Think of it as O(n*log(n)), i.e. “doing log(n) work n times”. For example, searching for an element in a sorted list of length n is O(log(n)). Searching for the element in n different sorted lists, each of length n is O(n*log(n)). Remember that O(n) is defined relative to some real quantity n. This might be … Read more
The big O notation, and its relatives, the big Theta, the big Omega, the small o and the small omega are ways of saying something about how a function behaves at a limit point (for example, when approaching infinity, but also when approaching 0, etc.) without saying much else about the function. They are commonly … Read more
A useful way to approach these types of problems is by thinking of the recursion tree. The two features of a recursive function to identify are: The tree depth (how many total return statements will be executed until the base case) The tree breadth (how many total recursive function calls will be made) Our recurrence … Read more
Asymptotic complexity (which is what both big-O and big-Theta represent) completely ignores the constant factors involved – it’s only intended to give an indication of how running time will change as the size of the input gets larger. So it’s certainly possible that an Θ(n) algorithm can take longer than an Θ(n2) one for some … Read more
Yes, nested loops are one way to quickly get a big O notation. Typically (but not always) one loop nested in another will cause O(n²). Think about it, the inner loop is executed i times, for each value of i. The outer loop is executed n times. thus you see a pattern of execution like … Read more
Yep, it’s still O(n^2), it has a smaller constant factor, but that doesn’t affect O notation.
Big O is the upper bound, while Omega is the lower bound. Theta requires both Big O and Omega, so that’s why it’s referred to as a tight bound (it must be both the upper and lower bound). For example, an algorithm taking Omega(n log n) takes at least n log n time, but has … Read more
You’re going to want to read up on Order of complexity. http://en.wikipedia.org/wiki/Big_O_notation In short, O(1) means that it takes a constant time, like 14 nanoseconds, or three minutes no matter the amount of data in the set. O(n) means it takes an amount of time linear with the size of the set, so a set … Read more