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docs-en/chapter_computational_complexity/time_complexity.md
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@ -2326,7 +2326,7 @@ The following image and code simulate the "halving each round" process, with a t
=== "Python"
```python title="time_complexity.py"
def logarithmic(n: float) -> int:
def logarithmic(n: int) -> int:
"""对数阶（循环实现）"""
count = 0
while n > 1:
@ -2339,7 +2339,7 @@ The following image and code simulate the "halving each round" process, with a t
```cpp title="time_complexity.cpp"
/* 对数阶（循环实现） */
int logarithmic(float n) {
int logarithmic(int n) {
int count = 0;
while (n > 1) {
n = n / 2;
@ -2353,7 +2353,7 @@ The following image and code simulate the "halving each round" process, with a t
```java title="time_complexity.java"
/* 对数阶（循环实现） */
int logarithmic(float n) {
int logarithmic(int n) {
int count = 0;
while (n > 1) {
n = n / 2;
@ -2367,7 +2367,7 @@ The following image and code simulate the "halving each round" process, with a t
```csharp title="time_complexity.cs"
/* 对数阶（循环实现） */
int Logarithmic(float n) {
int Logarithmic(int n) {
int count = 0;
while (n > 1) {
n /= 2;
@ -2381,7 +2381,7 @@ The following image and code simulate the "halving each round" process, with a t
```go title="time_complexity.go"
/* 对数阶（循环实现）*/
func logarithmic(n float64) int {
func logarithmic(n int) int {
count := 0
for n > 1 {
n = n / 2
@ -2395,7 +2395,7 @@ The following image and code simulate the "halving each round" process, with a t
```swift title="time_complexity.swift"
/* 对数阶（循环实现） */
func logarithmic(n: Double) -> Int {
func logarithmic(n: Int) -> Int {
var count = 0
var n = n
while n > 1 {
@ -2438,10 +2438,10 @@ The following image and code simulate the "halving each round" process, with a t
```dart title="time_complexity.dart"
/* 对数阶（循环实现） */
int logarithmic(num n) {
int logarithmic(int n) {
int count = 0;
while (n > 1) {
n = n / 2;
n = n ~/ 2;
count++;
}
return count;
@ -2452,10 +2452,10 @@ The following image and code simulate the "halving each round" process, with a t
```rust title="time_complexity.rs"
/* 对数阶（循环实现） */
fn logarithmic(mut n: f32) -> i32 {
fn logarithmic(mut n: i32) -> i32 {
let mut count = 0;
while n > 1.0 {
n = n / 2.0;
while n > 1 {
n = n / 2;
count += 1;
}
count
@ -2466,7 +2466,7 @@ The following image and code simulate the "halving each round" process, with a t
```c title="time_complexity.c"
/* 对数阶（循环实现） */
int logarithmic(float n) {
int logarithmic(int n) {
int count = 0;
while (n > 1) {
n = n / 2;
@ -2480,7 +2480,7 @@ The following image and code simulate the "halving each round" process, with a t
```zig title="time_complexity.zig"
// 对数阶（循环实现）
fn logarithmic(n: f32) i32 {
fn logarithmic(n: i32) i32 {
var count: i32 = 0;
var n_var = n;
while (n_var > 1)
@ -2494,8 +2494,8 @@ The following image and code simulate the "halving each round" process, with a t
??? pythontutor "Code Visualization"
<div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20float%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&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%20logarithmic%28n%3A%20float%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
<div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&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%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
![Logarithmic Order Time Complexity](time_complexity.assets/time_complexity_logarithmic.png){ class="animation-figure" }
@ -2506,7 +2506,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
=== "Python"
```python title="time_complexity.py"
def log_recur(n: float) -> int:
def log_recur(n: int) -> int:
"""对数阶（递归实现）"""
if n <= 1:
return 0
@ -2517,7 +2517,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```cpp title="time_complexity.cpp"
/* 对数阶（递归实现） */
int logRecur(float n) {
int logRecur(int n) {
if (n <= 1)
return 0;
return logRecur(n / 2) + 1;
@ -2528,7 +2528,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```java title="time_complexity.java"
/* 对数阶（递归实现） */
int logRecur(float n) {
int logRecur(int n) {
if (n <= 1)
return 0;
return logRecur(n / 2) + 1;
@ -2539,7 +2539,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```csharp title="time_complexity.cs"
/* 对数阶（递归实现） */
int LogRecur(float n) {
int LogRecur(int n) {
if (n <= 1) return 0;
return LogRecur(n / 2) + 1;
}
@ -2549,7 +2549,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```go title="time_complexity.go"
/* 对数阶（递归实现）*/
func logRecur(n float64) int {
func logRecur(n int) int {
if n <= 1 {
return 0
}
@ -2561,7 +2561,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```swift title="time_complexity.swift"
/* 对数阶（递归实现） */
func logRecur(n: Double) -> Int {
func logRecur(n: Int) -> Int {
if n <= 1 {
return 0
}
@ -2593,9 +2593,9 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```dart title="time_complexity.dart"
/* 对数阶（递归实现） */
int logRecur(num n) {
int logRecur(int n) {
if (n <= 1) return 0;
return logRecur(n / 2) + 1;
return logRecur(n ~/ 2) + 1;
}
```
@ -2603,11 +2603,11 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```rust title="time_complexity.rs"
/* 对数阶（递归实现） */
fn log_recur(n: f32) -> i32 {
if n <= 1.0 {
fn log_recur(n: i32) -> i32 {
if n <= 1 {
return 0;
}
log_recur(n / 2.0) + 1
log_recur(n / 2) + 1
}
```
@ -2615,7 +2615,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```c title="time_complexity.c"
/* 对数阶（递归实现） */
int logRecur(float n) {
int logRecur(int n) {
if (n <= 1)
return 0;
return logRecur(n / 2) + 1;
@ -2626,7 +2626,7 @@ Like exponential order, logarithmic order also frequently appears in recursive f
```zig title="time_complexity.zig"
// 对数阶（递归实现）
fn logRecur(n: f32) i32 {
fn logRecur(n: i32) i32 {
if (n <= 1) return 0;
return logRecur(n / 2) + 1;
}
@ -2634,8 +2634,8 @@ Like exponential order, logarithmic order also frequently appears in recursive f
??? pythontutor "Code Visualization"
<div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20log_recur%28n%3A%20float%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20return%20log_recur%28n%20/%202%29%20%2B%201%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20log_recur%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&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%20log_recur%28n%3A%20float%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20return%20log_recur%28n%20/%202%29%20%2B%201%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20log_recur%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
<div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20return%20log_recur%28n%20/%202%29%20%2B%201%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20log_recur%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&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%20log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20return%20log_recur%28n%20/%202%29%20%2B%201%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20log_recur%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
Logarithmic order is typical in algorithms based on the divide-and-conquer strategy, embodying the "split into many" and "simplify complex problems" approach. It's slow-growing and is the most ideal time complexity after constant order.
@ -2656,7 +2656,7 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
=== "Python"
```python title="time_complexity.py"
def linear_log_recur(n: float) -> int:
def linear_log_recur(n: int) -> int:
"""线性对数阶"""
if n <= 1:
return 1
@ -2670,7 +2670,7 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```cpp title="time_complexity.cpp"
/* 线性对数阶 */
int linearLogRecur(float n) {
int linearLogRecur(int n) {
if (n <= 1)
return 1;
int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
@ -2685,7 +2685,7 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```java title="time_complexity.java"
/* 线性对数阶 */
int linearLogRecur(float n) {
int linearLogRecur(int n) {
if (n <= 1)
return 1;
int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
@ -2700,7 +2700,7 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```csharp title="time_complexity.cs"
/* 线性对数阶 */
int LinearLogRecur(float n) {
int LinearLogRecur(int n) {
if (n <= 1) return 1;
int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);
for (int i = 0; i < n; i++) {
@ -2714,12 +2714,12 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```go title="time_complexity.go"
/* 线性对数阶 */
func linearLogRecur(n float64) int {
func linearLogRecur(n int) int {
if n <= 1 {
return 1
}
count := linearLogRecur(n/2) + linearLogRecur(n/2)
for i := 0.0; i < n; i++ {
for i := 0; i < n; i++ {
count++
}
return count
@ -2730,7 +2730,7 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```swift title="time_complexity.swift"
/* 线性对数阶 */
func linearLogRecur(n: Double) -> Int {
func linearLogRecur(n: Int) -> Int {
if n <= 1 {
return 1
}
@ -2774,9 +2774,9 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```dart title="time_complexity.dart"
/* 线性对数阶 */
int linearLogRecur(num n) {
int linearLogRecur(int n) {
if (n <= 1) return 1;
int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);
for (var i = 0; i < n; i++) {
count++;
}
@ -2788,11 +2788,11 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```rust title="time_complexity.rs"
/* 线性对数阶 */
fn linear_log_recur(n: f32) -> i32 {
if n <= 1.0 {
fn linear_log_recur(n: i32) -> i32 {
if n <= 1 {
return 1;
}
let mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);
let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);
for _ in 0..n as i32 {
count += 1;
}
@ -2804,7 +2804,7 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```c title="time_complexity.c"
/* 线性对数阶 */
int linearLogRecur(float n) {
int linearLogRecur(int n) {
if (n <= 1)
return 1;
int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
@ -2819,10 +2819,10 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
```zig title="time_complexity.zig"
// 线性对数阶
fn linearLogRecur(n: f32) i32 {
fn linearLogRecur(n: i32) i32 {
if (n <= 1) return 1;
var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);
var i: f32 = 0;
var i: i32 = 0;
while (i < n) : (i += 1) {
count += 1;
}
@ -2832,8 +2832,8 @@ Linear-logarithmic order often appears in nested loops, with the complexities of
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The image below demonstrates how linear-logarithmic order is generated. Each level of a binary tree has $n$ operations, and the tree has $\log_2 n + 1$ levels, resulting in a time complexity of $O(n \log n)$.