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Unless you've read a book on the Analysis of Algorithms, or have managed to pick up the basics along the way, the "O(N)" (AKA "Big O") notation that sometimes gets tossed around when comparing algorithms might seem obtuse. But the ideas that the notation expresses can keep you out of trouble, so it's worth picking up the basics. Here are enough of the informal basics to get you comfortable when O(N) notation starts getting tossed around informally. Analyzing algorithms is a separate discussion.

O(N) notation is used to express the worst-case order of growth of an algorithm. That is, how the algorithm's worst-case performance changes as the size of the data set it operates on increases.

When being informal, people often conflate general-case with worst-case behavior, though the more formal folk will throw in some additional symbols to distinguish the two. Order of growth characterizes the shape of the growth curve, not its slope. That is, order of growth is one level removed from notions of efficiency. More on this later. Growth can be in terms of speed or space, but unless people say space, they almost always mean speed.

Common Orders of Growth

O(1) is the no-growth curve. An O(1) algorithm's performance is conceptually independent of the size of the data set on which it operates. Array element access is O(1), if you ignore implementation details like virtual memory and page faults. Ignoring the data set entirely and returning undef is also O(1), though this is rarely useful.

O(N) says that the algorithm's performance is directly proportional to the size of the data set being processed. Scanning an array or linked list takes O(N) time. Probing an array is still O(N), even if statistically you only have to scan half the array to find a value. Because computer scientists are only interested in the shape of the growth curve at this level, when you see O(2N) or O(10 + 5N), someone is blending implementation details into the conceptual ones.

Depending on the algorithm used, searching a hash is O(N) in the worst case. Insertion is also O(N) in the worst case, but considerably more efficient in the general case.

O(N+M) is just a way of saying that two data sets are involved, and that their combined size determines performance.

O(N2) says that the algorithm's performance is proportional to the square of the data set size. This happens when the algorithm processes each element of a set, and that processing requires another pass through the set. The infamous Bubble Sort is O(N2).

O(N•M) indicates that two data sets are involved, and the processing of each element of one involves processing the second set. If the two set sizes are roughly equivalent, some people get sloppy and say O(N2) instead. While technically incorrect, O(N2) still conveys useful information.

"I've got this list of regular expressions, and I need to apply all of them to this chunk of text" is potentially O(N•M), depending on the regexes.

O(N3) and beyond are what you would expect. Lots of inner loops.

O(2N) means you have an algorithm with exponential time (or space, if someone says space) behavior. In the 2 case, time or space double for each new element in data set. There's also O(10N), etc. In practice, you don't need to worry about scalability with exponential algorithms, since you can't scale very far unless you have a very big hardware budget.

O(log N) and O(N log N) might seem a bit scary, but they're really not. These generally mean that the algorithm deals with a data set that is iteratively partitioned, like a balanced binary tree. (Unbalanced binary trees are O(N2) to build, and O(N) to probe.) Generally, but not always, log N implies log2N, which means, roughly, the number of times you can partition a set in half, then partition the halves, and so on, while still having non-empty sets. Think powers of 2, but worked backwards.

210 = 1024
log21024 = 10
The key thing to note is that log2N grows slowly. Doubling N has a relatively small effect. Logarithmic curves flatten out nicely.

It takes O(log N) time to probe a balanced binary tree, but building the tree is more expensive. If you're going to be probing a data set a lot, it pays to take the hit on construction to get fast probe time.

Quite often, when an algorithm's growth rate is characterized by some mix of orders, the dominant order is shown, and the rest are dropped. O(N2) might really mean O(N2 + N).

Scalability and Efficiency

An O(1) algorithm scales better than an O(log N) algorithm,
which scales better than an O(N) algorithm,
which scales better than an O(N log N) algorithm,
which scales better than an O(N2) algorithm,
which scales better than an O(2N) algorithm.

This pops right out when you look at a graph of their growth curves.

But scalability isn't efficiency. A well-coded, O(N2) algorithm can outperform a sloppily-coded O(N log N) algorithm, but only for a while. At some point their performance curves will cross. Hopefully, this happens before your code goes into production with a customer who is using a lot more data than you tested with.

Once you have a characterization of your algorithm, which might involve a mixture of the orders shown above, you're in a position to start plugging in numbers to predict how the algorithm will scale in your environment, on your hardware, with your data. Keep in mind that algorithms might have a characteristic, constant startup overhead, and a per-step overhead. So you need to move from

      O(N log N)
      k1 + k2(N log N)

and then determine values for the constants by taking some samples an doing some math. This is left as an exercise for the motivated reader.

Common Pitfalls

By far, the most common pitfall when dealing with algorithmic complexity is the naive belief (or blind hope) that the algorithm that was used successfully on a small project is going to scale to deal with 10x or 100x the data.

The inverse of this problem is not leaving "good enough" alone. In a given situation, an O(N2) algorithm will often work just fine. Giving into the temptation to switch to a more complex O(N log N) algorithm can needlessly complicate your code. And there's an opportunity cost: the time you spent switching to a "better" algorithm might better have been applied elsewhere.

A more subtle pitfall is when you've hung on to incomplete knowledge, inadvertently limiting your available choices. If you ask for a show of hands for who thinks Quicksort (which is O(N log N) is the fastest sort, you'll see a lot of arms go up. Refer these people to Knuth's Art of Computer Programming, volume 3: Sorting and Searching, where they'll find that Radix sort takes O(N) time (but requires O(N) space, and places constraints on keys). Spending some time with a good Algorithms book can expand your options.

This concludes the informal introduction to the "Big O" notation as it is used informally. There's a lot more rigor (and more symbols) if you want the full formal treatment.

See Also

Mastering Algorithms with Perl has a brief coverage of O(N) notation on pages 17-20, with some neat graphs, including one that shows how the characteristic order curves differ.

Thanks to tye, BrowserUK, zdog, and Abigail-II for reviewing a draft of this post. The mistakes and sloppiness that remain are mine. And thanks in advance to anyone who can point out errors or plug in gaps.

In reply to An informal introduction to O(N) notation by dws

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