[markisus]

Interesting article. It’s actually very strange that the dataset needs to be “big” for the O(n log n) algorithm to beat the O(n). Usually you’d expect the big O analysis to be “wrong” for small datasets.

I expect that in this case, like in all cases, as the datasets become gallactically large, the O(n) algorithm will start winning again.

[Anon_troll]

The hash-based algorithm is only O(n) because the entry size has a limit. In a more general case, it would be something more like O(m(n * e)). Here n is the number of entries, e is the maximum entry size and m is a function describing how caching and other details affect the computation. With small enough data, the hash is very fast due to CPU caches, even if it takes more steps, as the time taken by a step is smaller. The article explains this topic in a less handwavey manner.

Also, memory access is constant time only to some upper limit allowed by the hardware, which requires significant changes to the implementation when the data does not fit the system memory. So, the hash algorithm will not stay O(n) once you go past the available memory.

The sorting algorithms do not suffer from these complexities quite as much, and similar approaches can be used with data sets that do not fit a single system's memory. The sorting-based algorithms will likely win in the galactically large cases.

Edit: Also, once the hash table would need to grow beyond what the hash function can describe (e.g. beyond 64 bit integers), you need to grow the function's data type. This is essentially a hidden log(n) factor, as the required length of the data type is log(n) of the maximum data size.

[shiandow]

Interestingly you need a hash function big enough to be unique for all data points with high probability, it doesn't take much to point out that this is at least O(log(n)) if all items are unique.

Also if items take up k bytes then the hash must typically be O(k), and both the hashing and radix sort are O(n k).

Really radix sort should be considered O(N) where N is the total amount of data in bytes. It can beat the theoretical limit because it sorts lexicographically, which is not always an option.

[zahlman]

Radix sort isn't a comparison-based sort, so it isn't beholden to the O(n log n) speed limit in the same way. It's basically O(n log k), where k is the number of possible different values in the dataset. If we're using machine data types (TFA is discussing 64-bit integers) then k is a constant factor and drops out of the analysis. Comparison-based sorts assume, basically, that every element in the input could in principle be distinct.

Basically, the hashed sorting approach is effectively actually O(n), and is so for the same reason that the "length of hash table" approach is. The degenerate case is counting sort, which is a single pass with a radix base of k. (Which is analogous to pointing out that you don't get hash collisions when the hash table is as big as the hashed key space.)

[daemontus]

One detail most comments seem to be missing is that the O(1) complexity of get/set in hash tables depends on memory access being O(1). However, if you have a memory system operating in physical space, that's just not possible (you'd have to break the speed of light). Ultimately, the larger your dataset, the more time it is going to take (on average) to perform random access on it. The only reason why we "haven't noticed" this yet that much in practice is that we mostly grow memory capacity by making it more compact (the same as CPU logic), not by adding more physical chips/RAM slots/etc. Still, memory latency has been slowly rising since the 2000s, so even shrinking can't save us indefinitely.

One more fun fact: this is also the reason why Turing machines are a popular complexity model. The tape on a Turing machine does not allow random access, so it simulates the act of "going somewhere to get your data". And as you might expect, hash table operations are not O(1) on a Turing machine.

[dgreensp]

No, because radix sort is not a comparison-based sort and is not O(n log n).

[spott]

The key here is in the cache lines.

Sorting (really scanning through an array) is very efficient in cache lines, while hash tables are very inefficient.

In the example, every memory access in hashing gets 8 byte of useful data, while every memory access in sorting gets 64 bytes of useful data.

So you have to be 8x more efficient to win out.

For this problem, the radix sort is log_1024(n) passes, which for a key size of 64 bits can’t ever get above 7 passes.

If you increase the key size then the efficiency win of sorting drops (128 bit keys means you have to run less than 4 passes to win).

Essentially the size of the key compared to the cache line size is a (hidden) part of the big O.

[aDyslecticCrow]

That is what baffles me. The difference in big O complexity should be more visible with size, but thats where it looses to the "worse" algorithm.

I could imagine the hash table wins again beyond a even greater threshold. Like what about 120GB and beyond?

[ants_everywhere]

As others have pointed out, radix sort is O(64N) = O(N) for a fixed key length of uint64s as in this article

So it comes down to the cost of hashing, hash misses, and other factors as discussed in the article.

I remember learning that radix sort is (almost?) always fastest if you're sorting integers.

A related fun question is to analyze hash table performance for strings where you have no bound on string length.

[shoo]

The analysis in the "Why does sorting win?" section of this article gave us a way to estimate that threshold.

Here's my attempt at it, based on that analysis:

Suppose each item key requires s bytes

For the hash table, assuming s <= 64, our cache line size, then we need to read one cache line and write one cache line.

The bandwidth to sort one key is p(N) * 2 * s where p(N) is the number of passes of 1024-bucket radix sort required to sort N elements, and 2 comes from 1 read + 1 write per 1024-bucket radix sort pass

p(N) = ceil(log2(N) / log2(buckets))

Suppose N is the max number of items we can distinguish with an s=8 byte item key, so N = 2^64

then p(N) = ceil(64 / 10) = 7 passes

7 radix passes * (1 read + 1 write) * 8 bytes per item key = 112 bytes of bandwidth consumed, this is still less than the bandwidth of the hash table 2 * 64.

We haven't found the threshold yet.

We need to either increase the item count N or increase the item key size s beyond 8 bytes to find a threshold where this analysis estimates that the hash map uses less bandwidth. But we cant increase the item count N without first increasing the key size s. Suppose we just increase the key size and leave N as-is.

Assuming N=2^64 items and an item size of b=10 bytes gives an estimate of 140 bytes of bandwidth for sorting vs 128 bytes of bandwidth. we expect sorting to be slower for these values, and increasing b or N further should make it even worse.

(the bandwidth required for hash map hasnt increased as our 10 byte b is still less than the size of a cache line)

edit: above is wrong as i'm getting mixed up with elements vs items. elements are not required to be unique items. if elements are unique items then the problem is trivial. So N is not bounded by the key size, and we can increase N beyond 2^64 without increasing the key size s beyond 8 bytes.

keeping key size s fixed at 8 bytes per item suggests the threshold is at N=2^80 items

given by solving for N for the threshold where bandwidth estimate for sort = bandwidth estimate for hash table

2 * 8 * (log2(N) / log2(1024)) = 2 * 64

[cdavid]

the big O copmlexity makes assumptions that break down in this case. E.g. it "ignores" memory access cost, which seems to be a key factor here.

[edit] I should have said "basic big O complexity" makes assumptions that break down. You can ofc decide to model memory access as part of "big O" which is a mathematical model

[dgreensp]

Radix sort is not a comparison-based sort and is not O(n log n).

[Ar-Curunir]

why is it strange? the case where asymptotically better = concretely worse is usually the exception, not the norm.

[simiones]

We expect asymptotically better algorithms to be, well, asymptotically better. Even if they lose out for small N, they should win for larger N - and we typically expect this "larger N" to not be that large - just big enough so that data doesn't fit in any cache or something like that.

However, in this particular case, the higher-complexity sorting algorithms gets better than the hash algorithm as N gets larger, even up to pretty large values of N. This is the counter-intuitive part. No one is surprised if an O(n log n) algorithm beats an O(n) algorithm for n = 10. But if the O(n log n) algorithm wins for n = 10 000 000, that's quite surprising.