Big O Notation

Big O Notation

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Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. (Sestoft, p. 40) Its purpose is to characterize a function's behavior for very large (or very small) inputs in a simple but rigorous way that enables comparison to other functions. More precisely, the symbol O is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler, function. It has two main areas of application in mathematics, it is usually used to characterize the residual term of a truncated infinite series, especially an asymptotic series, and in computer science, it is useful in the analysis of the complexity of algorithms.
Big-O Notation is short for order of growth notation. It is defined as given two function t(n) and g(n), we say that t(n) = O(g(n)) if there exist a positive constant A and some number N such that t(n) <= A g(n) for all n > n T(n) means the running time of the algorithm on a problem of size N. (Sestoft, p. 105) Big-O basically means that t(n) asymptotically (only for large n) grows not faster than g(n) (give or take a constant factor). The order of growth of t(n) is not larger than g(n).
When put in terms of order of growth, f = O(g) is like "f <= g." When two searching algorithms are compared, both the worst and average time is O(n) for a sequential search and O(log2 n) for the binary search. (Sestoft, p. 110)
Big-O notation is concerned with what happens for very large values of N, therefore only the largest term in a polynomial is needed. All smaller terms are dropped.
For example, the number of operations in some sorts is N2 - N. For large values of N, the single N term is insignificant compared to N2, therefore one of these sorts would be described as an O(N2) algorithm. (Sestoft, p. 80)
Similarly, constant multipliers are ignored. So a O(4*N) algorithm is equivalent to O(N), which is how it should be written. Ultimately you want to pay attention to these multipliers in determining the performance, but for the first round of analysis using Big-O, you simply ignore constant factors.
Complexity analysis can be very useful, but there are problems with it too. Many algorithms are simply too hard to analyze mathematically.

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There may not be sufficient information to know what the most important "average" case really is, therefore analysis is impossible. Also there may be unknown constants. For example, both walking and traveling at the speed of light have a time as function of distance big oh complexity of O(N). Although they have the same big-oh characteristics, one is rather faster than the other. Big-oh analysis only tells you how it grows with the size of the problem, not how efficient it is. Last but not least there may be small data sets. If there are no large amounts of data, algorithm efficiency may not be important.
In an asymptotic analysis, we care more about the order of magnitude of a function rather than the actual value of a function itself. In terms of the abstract time of an algorithm, this should make some sense. “After all, we call it "abstract time" because we use "abstract" operations such as "number of math operations" in f (n), or "number of comparisons" in f (n). In addition, we don't know exactly how long each operation might actually take on a certain computer.” (Sestoft, p. 95) The order of magnitude appeals to our sense that n2 is a faster-growing function than a linear function like n. To describe the order of magnitude of a function, we use Big-O notation. If we had an algorithm that did 7n4 +35n3 - 19n2 + 3 operations, its big-O notation would be O(n4). (Sestoft, p. 89) If we had an algorithm that did 2n + 5 operations, the big-O notation would be O(n). A function g(n) is in the class of functions of the order f (n) if, and only if, we can multiply f (n) by some constant c, and ignore all the n below some constant n0, and have the function c*f (n) be greater (for each n > n0) than g(n). There are a few basic big-Os that are used frequently such as:
1. O(1) - constant time
2. O(logn) - logarithmic time
3. O(n) - linear time
4. O(nlogn)
5. O(nc) - polynomial
6. O(cn) - exponential
7. O(n!) - factorial
When discussing the efficiency of an algorithm, there are three cases that need to be considered, the best case, the worst case, and the average case. The best case is how the algorithm will work on the best possible input. The worst case is how the algorithm runs on the worst possible input. And the average case is how it runs on most inputs. When comparing algorithms we very rarely use the best case, often use the average case and sometimes use the worst case.
This is how we measure the efficiency of an algorithm


Big-O Notation and Algorithm Analysis Article: Algorithmic Efficiency and Big-O notation
Sestoft, P (2002). Java Precisely. Cambridge, Mass: Massachusetts
Institute of Technology.
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