Python sets and dictionaries can have quadratic-time performance

I call shenanigans on this.

    import timeit
    def test(M,n):
        values = [i * M for i in range(1, n + 1)]
        s = set(values)
        sum(v in s for v in values)
    M = (1 << 61) - 1
    for n in [1000, 2000, 4000, 8000, 16000]:
        print(f"M=2^61-1, {n=:5d} ->", timeit.timeit(lambda: test(M,n), number=3))
    for n in [1000, 2000, 4000, 8000, 16000]:
        print(f"M=1,      {n=:5d} ->", timeit.timeit(lambda: test(1,n), number=3))

    M=2^61-1, n= 1000 -> 0.13144792200182565
    M=2^61-1, n= 2000 -> 0.48016051898594014
    M=2^61-1, n= 4000 -> 2.058760045998497
    M=2^61-1, n= 8000 -> 7.843778470996767
    M=2^61-1, n=16000 -> 40.01485426299041
    M=1,      n= 1000 -> 0.000748768012272194
    M=1,      n= 2000 -> 0.0015750699967611581
    M=1,      n= 4000 -> 0.003037029004190117
    M=1,      n= 8000 -> 0.006664915999863297
    M=1,      n=16000 -> 0.012693285010755062

Raymond Hettinger has a great talk about how much python's dict has improved over the years. So this is super interesting and will probably just make the builtin dict better eventually.

The lesson of the talk is that if you are idiomatic then you will benefit as the language improves.

https://www.youtube.com/watch?v=npw4s1QTmPg

That's usually true of all common hash table implementations (when objects don't have a defined order, if they have you can get O(1) average and O(log N) worst case), regardless of language.

The O(1) is the expected average case, which usually holds.

Yeah, O(N^2) is theoretically possible, but unless you're defending against some sort of denial of service attack, in practice it rarely matters.

Still, if you can guess a sensible initial size for a hash table you can avoid a lot of the overhead of rehashing.

Most algorithm analyses don't incorporate memory hierarchies. I'm not sure what the point of this post was.

If you are really concerned about it, compute the empirical roofline for your machine.

Also, if the point is to point out memory hierarchies, it's not 'quadratic performance.' The algorithm doesn't behave differently once it spills over. The costs just get bigger.

Reminds me of the article about random memory access being O(√N): https://www.ilikebigbits.com/2014_04_21_myth_of_ram_1.html

But could we create a hash table that would be truly constant-time? No. As the size of your data structure grows, it requires progressively slower memory.

At a large enough size to be interesting, everything is dominated by IO and because IO is slow, at any interesting size performance is a matter of tailoring the implementation to the details of the data {0}.

Engineering is hard work, not naive math.

[0] Data might be arbitrary but it is never random. Not being random is what makes it data.

That is why I like to use __slots__ when defining a class. Unlike dicts, using __slots__ is a tuple so using it to store class attributes is much faster.

Which statement in this article applies only to Python?

> To put it differently, saying that a hash table is O(1) or constant time is a model

Nobody really says that, nor is it a model. It is the expected time complexity.

Time complexity is something that applies to algorithms, and is determined analytically. I don't think it's useful to equivocate the definition with performance of implementations of algorithms determined through real world data. Both of these things are important, but they are not the same. The fact that they differ is not very surprising and does not necessarily mean that a misunderstanding has occurred.

Real big brain moment in the comments

>Can we say that since every real-life data structure size is bounded by some constant, it is O(1)? If not, why?

>>You may, yes.

Very powerful thinking coming from someone who is "a software performance expert. He ranks among the top 2% of scientists globally (Stanford/Elsevier 2025) and is one of GitHub's top 1000 most followed developers"

Java's HashMap also has O(log(N)) complexity on hash collision, and that is before memory/cache details.

https://docs.oracle.com/javase/8/docs/api/java/util/HashMap....

In fact, some studying on data structures probably leads to the conclusion that it is impossible to guarantee that an unbounded set/map to have access performance under O(log(N)).

Once a hash table has outgrown all caches, it should have linear performance. It's just that caches accelerate it at sufficiently small sizes.

Uh, what is going on with this benchmark?

Why is M so big? Why does it cross the maxint boundary? Why is constructing the list comprehension part of the benchmark? Why are we summing the set? Why are we only measuring 5 values for n?

This "quadratic-time performance" is incredibly disingenuous. First, it's doing n operations that are each O(n), so it's more like "can have linear time performance, but done n times so I can give you a scary title".

Edit: A charitable take is constructing a set/dict from a list is indeed a common operation so it's worthwhile to think about its complexity, but it's not really one of the standard operations when discussing the performance of a hashset/hashmap, so really shouldn't be this handwavy.

And instead of attacking some straw man "It is indeed widely believed that ..." claim (widely believed by who?), why not attack what's literally on docs.python.org? https://docs.python.org/3/library/time-complexity.html:

> dict

> The times listed for dict objects are average-case times, as they assume the hash function for the objects is sufficiently robust to make collisions uncommon. They also assume the keys are well-distributed among the set of possible keys. In the worst case, when every key hashes to the same value, each of the O(1) operations below instead takes O(n) time. They also assume that hashing and comparing a key is O(1). For more detail on the implementation, see How are dictionaries implemented in CPython?.

> ...

> set, frozenset

> See dict as the set and frozenset implementations are similar, and the same caveats apply. In the worst case, O(1) operations instead take O(n) time, and operations that look up every element degrade accordingly.

  +--------------------------------------+------------+
  | Operation                            | Complexity |
  +--------------------------------------+------------+
  | x in s                               | O(1)       |
  | Copy (s.copy()) [6] [7]              | O(n)       |
  | Add (s.add(x)) [1]                   | O(1)       |
  | Discard (s.discard(x), s.remove(x))  | O(1)       |
  | ...                                  | ...        |
  +--------------------------------------+------------+

The discussion of CPU cache is good though, so why hide that behind this clickbait.

The map is not the territory.

Lemire back again with CS101 insights.

Next, he will teach us doubles have finite precision.

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Generally for performance, the rule is you _measure_ it, you don't rely on big-O. So if you're really concerned, please just use a profiler.

Also hashmaps (python Dicts) have amortized O(1) and not O(1) big-O.

O(1) doesn't mean constant, it means bounded by a constant. An algorithm can be faster with small n and converge to a horizontal asymptote as n goes to infinity, and it would still be O(1).

In a real machine there is no infinity, but hundreds of GBs of memory are "infinity enough" compared to the cache size [1]. So asymptotic analysis is still a decent model.

I'm surprised that even a CS professor confuses this.

[1] Ok if we want to be pedantic memory access is logarithmic due to the traversal of page tables, but you can use huge pages.