1.5.3 Time Complexity of While and if #3 | Highlights and Annotations by Gistr.

This segment presents a comparative analysis of equivalent code snippets implemented using both `for` and `while` loops in C. The speaker demonstrates how to analyze the time complexity of a `while` loop by counting the number of times the loop body executes, emphasizing that the focus is on the order of the function rather than the exact formula. The equivalence of `for` and `while` loops in terms of expressiveness is also highlighted.This segment delves into the analysis of `while` loops where the loop variable is not incremented linearly. The speaker uses an example where the variable is multiplied by 2 in each iteration, demonstrating how to determine the number of iterations by solving a logarithmic equation. The resulting time complexity is shown to be logarithmic (O(log n)), contrasting with the linear time complexity of loops with linear increments. This segment traces the historical evolution of loops from Pascal's structured `for` loop to the more flexible `while` loop in C, highlighting the differences in syntax and how this impacts time complexity analysis. The discussion emphasizes the linear increment nature of early `for` loops and contrasts it with the conditional nature of `while` loops, setting the stage for a deeper dive into time complexity analysis. This segment focuses on the challenges of analyzing `while` loops compared to `for` loops. It explains that the number of iterations in a `while` loop is not immediately apparent and depends on the loop's termination condition. The speaker emphasizes the need for careful study of the loop's condition to determine the number of iterations and, consequently, the time complexity. Analyzing time complexity of loops (while, do-while, for) in C requires examining iteration counts. For loops with linear increments are O(n). While loops depend on their conditions; some are O(log n) (e.g., repeated halving), others O(√n) (e.g., summing series), and some O(n). Conditional statements introduce best-case and worst-case scenarios, affecting overall complexity. This segment tackles a more complex `while` loop where the loop variable's update depends on the values of two variables. The speaker meticulously traces the execution of the loop, showing how the number of iterations relates to the sum of an arithmetic series. The analysis reveals a time complexity of O(√n), demonstrating a more intricate approach to analyzing non-linear loop behavior.