The runtime of an algorithm depends on many things:
The size of the input
The content/structure of the input
They type of computer you are using (huge work station vs laptop)
The amount of memory the computer has (space available)
How the program was implemented
The programming language used.
The biggest factor we are considering is how the program was implemented, most efficient implementation, any algorithm with exponential runtime is essentially said to be an impractical algorithm. and that is where Big O Notation comes into analyzing algorithms.
The Big-O of an algorithm is determined by analyzing the the algorithm and determining how many time steps the algorithm will require, dependent on the input size of the problem. “Simple” operations such as Arithmetic, executing IF statements, return statement etc take 1 timestep. Executing for and while loops takes n steps, 1 per iteration. We count steps as a way to mathematically express the number of steps required to solve the problem. Since the exact number of steps does not matter using Big O Notation we can ignore “Unnecessary Details” such as constant steps or single operations that don’t have a big effect in runtime and focus on the dominant process. I also learned that to most accurately analyze an algorithm’s efficiency we focus on the “worst possible case” to do this we look at the worst possible inputs. By knowing the big O of we can guarantee that no matter the input size n, the time complexity to complete the process will always take at most the worst possible time. For Example consider Selection sort its best, worst an average case iwill always grow O(n * n) to the input size because no matter what it always has to check the entire list, so it is considered a stable sort that is very predictable.
I’ve learned that the goal of Big O Notation is to show us how the algorithm execution time grows with respect to the size of the problem(input). Professor Heap told me to ask myself, does the runtime of a program grow proportional to the size of the input?
Here is a table of the Big O of sorting algorithms with discussed in class and their worst, average and best case performances, but remember there are other complexities such as O(N^2), O(N^3), O(N * N), O(N log N), O(N!) or even O(∞) (which will require an infinite amount of time and that’s where we reach computational limits).
next time I will tell you about proving Big O
Until next time, yours truly, – CodeShark