Logarithmic Time Algorithms: The Magic of Scaling
Logarithmic time algorithms are one of those beautiful little hacks of computer science that feel almost unfair. No, they're not technically constant time—but in practice, they’re so fast it barely matters.
The magic trick? Every time your input size doubles, a logarithmic algorithm takes just one more step. That’s it. If your input is 1,000 items, it might take about 10 steps. A million? 20 steps. A billion? Around 30. You could have an array the size of the entire internet, and your algorithm is still wrapping up before your coffee even cools.
To put it in perspective: the largest array you can reasonably create in JavaScript is around 2³² elements (about 4.3 billion). Binary search on that? Just 32 steps. Your average SSD might store a few terabytes—call it a few trillion bytes—and even if you’re indexing every single byte (which you’re not), that’s about 40 steps. Try to push it further with a multi-petabyte data warehouse, and you might hit 50.
It’s practically impossible to get a logarithmic algorithm to take more than, say, 60 or 70 steps, even with the biggest inputs modern hardware can handle. You could build a computer the size of the moon and you’d still be done in under 100.
So while log-time algorithms aren’t constant in the strictest sense, they’re constant enough that you’ll never notice. They scale so gracefully it’s like they’re ignoring the input size entirely.
This chart says it all. You'll never take a large number of steps on a log-time process.
Published on: April 1, 2025