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@@ -11,33 +11,287 @@ comments: true
下面兩段程式碼都計算 $1 + 2 + \dots + n$（設 $n \ge 1$）。請把 `n` 設為 4，
按照程式實際執行的順序回答問題，然後比較兩種寫法的效率。
```python
def sum_iter(n):
s = 0
for i in range(1, n + 1):
s += i
return s
=== "Python"
def sum_recur(n):
if n == 1:
return 1
return n + sum_recur(n - 1)
```
```python title="complexity_exercises.py"
def sum_iter(n: int) -> int:
"""迭代求和"""
res = 0
for i in range(1, n + 1):
res += i
return res
def sum_recur(n: int) -> int:
"""遞迴求和"""
if n == 1:
return 1
return n + sum_recur(n - 1)
```
=== "C++"
```cpp title="complexity_exercises.cpp"
/* 迭代求和 */
int sumIter(int n) {
int res = 0;
for (int i = 1; i <= n; ++i) {
res += i;
}
return res;
}
/* 遞迴求和 */
int sumRecur(int n) {
if (n == 1) {
return 1;
}
return n + sumRecur(n - 1);
}
```
=== "Java"
```java title="complexity_exercises.java"
/* 迭代求和 */
int sumIter(int n) {
int res = 0;
for (int i = 1; i <= n; i++) {
res += i;
}
return res;
}
/* 遞迴求和 */
int sumRecur(int n) {
if (n == 1) {
return 1;
}
return n + sumRecur(n - 1);
}
```
=== "C#"
```csharp title="complexity_exercises.cs"
/* 迭代求和 */
int SumIter(int n) {
int res = 0;
for (int i = 1; i <= n; i++) {
res += i;
}
return res;
}
/* 遞迴求和 */
int SumRecur(int n) {
if (n == 1) {
return 1;
}
return n + SumRecur(n - 1);
}
```
=== "Go"
```go title="complexity_exercises.go"
/* 迭代求和 */
func sumIter(n int) int {
res := 0
for i := 1; i <= n; i++ {
res += i
}
return res
}
/* 遞迴求和 */
func sumRecur(n int) int {
if n == 1 {
return 1
}
return n + sumRecur(n-1)
}
```
=== "Swift"
```swift title="complexity_exercises.swift"
/* 迭代求和 */
func sumIter(n: Int) -> Int {
var res = 0
for i in 1 ... n {
res += i
}
return res
}
/* 遞迴求和 */
func sumRecur(n: Int) -> Int {
if n == 1 {
return 1
}
return n + sumRecur(n: n - 1)
}
```
=== "JS"
```javascript title="complexity_exercises.js"
/* 迭代求和 */
function sumIter(n) {
let res = 0;
for (let i = 1; i <= n; i++) {
res += i;
}
return res;
}
/* 遞迴求和 */
function sumRecur(n) {
if (n === 1) {
return 1;
}
return n + sumRecur(n - 1);
}
```
=== "TS"
```typescript title="complexity_exercises.ts"
/* 迭代求和 */
function sumIter(n: number): number {
let res = 0;
for (let i = 1; i <= n; i++) {
res += i;
}
return res;
}
/* 遞迴求和 */
function sumRecur(n: number): number {
if (n === 1) {
return 1;
}
return n + sumRecur(n - 1);
}
```
=== "Dart"
```dart title="complexity_exercises.dart"
/* 迭代求和 */
int sumIter(int n) {
int res = 0;
for (int i = 1; i <= n; i++) {
res += i;
}
return res;
}
/* 遞迴求和 */
int sumRecur(int n) {
if (n == 1) {
return 1;
}
return n + sumRecur(n - 1);
}
```
=== "Rust"
```rust title="complexity_exercises.rs"
/* 迭代求和 */
fn sum_iter(n: i32) -> i32 {
let mut res = 0;
for i in 1..=n {
res += i;
}
res
}
/* 遞迴求和 */
fn sum_recur(n: i32) -> i32 {
if n == 1 {
return 1;
}
n + sum_recur(n - 1)
}
```
=== "C"
```c title="complexity_exercises.c"
/* 迭代求和 */
int sumIter(int n) {
int res = 0;
for (int i = 1; i <= n; i++) {
res += i;
}
return res;
}
/* 遞迴求和 */
int sumRecur(int n) {
if (n == 1) {
return 1;
}
return n + sumRecur(n - 1);
}
```
=== "Kotlin"
```kotlin title="complexity_exercises.kt"
/* 迭代求和 */
fun sumIter(n: Int): Int {
var res = 0
for (i in 1..n) {
res += i
}
return res
}
/* 遞迴求和 */
fun sumRecur(n: Int): Int {
if (n == 1) {
return 1
}
return n + sumRecur(n - 1)
}
```
=== "Ruby"
```ruby title="complexity_exercises.rb"
### 迭代求和 ###
def sum_iter(n)
res = 0
for i in 1..n
res += i
end
res
end
### 遞迴求和 ###
def sum_recur(n)
return 1 if n == 1
n + sum_recur(n - 1)
end
```
<!-- numbered-subquestions -->
1. 執行 `sum_iter(4)` 時，每輪迴圈結束後，變數 `s` 的值分別是多少？
2. 執行 `sum_recur(4)` 時，會依次呼叫哪些函式？從最深的一層開始返回時，結果怎樣得到？
1. 輸入 `n = 4` 執行迭代函式時，每輪迴圈結束後，累加變數 `res` 的值分別是多少？
2. 輸入 `n = 4` 執行遞迴函式時，參數 `n` 會依次取哪些值？從最深的一層開始返回時，結果怎樣得到？
3. 兩種寫法的時間複雜度和空間複雜度分別是多少？結合第 1、2 問的執行過程說明理由。
??? success "參考答案"
1. 迴圈變數 `i` 依次為 `1、2、3、4`，每輪結束後，`s` 依次變為
`1、3、6、10`，所以 `sum_iter(4)` 返回 10。
1. 迴圈變數 `i` 依次為 `1、2、3、4`，每輪結束後，`res` 依次變為
`1、3、6、10`，所以迭代函式返回 10。
2. 函式依次呼叫
`sum_recur(4) → sum_recur(3) → sum_recur(2) → sum_recur(1)`。
`sum_recur(1)` 返回 1，隨後各層依次得到 `2 + 1 = 3`、`3 + 3 = 6`、`4 + 6 = 10`。
2. 參數 `n` 依次為 `4 → 3 → 2 → 1`。
最深一層返回 1，隨後各層依次得到 `2 + 1 = 3`、`3 + 3 = 6`、`4 + 6 = 10`。
在最深處，4 次函式呼叫都尚未結束。
3. 兩段程式碼都進行與 $n$ 成正比的迴圈或呼叫，因此時間複雜度均為 $O(n)$ 。
@@ -51,22 +305,414 @@ def sum_recur(n):
以下三個程式碼片段的輸入均為正整數 $n$ 。請按時間複雜度從低到高排序，並寫出各自的複雜度。
```python
# 片段一
s = 0
for i in range(n):
s += i
=== "Python"
# 片段二
s = 0
for i in range(n):
for j in range(i, n):
s += j
```python title="complexity_exercises.py"
def linear_loop(n: int) -> int:
"""線性階迴圈"""
res = 0
for i in range(n):
res += i
return res
# 片段三
while n > 1:
n = n // 2
```
def quadratic_loop(n: int) -> int:
"""平方階迴圈"""
res = 0
for i in range(n):
for j in range(i, n):
res += j
return res
def logarithmic_loop(n: int) -> int:
"""對數階迴圈"""
while n > 1:
n //= 2
return n
```
=== "C++"
```cpp title="complexity_exercises.cpp"
/* 線性階迴圈 */
int linearLoop(int n) {
int res = 0;
for (int i = 0; i < n; ++i) {
res += i;
}
return res;
}
/* 平方階迴圈 */
int quadraticLoop(int n) {
int res = 0;
for (int i = 0; i < n; ++i) {
for (int j = i; j < n; ++j) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
int logarithmicLoop(int n) {
while (n > 1) {
n /= 2;
}
return n;
}
```
=== "Java"
```java title="complexity_exercises.java"
/* 線性階迴圈 */
int linearLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
res += i;
}
return res;
}
/* 平方階迴圈 */
int quadraticLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
for (int j = i; j < n; j++) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
int logarithmicLoop(int n) {
while (n > 1) {
n /= 2;
}
return n;
}
```
=== "C#"
```csharp title="complexity_exercises.cs"
/* 線性階迴圈 */
int LinearLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
res += i;
}
return res;
}
/* 平方階迴圈 */
int QuadraticLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
for (int j = i; j < n; j++) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
int LogarithmicLoop(int n) {
while (n > 1) {
n /= 2;
}
return n;
}
```
=== "Go"
```go title="complexity_exercises.go"
/* 線性階迴圈 */
func linearLoop(n int) int {
res := 0
for i := 0; i < n; i++ {
res += i
}
return res
}
/* 平方階迴圈 */
func quadraticLoop(n int) int {
res := 0
for i := 0; i < n; i++ {
for j := i; j < n; j++ {
res += j
}
}
return res
}
/* 對數階迴圈 */
func logarithmicLoop(n int) int {
for n > 1 {
n /= 2
}
return n
}
```
=== "Swift"
```swift title="complexity_exercises.swift"
/* 線性階迴圈 */
func linearLoop(n: Int) -> Int {
var res = 0
for i in 0 ..< n {
res += i
}
return res
}
/* 平方階迴圈 */
func quadraticLoop(n: Int) -> Int {
var res = 0
for i in 0 ..< n {
for j in i ..< n {
res += j
}
}
return res
}
/* 對數階迴圈 */
func logarithmicLoop(n: Int) -> Int {
var n = n
while n > 1 {
n /= 2
}
return n
}
```
=== "JS"
```javascript title="complexity_exercises.js"
/* 線性階迴圈 */
function linearLoop(n) {
let res = 0;
for (let i = 0; i < n; i++) {
res += i;
}
return res;
}
/* 平方階迴圈 */
function quadraticLoop(n) {
let res = 0;
for (let i = 0; i < n; i++) {
for (let j = i; j < n; j++) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
function logarithmicLoop(n) {
while (n > 1) {
n = Math.floor(n / 2);
}
return n;
}
```
=== "TS"
```typescript title="complexity_exercises.ts"
/* 線性階迴圈 */
function linearLoop(n: number): number {
let res = 0;
for (let i = 0; i < n; i++) {
res += i;
}
return res;
}
/* 平方階迴圈 */
function quadraticLoop(n: number): number {
let res = 0;
for (let i = 0; i < n; i++) {
for (let j = i; j < n; j++) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
function logarithmicLoop(n: number): number {
while (n > 1) {
n = Math.floor(n / 2);
}
return n;
}
```
=== "Dart"
```dart title="complexity_exercises.dart"
/* 線性階迴圈 */
int linearLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
res += i;
}
return res;
}
/* 平方階迴圈 */
int quadraticLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
for (int j = i; j < n; j++) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
int logarithmicLoop(int n) {
while (n > 1) {
n ~/= 2;
}
return n;
}
```
=== "Rust"
```rust title="complexity_exercises.rs"
/* 線性階迴圈 */
fn linear_loop(n: i32) -> i32 {
let mut res = 0;
for i in 0..n {
res += i;
}
res
}
/* 平方階迴圈 */
fn quadratic_loop(n: i32) -> i32 {
let mut res = 0;
for i in 0..n {
for j in i..n {
res += j;
}
}
res
}
/* 對數階迴圈 */
fn logarithmic_loop(mut n: i32) -> i32 {
while n > 1 {
n /= 2;
}
n
}
```
=== "C"
```c title="complexity_exercises.c"
/* 線性階迴圈 */
int linearLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
res += i;
}
return res;
}
/* 平方階迴圈 */
int quadraticLoop(int n) {
int res = 0;
for (int i = 0; i < n; i++) {
for (int j = i; j < n; j++) {
res += j;
}
}
return res;
}
/* 對數階迴圈 */
int logarithmicLoop(int n) {
while (n > 1) {
n /= 2;
}
return n;
}
```
=== "Kotlin"
```kotlin title="complexity_exercises.kt"
/* 線性階迴圈 */
fun linearLoop(n: Int): Int {
var res = 0
for (i in 0 until n) {
res += i
}
return res
}
/* 平方階迴圈 */
fun quadraticLoop(n: Int): Int {
var res = 0
for (i in 0 until n) {
for (j in i until n) {
res += j
}
}
return res
}
/* 對數階迴圈 */
fun logarithmicLoop(n: Int): Int {
var value = n
while (value > 1) {
value /= 2
}
return value
}
```
=== "Ruby"
```ruby title="complexity_exercises.rb"
### 線性階迴圈 ###
def linear_loop(n)
res = 0
for i in 0...n
res += i
end
res
end
### 平方階迴圈 ###
def quadratic_loop(n)
res = 0
for i in 0...n
for j in i...n
res += j
end
end
res
end
### 對數階迴圈 ###
def logarithmic_loop(n)
n /= 2 while n > 1
n
end
```
??? success "參考答案"