The idea behind big O notation
Big O notation is the language we use for articulating how long an algorithm takes to run. It's how we compare the efficiency of different approaches to a problem.
With big O notation we express the runtime in terms of—brace yourself—how quickly it grows relative to the input, as the input gets arbitrarily large.
Let's break that down:
- how quickly the runtime grows—Some external factors affect the time it takes for a function to run: the speed of the processor, what else the computer is running, etc. So it's hard to make strong statements about the exact runtime of an algorithm. Instead we use big O notation to express how quickly its runtime grows.
- relative to the input—Since we're not looking at an exact number, we need something to phrase our runtime growth in terms of. We use the size of the input. So we can say things like the runtime grows "on the order of the size of the input" () or "on the order of the square of the size of the input" ().
- as the input gets arbitrarily large—Our algorithm may have steps that seem expensive when n is small but are eclipsed eventually by other steps as n gets huge. For big O analysis, we care most about the stuff that grows fastest as the input grows, because everything else is quickly eclipsed as n gets very large. If you know what an asymptote is, you might see why "big O analysis" is sometimes called "asymptotic analysis."
Big O notation is like math except it's an awesome, not-boring kind of math where you get to wave your hands through the details and just focus on what's basically happening.
If this seems abstract so far, that's because it is. Let's look at some examples.
This function runs in time (or "constant time") relative to its input. The input list could be 1 item or 1,000 items, but this function would still just require one "step."
This function runs in time (or "linear time"), where n is the number of items in the list. If the list has 10 items, we have to print 10 times. If it has 1,000 items, we have to print 1,000 times.
Here we're nesting two loops. If our list has n items, our outer loop runs n times and our inner loop runs n times for each iteration of the outer loop, giving us n^2 total prints. Thus this function runs in time (or "quadratic time"). If the list has 10 items, we have to print 100 times. If it has 1,000 items, we have to print 1,000,000 times.
N could be the actual input, or the size of the input
Both of these functions have runtime, even though one takes an integer as its input and the other takes a list:
So sometimes n is an actual number that's an input to our function, and other times n is the number of items in an input list (or an input map, or an input object, etc.).
Drop the constants
This is why big O notation rules. When you're calculating the big O complexity of something, you just throw out the constants. So like:
This is , which we just call .
This is , which we just call .
Why can we get away with this? Remember, for big O notation we're looking at what happens as n gets arbitrarily large. As n gets really big, adding 100 or dividing by 2 has a decreasingly significant effect.
Drop the less significant terms
Here our runtime is , which we just call . Even if it was , it would still be .
Again, we can get away with this because the less significant terms quickly become, well, less significant as n gets big.
We're usually talking about the "worst case"
Often this "worst case" stipulation is implied. But sometimes you can impress your interviewer by saying it explicitly.
Sometimes the worst case runtime is significantly worse than the best case runtime:
Here we might have 100 items in our haystack, but the first item might be the needle, in which case we would return in just 1 iteration of our loop.
In general we'd say this is runtime and the "worst case" part would be implied. But to be more specific we could say this is worst case and best case runtime. For some algorithms we can also make rigorous statements about the "average case" runtime.
Space complexity: the final frontier
Sometimes we want to optimize for using less memory instead of (or in addition to) using less time. Talking about memory cost (or "space complexity") is very similar to talking about time cost. We simply look at the total size (relative to the size of the input) of any new variables we're allocating.
This function takes space (we aren't allocating any new variables):
This function takes space (the size of hi_list scales with the size of the input):
Usually when we talk about space complexity, we're talking about additional space, so we don't include space taken up by the inputs. For example, this function takes constant space even though the input has n items:
Sometimes there's a tradeoff between saving time and saving space, so you have to decide which one you're optimizing for.
Big O analysis is awesome except when it's not
You should make a habit of thinking about the time and space complexity of algorithms as you design them. Before long this'll become second nature, allowing you to see optimizations and potential performance issues right away.
Asymptotic analysis is a powerful tool, but wield it wisely.
Big O ignores constants, but sometimes the constants matter. If we have a script that takes 5 hours to run, an optimization that divides the runtime by 5 might not affect big O, but it still saves you 4 hours of waiting.
Beware of premature optimization. Sometimes optimizing time or space negatively impacts readability or coding time. For a young startup it might be more important to write code that's easy to ship quickly or easy to understand later, even if this means it's less time and space efficient than it could be.
But that doesn't mean startups don't care about big O analysis. A great engineer (startup or otherwise) knows how to strike the right balance between runtime, space, implementation time, maintainability, and readability.
You should develop the skill to see time and space optimizations, as well as the wisdom to judge if those optimizations are worthwhile.