1.5.1 Time Complexity #1

Summary

The video explains how to estimate an algorithm’s time complexity by counting how many times its main statement executes and expressing the growth using Big-O notation.

Examples Covered

• A basic loop from 1 to n: The body runs about n times, so its complexity is O(n). The loop condition may be checked one extra time, but this does not change the asymptotic order.
• A loop counting down: Counting from n toward zero still takes a number of steps proportional to n, so the complexity is O(n).
• A loop with a fixed increment greater than 1: If the counter advances by 2, 20, or another fixed constant, the body runs roughly n divided by that constant times. Since fixed factors are ignored in asymptotic analysis, the complexity remains O(n).
• Two independent nested loops: If each loop runs about n times, the inner statement executes about n × n times, giving O(_n_²).
• A nested loop whose inner limit depends on the outer counter: For each outer-loop value i, the inner loop runs i times. The total is a sum such as 0 + 1 + 2 + … + (n − 1), which grows proportionally to _n_². The complexity is therefore O(_n_²), even though the loop structure differs from the previous example.
• A loop that accumulates values until a threshold is exceeded: The loop adds 1, then 2, then 3, and so on to a running total p. After k iterations, the total is approximately k(k + 1)/2. The loop stops when this total exceeds n, so k grows roughly like √n. Its complexity is O(√n).

Method for Analyzing Loops

1. Focus on how many times the loop’s main statement executes.
2. For nested loops, count the inner statement across all outer-loop iterations.
3. If the loop count is not immediately clear, trace the relevant variables for several iterations.
4. Write the resulting count as a function of n, using a sum or equation when needed.
5. Keep the dominant growth term; ignore fixed multipliers and lower-order terms.

Speakers and Sources

• Abdul Bari — presenter and instructor.
• No other speakers or sources are featured in the subtitles.