The previous slide established that the RAM model allows us to quantify the cost of an algorithm step by counting atomic operations. However, precisely counting every operation for an entire algorithm is highly tedious and dependent on minute implementation details, making it unsuitable for general analysis.
- Instead of absolute counts, Algorithm Analysis seeks a measure that is hardware-independent and focuses on the relationship between input size and execution time: "How does performance scale as the input size $n$ grows?"
- The Running Time Cost $f(n)$ is a function representing the total number of basic operations for an input of size $n$, measured in abstract time units.
- The primary interest is in the Order of Growth—the pattern by which $f(n)$ increases as $n$ approaches infinity (the asymptotic view).
- To find this, we ignore constant factors (e.g., the $k$ in $k \cdot n$) as they represent minor hardware differences.
- We also ignore lower-order terms (e.g., in $n^2 + 5n + 100$), because the highest power term dominates performance for large $n$.
- This simplification allows us to express complexity using $O(...)$ notation, which captures the dominant term. For example, Linear Search is $O(n)$, meaning its cost grows linearly with $n$.
Key Takeaways
This slide introduces the core concepts of algorithm analysis, focusing on how an algorithm's performance scales with input size $n$ rather than its exact operation count on specific hardware.
Code Implementation (Python)
1# Calculates the sum of integers from 1 to n.
2def calculate_sum(n):
3 # 1. Initialization (Cost: Constant)
4 current_sum = 0
5
6 # 2. Loop Execution (Cost: Directly tied to n)
7 # The body of this loop runs exactly n times.
8 for i in range(1, n + 1):
9 # Inside the loop, a constant number of operations occur
10 current_sum += i
11
12 # 3. Return (Cost: Constant)
13 return current_sum
14
15# If n=100, f(100) is approximately proportional to 100.
Knowledge Check
When analyzing the running time cost $f(n)$ for very large inputs $n$, we typically focus on constant factors (like initialization cost) rather than the dominant term.