Having established the rapid reduction of the search space which yields the $O(\log_2(n))$ complexity of Binary Search, we must now directly compare this growth rate to the $O(n)$ complexity of Linear Search.

  • This comparison highlights why Big O notation ignores constants and focuses purely on the scale of growth.
  • Consider two hypothetical implementations where the running time functions are known:
    • Algorithm 1 (Linear): $f(n) = 3n + 1$
    • Algorithm 2 (Logarithmic): $g(n) = 8\log_2(n) + 10$
  • For very small input sizes $n$, Algorithm 1 might actually execute faster (e.g., for $n=1$, $f(1)=4$ while $g(1)=10$). However, this is irrelevant for asymptotic analysis.
  • Why the Growth Rate Matters: The constants (like 3, 1, 8, 10) represent overhead (setup time, hardware speed, etc.). We focus only on the term that grows fastest as $n \rightarrow \infty$.
  • The Crossover Point: Even though Algorithm 2 has larger constants, its superior order of growth means it will dominate for all sufficiently large values of $n$. This is the core principle of using $O(\ldots)$.

Key Takeaway

Asymptotic analysis ($O$-notation) prioritizes the long-term growth rate of an algorithm over constant factors and initial overhead, as this is what determines performance on large-scale problems.