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Comment by joshuamorton | Hacker News Reader

No, radix sort is worst case O(n * k). In many common cases, k ~= log(n). IN certain specific cases, k < log(n), and specifically for cases where you have a very large n, but a bounded number of values (say, you're sorting 10 billion 4-bit ints), k can be considered a constant. But that is by no means generally true.

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In most cases k << n. For 64-bit integers, byte wise radix sort k is 8, which is less than log n whenever n is more than 256. So, radix sort is typically much faster than an O(n log n) sort of your data support it. It just isn’t as widely used because it is not as general as a comparison based sort.

After a byte-wise radix sort you still have to sort within each bucket.

In the worst case, every element falls into a single bucket, at which point your best best is to do a bit wise radix sort over the low 8 bits.

This ends up being equivalent to k=log(n).

Jumping in this fascinating thread about sorting to ask what the double less than chevrons mean? I know what one means but what do two of them mean?

https://www.google.com/search?q=what+does+<<+mean

Basically “much less than”.

In this context, it means "much less." It's used that way sometimes in mathematical discussions.

That logic isn't right. You can implement an O(n log n) sort with whatever log base/radix you want. It could be 2, it could be 256, it could be 4096. If you use the same base/radix for both n and k, then log n is close to k and probably smaller.

Which version ends up faster is mostly based on factors that big O notation doesn't capture easily. You have to push things to impractical numbers like n=2^1000 or 2^1000000 to properly distinguish between constant factors and log factors, and at that point it doesn't actually help you write better code.

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