Write Big-O notation, and show proof with steps that show their time complexity function
Write Big-O notation, and show proof with steps that show their time complexity function
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
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Write Big-O notation, and show proof with steps that show their time complexity function
Transcribed Image Text:The image contains a piece of pseudocode for a nested loop structure, often used in programming education to illustrate algorithmic concepts.
1. **Outer Loop:**
- **Initialization:** `i = 1`
- **Condition:** `i <= n`
- **Increment:** `i++`
- This loop will iterate with `i` starting at 1 and continue as long as `i` is less than or equal to `n`.
2. **Inner Loop:**
- **Initialization:** `j = 2i`
- **Condition:** `j <= n`
- **Increment:** `j++`
- This loop will start with `j` set to twice the current value of `i` and continue as long as `j` is less than or equal to `n`.
3. **Body of the Inner Loop:**
- The comment `// 3 assignment instructions` suggests that there are three assignment operations to be carried out inside this loop, although the specific operations are not detailed.
This pseudocode is likely used to discuss concepts such as nested loops, iteration, and complexity analysis in computer science.
![### Algorithm Analysis
#### Code Snippet
```c
for (i = 1; i <= n; i++) {
for (j = 2i; j <= n; j++) {
// 3 assignment instructions
}
}
```
#### Explanation
- **Outer Loop**: Iterates `i` from 1 to `n`.
- **Inner Loop**:
- Starts `j` at `2i` and iterates while `j` is less than or equal to `n`.
- Executes 3 assignment instructions per iteration.
#### Hint
- Initialize `j` as `2*i`. The loop increments `j` until it exceeds `n`.
- The iterations of `j` are calculated as:
- `j = 2i, (2i+1), (2i+2), ..., n`
- Number of iterations for `j`: `(n - 2i) + 1`
#### Calculations
- **Iteration Examples**:
- For `i = 1`: `j = (n - 2(1) + 1) = (n + 1 - 2(1))`
- For `i = 2`: `j = (n + 1 - 2(2))`
- For `i = 3`: `j = (n + 1 - 2(3))`
- Continue this pattern up to `i = n/2`.
#### Function Calculation
\[ f(n) = 3 \times ((n+1) \times n/2 - 2 \times (1+2+3+...+ n/2)) \]
**Expanded Form**:
\[ = 3 \times ((n+1) \times n/2 - 2 \times (n/2 \times (n/2 + 1)/2)) \]
This derivation calculates the total number of assignments made by the loop across all iterations. The formula considers the decreasing number of times the inner loop runs as `i` increases, due to the increase in the starting value of `j`.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc78cb2d6-68a5-41e0-b558-8f8bf6426ef5%2F8b1225f5-14df-4fab-9567-4b3b0a840750%2Fy79ua4u_processed.png&w=3840&q=75)
Transcribed Image Text:### Algorithm Analysis
#### Code Snippet
```c
for (i = 1; i <= n; i++) {
for (j = 2i; j <= n; j++) {
// 3 assignment instructions
}
}
```
#### Explanation
- **Outer Loop**: Iterates `i` from 1 to `n`.
- **Inner Loop**:
- Starts `j` at `2i` and iterates while `j` is less than or equal to `n`.
- Executes 3 assignment instructions per iteration.
#### Hint
- Initialize `j` as `2*i`. The loop increments `j` until it exceeds `n`.
- The iterations of `j` are calculated as:
- `j = 2i, (2i+1), (2i+2), ..., n`
- Number of iterations for `j`: `(n - 2i) + 1`
#### Calculations
- **Iteration Examples**:
- For `i = 1`: `j = (n - 2(1) + 1) = (n + 1 - 2(1))`
- For `i = 2`: `j = (n + 1 - 2(2))`
- For `i = 3`: `j = (n + 1 - 2(3))`
- Continue this pattern up to `i = n/2`.
#### Function Calculation
\[ f(n) = 3 \times ((n+1) \times n/2 - 2 \times (1+2+3+...+ n/2)) \]
**Expanded Form**:
\[ = 3 \times ((n+1) \times n/2 - 2 \times (n/2 \times (n/2 + 1)/2)) \]
This derivation calculates the total number of assignments made by the loop across all iterations. The formula considers the decreasing number of times the inner loop runs as `i` increases, due to the increase in the starting value of `j`.
Expert Solution
Step 1
Here we drive simple method to find time complexity:
we not drive look above. we use hint and drive it.
the time complexity is O(n^2).
Step by step
Solved in 2 steps with 1 images
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