA%20target%22%22%22%0A%20%20%20%20%23%20%E7%AD%89%E4%BB%B7%E4%BA%8E%E6%9F%A5%E6%89%BE%20target%20%E7%9A%84%E6%8F%92%E5%85%A5%E7%82%B9%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%29%0A%20%20%20%20%23%20%E6%9C%AA%E6%89%BE%E5%88%B0%20target%20%EF%BC%8C%E8%BF%94%E5%9B%9E%20-1%0A%20%20%20%20if%20i%20%3D%3D%20len%28nums%29%20or%20nums%5Bi%5D%20!%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20%E6%89%BE%E5%88%B0%20target%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E7%B4%A2%E5%BC%95%20i%0A%20%20%20%20return%20i%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%8C%85%E5%90%AB%E9%87%8D%E5%A4%8D%E5%85%83%E7%B4%A0%E7%9A%84%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E5%B7%A6%E8%BE%B9%E7%95%8C%E5%92%8C%E5%8F%B3%E8%BE%B9%E7%95%8C%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_left_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22%E6%9C%80%E5%B7%A6%E4%B8%80%E4%B8%AA%E5%85%83%E7%B4%A0%20%7Btarget%7D%20%E7%9A%84%E7%B4%A2%E5%BC%95%E4%B8%BA%20%7Bindex%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E6%8F%92%E5%85%A5%E7%82%B9%EF%BC%88%E5%AD%98%E5%9C%A8%E9%87%8D%E5%A4%8D%E5%85%83%E7%B4%A0%EF%BC%89%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E9%97%AD%E5%8C%BA%E9%97%B4%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%AD%E7%82%B9%E7%B4%A2%E5%BC%95%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bm%2B1,%20j%5D%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m-1%5D%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20%E9%A6%96%E4%B8%AA%E5%B0%8F%E4%BA%8E%20target%20%E7%9A%84%E5%85%83%E7%B4%A0%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m-1%5D%20%E4%B8%AD%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E6%8F%92%E5%85%A5%E7%82%B9%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_left_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E6%9C%80%E5%B7%A6%E4%B8%80%E4%B8%AA%20target%22%22%22%0A%20%20%20%20%23%20%E7%AD%89%E4%BB%B7%E4%BA%8E%E6%9F%A5%E6%89%BE%20target%20%E7%9A%84%E6%8F%92%E5%85%A5%E7%82%B9%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%29%0A%20%20%20%20%23%20%E6%9C%AA%E6%89%BE%E5%88%B0%20target%20%EF%BC%8C%E8%BF%94%E5%9B%9E%20-1%0A%20%20%20%20if%20i%20%3D%3D%20len%28nums%29%20or%20nums%5Bi%5D%20!%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20%E6%89%BE%E5%88%B0%20target%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E7%B4%A2%E5%BC%95%20i%0A%20%20%20%20return%20i%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%8C%85%E5%90%AB%E9%87%8D%E5%A4%8D%E5%85%83%E7%B4%A0%E7%9A%84%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E5%B7%A6%E8%BE%B9%E7%95%8C%E5%92%8C%E5%8F%B3%E8%BE%B9%E7%95%8C%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_left_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22%E6%9C%80%E5%B7%A6%E4%B8%80%E4%B8%AA%E5%85%83%E7%B4%A0%20%7Btarget%7D%20%E7%9A%84%E7%B4%A2%E5%BC%95%E4%B8%BA%20%7Bindex%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
## 10.3.2 &nbsp; 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.
Below we introduce two more cunning methods.
### 1. &nbsp; Reusing the search for the left boundary
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`**.
As shown in the Figure 10-7 , 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**.
![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){ class="animation-figure" }
<p align="center"> Figure 10-7 &nbsp; Transforming the search for the right boundary into the search for the left boundary </p>
Please note, the insertion point returned is $i$, therefore, it should be subtracted by $1$ to obtain $j$:
=== "Python"
```python title="binary_search_edge.py"
def binary_search_right_edge(nums: list[int], target: int) -> int:
"""二分查找最右一个 target"""
# 转化为查找最左一个 target + 1
i = binary_search_insertion(nums, target + 1)
# j 指向最右一个 target ，i 指向首个大于 target 的元素
j = i - 1
# 未找到 target ，返回 -1
if j == -1 or nums[j] != target:
return -1
# 找到 target ，返回索引 j
return j
```
=== "C++"
```cpp title="binary_search_edge.cpp"
/* 二分查找最右一个 target */
int binarySearchRightEdge(vector<int> &nums, int target) {
// 转化为查找最左一个 target + 1
int i = binarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
int j = i - 1;
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "Java"
```java title="binary_search_edge.java"
/* 二分查找最右一个 target */
int binarySearchRightEdge(int[] nums, int target) {
// 转化为查找最左一个 target + 1
int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
int j = i - 1;
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "C#"
```csharp title="binary_search_edge.cs"
/* 二分查找最右一个 target */
int BinarySearchRightEdge(int[] nums, int target) {
// 转化为查找最左一个 target + 1
int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
int j = i - 1;
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "Go"
```go title="binary_search_edge.go"
/* 二分查找最右一个 target */
func binarySearchRightEdge(nums []int, target int) int {
// 转化为查找最左一个 target + 1
i := binarySearchInsertion(nums, target+1)
// j 指向最右一个 target ，i 指向首个大于 target 的元素
j := i - 1
// 未找到 target ，返回 -1
if j == -1 || nums[j] != target {
return -1
}
// 找到 target ，返回索引 j
return j
}
```
=== "Swift"
```swift title="binary_search_edge.swift"
/* 二分查找最右一个 target */
func binarySearchRightEdge(nums: [Int], target: Int) -> Int {
// 转化为查找最左一个 target + 1
let i = binarySearchInsertion(nums: nums, target: target + 1)
// j 指向最右一个 target ，i 指向首个大于 target 的元素
let j = i - 1
// 未找到 target ，返回 -1
if j == -1 || nums[j] != target {
return -1
}
// 找到 target ，返回索引 j
return j
}
```
=== "JS"
```javascript title="binary_search_edge.js"
/* 二分查找最右一个 target */
function binarySearchRightEdge(nums, target) {
// 转化为查找最左一个 target + 1
const i = binarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
const j = i - 1;
// 未找到 target ，返回 -1
if (j === -1 || nums[j] !== target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "TS"
```typescript title="binary_search_edge.ts"
/* 二分查找最右一个 target */
function binarySearchRightEdge(nums: Array<number>, target: number): number {
// 转化为查找最左一个 target + 1
const i = binarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
const j = i - 1;
// 未找到 target ，返回 -1
if (j === -1 || nums[j] !== target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "Dart"
```dart title="binary_search_edge.dart"
/* 二分查找最右一个 target */
int binarySearchRightEdge(List<int> nums, int target) {
// 转化为查找最左一个 target + 1
int i = binarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
int j = i - 1;
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "Rust"
```rust title="binary_search_edge.rs"
/* 二分查找最右一个 target */
fn binary_search_right_edge(nums: &[i32], target: i32) -> i32 {
// 转化为查找最左一个 target + 1
let i = binary_search_insertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
let j = i - 1;
// 未找到 target ，返回 -1
if j == -1 || nums[j as usize] != target {
return -1;
}
// 找到 target ，返回索引 j
j
}
```
=== "C"
```c title="binary_search_edge.c"
/* 二分查找最右一个 target */
int binarySearchRightEdge(int *nums, int numSize, int target) {
// 转化为查找最左一个 target + 1
int i = binarySearchInsertion(nums, numSize, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
int j = i - 1;
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
```
=== "Kotlin"
```kotlin title="binary_search_edge.kt"
/* 二分查找最右一个 target */
fun binarySearchRightEdge(nums: IntArray, target: Int): Int {
// 转化为查找最左一个 target + 1
val i = binarySearchInsertion(nums, target + 1)
// j 指向最右一个 target ，i 指向首个大于 target 的元素
val j = i - 1
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1
}
// 找到 target ，返回索引 j
return j
}
```
=== "Ruby"
```ruby title="binary_search_edge.rb"
### 二分查找最右一个 target ###
def binary_search_right_edge(nums, target)
# 转化为查找最左一个 target + 1
i = binary_search_insertion(nums, target + 1)
# j 指向最右一个 target ，i 指向首个大于 target 的元素
j = i - 1
# 未找到 target ，返回 -1
return -1 if j == -1 || nums[j] != target
j # 找到 target ，返回索引 j
end
```
=== "Zig"
```zig title="binary_search_edge.zig"
[class]{}-[func]{binarySearchRightEdge}
```
??? pythontutor "Code Visualization"
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E6%8F%92%E5%85%A5%E7%82%B9%EF%BC%88%E5%AD%98%E5%9C%A8%E9%87%8D%E5%A4%8D%E5%85%83%E7%B4%A0%EF%BC%89%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E9%97%AD%E5%8C%BA%E9%97%B4%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%AD%E7%82%B9%E7%B4%A2%E5%BC%95%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bm%2B1,%20j%5D%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m-1%5D%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20%E9%A6%96%E4%B8%AA%E5%B0%8F%E4%BA%8E%20target%20%E7%9A%84%E5%85%83%E7%B4%A0%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m-1%5D%20%E4%B8%AD%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E6%8F%92%E5%85%A5%E7%82%B9%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_right_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E6%9C%80%E5%8F%B3%E4%B8%80%E4%B8%AA%20target%22%22%22%0A%20%20%20%20%23%20%E8%BD%AC%E5%8C%96%E4%B8%BA%E6%9F%A5%E6%89%BE%E6%9C%80%E5%B7%A6%E4%B8%80%E4%B8%AA%20target%20%2B%201%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%20%2B%201%29%0A%20%20%20%20%23%20j%20%E6%8C%87%E5%90%91%E6%9C%80%E5%8F%B3%E4%B8%80%E4%B8%AA%20target%20%EF%BC%8Ci%20%E6%8C%87%E5%90%91%E9%A6%96%E4%B8%AA%E5%A4%A7%E4%BA%8E%20target%20%E7%9A%84%E5%85%83%E7%B4%A0%0A%20%20%20%20j%20%3D%20i%20-%201%0A%20%20%20%20%23%20%E6%9C%AA%E6%89%BE%E5%88%B0%20target%20%EF%BC%8C%E8%BF%94%E5%9B%9E%20-1%0A%20%20%20%20if%20j%20%3D%3D%20-1%20or%20nums%5Bj%5D%20!%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20%E6%89%BE%E5%88%B0%20target%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E7%B4%A2%E5%BC%95%20j%0A%20%20%20%20return%20j%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%8C%85%E5%90%AB%E9%87%8D%E5%A4%8D%E5%85%83%E7%B4%A0%E7%9A%84%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%E5%B7%A6%E8%BE%B9%E7%95%8C%E5%92%8C%E5%8F%B3%E8%BE%B9%E7%95%8C%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_right_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22%E6%9C%80%E5%8F%B3%E4%B8%80%E4%B8%AA%E5%85%83%E7%B4%A0%20%7Btarget%7D%20%E7%9A%84%E7%B4%A2%E5%BC%95%E4%B8%BA%20%7Bindex%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
### 2. &nbsp; Transforming 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.
Thus, as shown in the Figure 10-8 , we can construct an element that does not exist in the array, to search for the left and right boundaries.
- To find the leftmost `target`: it can be transformed into searching for `target - 0.5`, and return the pointer $i$.
- To find the rightmost `target`: it can be transformed into searching for `target + 0.5`, and return the pointer $j$.
![Transforming the search for boundaries into the search for an element](binary_search_edge.assets/binary_search_edge_by_element.png){ class="animation-figure" }
<p align="center"> Figure 10-8 &nbsp; Transforming the search for boundaries into the search for an element </p>
The code is omitted here, but two points are worth noting.
- 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).