Big O Notation

Step 1: Understanding Big O Notation 📏

Big O notation describes how the runtime of an algorithm grows relative to the size of the input (denoted as nnn) as nnn becomes large. It's used to classify algorithms according to their growth rates, which helps in understanding the scalability of the algorithm.

Step 2: Analyzing the Expression O(n2+n+2)O(n^2 + n + 2)O(n2+n+2) 🧠

Given the expression:

O(n2+n+2)O(n^2 + n + 2)O(n2+n+2)

This expression represents the time complexity of an algorithm where:

• n2n^2n2 corresponds to some part of the algorithm that takes time proportional to the square of the input size.

• nnn corresponds to some part of the algorithm that takes time proportional to the input size.

• 222 corresponds to some constant time operations, independent of the input size.

Step 3: Identifying Growth Rates of Each Term 🚀

Each term in the expression has a different growth rate as nnn increases:

• n2n^2n2: This is a quadratic term. As nnn grows, n2n^2n2 grows much faster than nnn or any constant. For example, if n=10n = 10n=10, then n2=100n^2 = 100n2=100, and if n=100n = 100n=100, then n2=10,000n^2 = 10,000n2=10,000. The rate of growth is proportional to the square of nnn. 📈

• nnn: This is a linear term. It grows proportionally to nnn. If n=10n = 10n=10, then n=10n = 10n=10, and if n=100n = 100n=100, then n=100n = 100n=100. The rate of growth is directly proportional to nnn. ➡️

• 222: This is a constant term. No matter how large nnn gets, this term will always remain 2. It does not grow with nnn at all. 🔢

Step 4: Identifying the Dominant Term 🏆

In Big O notation, we are primarily interested in how the algorithm behaves as nnn becomes very large (approaches infinity). The term with the highest growth rate will dominate the performance of the algorithm, meaning it will have the greatest impact on the runtime as nnn increases.

In the expression n2+n+2n^2 + n + 2n2+n+2:

• n2n^2n2 dominates because it grows faster than both nnn and 222 as nnn increases. 🎯

Step 5: Simplifying the Expression ✂️

Since n2n^2n2 dominates the other terms, we can simplify the time complexity by ignoring the smaller terms (nnn and 222) because their contribution to the overall runtime becomes negligible for large nnn.

Therefore, the expression simplifies to:

O(n2)O(n^2)O(n2)

Step 6: Conclusion ✅

The Big O notation for the time complexity O(n2+n+2)O(n^2 + n + 2)O(n2+n+2) is O(n^2).

Summary of the Process:

1. List all terms in the expression: n2n^2n2, nnn, and 222. 📝

2. Determine the growth rate of each term:

   • n2n^2n2: Quadratic growth. 🚀

   • nnn: Linear growth. ➡️

   • 222: Constant. 🔢

3. Identify the dominant term: The term with the highest growth rate, which is n2n^2n2 in this case. 🏆

4. Simplify by ignoring lower-order terms and constants: Drop nnn and 222 as they have lower growth rates. ✂️

5. Result: The time complexity in Big O notation is O(n2)O(n^2)O(n2). ✅

This means that as the input size nnn increases, the algorithm's runtime grows at a rate proportional to n2n^2n2. 📈

Example 2: O(100n+300)

Step 1: List All Terms 📝

• 100n

• 300 (constant)

Step 2: Determine Growth Rates 🚀

• nnn: Linear growth. ➡️

• 300 Constant, doesn’t grow with nnn. 🔢

Step 3: Identify the Dominant Term 🏆

• 100n dominates because linear growth increases as nnn increases, while the constant 300 doesn’t change.

Step 4: Simplify ✂️

• Drop the constant 300

Result ✅

• The time complexity is O(n)

Example 2: O(3n3+2n2+10n+5)

Step 1: List All Terms 📝

• 3n3

• 2n2

• 10n

• 5(constant)

Step 2: Determine Growth Rates 🚀

• n^3: Cubic growth, fastest-growing term. 📈

• 2n2: Quadratic growth.

• n: Linear growth.

• 5: Constant, doesn’t grow with nnn. 🔢

Step 3: Identify the Dominant Term 🏆

• 3n3 dominates because 3n3 grows faster than the other terms as nnn increases.

Step 4: Simplify ✂️

• Drop the lower-order terms 2n2, 10n, and 5.

Result ✅

• The time complexity is O(n3)