[jfe]

"Hashes are always O(1) reads, inserts and writes."

Maybe, once you've found a location to read, insert, or write to. The author neglects the runtime cost required for the hash algorithm itself, which may not be trivial; computing the hash of a string key is typically an O(n) operation.

Furthermore, unless a suitable table size is selected, integer keys (should one use a map like an array) will eventually hash to the same value, requiring even more time to iterate through all the remaining values that match that key until the desired value is found.

"I don’t know why you would ever use [a linked list] over an array (or a hash)..."

Here's why: because arrays take up space whether you use it or not. Linked lists don't suffer from this problem, but at the cost of 1-2 pointers per item. Has the author seriously never managed memory before? Please tell me this article is a joke.

[seubert]

It's even tagged "bad-theory." I think it's pretty clearly a joke! A really good one!

[dhimes]

"And we can’t forget our favorite JavaScript interview question of all time: If you only had twenty-four hours to implement arrays in JavaScript, how would you do it?"

[mdaniel]

Aren't they just objects that have integer properties; that's why you can

    for (k in ['a', 'b']) {
      console.log(k);
    }

and get back 0, 1? However:

    var a = {0:'a', 1:'b'};
    for (k in a) {
      console.log(k);
    }

also outputs 0, 1.

Edit: turns out, you can even have doubles, too:

    var weird = {3.14:'hello', 6.28:'world'};
    // for loop above emits: 3.14, 6.28
    console.log(weird[3.14]); // emits 'hello'

[chriswarbo]

> Edit: turns out, you can even have doubles, too:

That's because Javascript doesn't actually have integers, it just has "Number":

> The Number type has exactly 18437736874454810627 (that is, 264−253+3) values, representing the double-precision 64-bit format IEEE 754 values as specified in the IEEE Standard for Binary Floating-Point Arithmetic

http://www.ecma-international.org/ecma-262/5.1/#sec-8.5

[tomjakubowski]

JavaScript objects can only have string keys (this may have changed in ES2015 with Symbols). If you pass a non-string value in an object literal or inside a []-accessor, that value is converted to a string.

[pkaye]

I'm not really versed on JavaScript. I presume this is a joke about the lack of a proper array structure in JavaScript? How would one answer this question anyway?

[aggronn]

This is probably intended to be a joke about how Javascript was originally developed over about 10 days, which has been the root of many of JS's problems.

http://www.computer.org/csdl/mags/co/2012/02/mco2012020007.p...

[roghummal]

But this heritage produced, years later, a great presentation about the evolution of JavaScript... that was best viewed without JavaScript.

[nawitus]

While it's probably joke, JavaScript's Array is fine (although it's more like a 'vector'). JavaScript VMs implement it as a real array (in fact, plain objects are often optimized into arrays too).

[mutagen]

I think I came around to that realization when he starts comparing Postgres to HTML tables.

[justaaron]

sounds reasonable. normalization vs denormalization

ok, the last paragraphs kind of give it away... but there are good points in terms of keeping the data flat

[SilasX]

>Maybe, once you've found a location to read, insert, or write to. The author neglects the runtime cost required for the hash algorithm itself, which may not be trivial; computing the hash of a string key is typically an O(n) operation.

Yeah, I know, it always came off to me as BS to repeat that hashes are O(1) lookup, like we're making a special exception. Anywhere else, you can look up the algorithm and derive its big-O without having to memorize, but not this one.

Pretty clearly, as the hash table grows linearly, the keyspace has to grow linearly and the key size logarithmically. Since you get the keys from the output of a (nice, well-mixed) hash function, then the computation for where to insert has to increase logarithmically -- a longer hash output requires more steps.

You only get O(1) by selectively ignoring one kind of asymptotic behavior (that of the computations needed for the hash).

Reddit discussion: http://www.reddit.com/r/compsci/comments/2z74z8/why_is_hasht...

[Cushman]

This is mostly an issue with terms. Most (all?) of the operations we call constant-time, say, comparison, are technically logarithmic in the number of bits on real computers.

Since that's usually not relevant to big-O analysis, we can sidestep the issue by specifying what we're counting: rather than say that mergesort is in O((log n) log (log n)) time, we say it takes O(n log n) comparisons.

Whether you think of that as a theoretical computer with a constant-time comparison instruction or as a real computer with a commonsense (2^64) upper bound, the upshot is that we don't care. As long as what we're studying is sorting, not comparison, it will fall out of every algorithm, so the difference isn't important even in theory.

It's still an important thing to notice, though-- there are plenty of cases where you definitely do care about counting bits.

In more detail: http://cs.stackexchange.com/questions/1643/how-can-we-assume...

[SilasX]

But those situations aren't the same -- comparison sorts take O(n log n) because it is expressed in terms of n, the number of elements, which is orthogonal to element size. Even if you did account for the max element value V, it would be constant with respect to that value. The full bound would then be n log(V) log n -- regardless of your choice of V, you don't affect the scaling with respect to n.

But for hashtables, what exactly are we studying that doesn't care about hash time? The whole reason that a hashtable is supposed to save time is that you locate the desired key's value by computing the location from the key itself. To whatever extent that computation requires more steps, that cannot itself be abstract away -- if only because a limitation on key size is a limitation on table size.

[Cushman]

That example may have misfired-- perhaps consider that sorting an array of n elements requires addressing n elements, which requires arithmetic on words of log n bits, which are therefore log time operations. Limiting the size of each element won't help your complexity.

But we really don't care about that very much. In this case.

It's kind of described in that SO post I linked-- you can assume a constant-time hashing operation, but if that offends your conscience, assume an "integer RAM" model where arithmetic operations up to a certain word size w are constant time. Then you can observe that any reasonable w is inconsequential to your problem domain, and go back to treating hashing as a constant-time operation :)

The idea is that the fact that w increases at least as the log of n is shared by all computations modeled in integer RAM, so it's not a useful contribution to analysis at that scale. It's in the model, it's just not generally useful to the model. If you ever find yourself asking, is the bit width relevant? You can do some math and get a straight answer.

Of course real world algorithms often hash strings which complicates the analysis, but that's orthogonal, like my misstep earlier in implying the size of an element rather than the size of n. The mathematical sense in which hashing time must increase with the log of n is a fact of all algorithms that have ns.

I think your surprise at this just stems from wanting a rigorous definition for something that's more of a modeling tool, with different models to choose from. In real life, there's the time on the wall since you clicked "Run", and everything else is theorygramming.

[SilasX]

You can't have it both ways: on the one hand, say that big-O is about arbitrarily large unrealizable values, but on the other hand make a bunch of exceptions for where things don't matter in practice, lest you be theory-gramming.

If you go the route that requires you to handle arbitrarily large tables, then computing a hash with a long enough value necessarily requires more steps, even if you assume all operations on length w integers are constant time -- because eventually your keys have to be larger than w, after which computing the hash increases in number of steps with the key size.

This is still different from the sorting problem. You can have a consistent computation model in which memory seek times are constant (even if unrealizable). That seems fundamentally different from a model where a hash computation is constant steps but also has arbitrarily long output.

The issue isn't whether the scaling of the hash computation matters in practice, but whether the big-O is properly O(1). And even if you did just care about "in practice" limitations, you're stuck with the problem of the hash computation taking longer than any realizable trie lookup would (as another poster mentioned is common), meaning that the O(1) claim misrepresents performance relative to another data structure.

[Cushman]

Edit: Ugh, the old version of this post turned out really lectury, and I'm still working on my own intuitive understanding here, so I'm just gonna start over.

Basically-- you're not wrong. It's a quirk of analysis that a hash table is allowed to take the modulo (for example) of an arbitrarily large number, producing an arbitrarily large number, in "constant time", while a trie is not allowed to descend to an arbitrary depth in constant time.

But saying that such a thing is possible is a simplifying assumption that we make, in general, for non-numeric algorithms: arithmetic is O(1).

Of course, real programs cannot do this for arbitrarily large numbers. So if you are actually concerned with arbitrarily large numbers, you must do something else. Thus, other sorts of analysis which would give you a different number for hash table complexity. Yay!

And if you're having difficulty using the reported big-O value under the canonical model to compare algorithms which have very similar amortized performance in the real world, your problem is just that big-O is not for that thing you're doing.

[brudgers]

Since we call cases where something that looks polynomial on the surface but actually performs in NP "pseudo-polynomial," does it makes sense to call the cases where something more or less takes constant time "pseudo-logarithmic?"

[robrenaud]

Well, there are terms linearithmic and quasilinear that already get some usage.

> https://en.wikipedia.org/wiki/Time_complexity#Quasilinear_ti...

[Cushman]

Or maybe, since O(1) is in O(log n), you can cover your bases by just calling everything logarithmic regardless :)

[bottled_poe]

How about we refer to this as O(k) instead of O(1)?

[pmalynin]

O(n) in the size of the string not the hash map.

Also, if the key space is known before hand, it is possible to build a perfect minimal hash table with 1.3n total keys (n is the size of the keyspace)

[pmelendez]

> Linked lists don't suffer from this problem, but at the cost of 1-2 pointers per item.

Another cost is the cache misses. If you are doing operations in a sequential fashion using an array would have a significant advantage if the number of elements is high. That's another tradeoff to take in consideration.

[adrusi]

It can be avoided by choosing a good allocation strategy for the linked list, allocating nodes in an arena can eliminate cache misses.

[yoklov]

You'll still have more than if you had used an array, due to the low information density (2 extra pointers / node).

[nhaehnle]

computing the hash of a string key is typically an O(n)

This is a good point, though to be fair, this aspect of it is usually also ignored when analyzing the main alternative data structures, which are balanced binary trees. At least every successful lookup in such a tree with strings as keys also has to read the entire string.

This means that hashes probably still have an advantage over balanced binary trees. Tries, on the other hand, could really beat hashes when the keys are strings. In practice, they might have a cache and branching disadvantage, though.

[tikhonj]

Actually, modern tries can have good cache behavior, especially at larger sizes. Hash maps inherently store data haphazardly inside themselves while tries keep it in order with similar keys near each other in memory.

At the very least, this makes it easier for your algorithms to be more cache friendly, because tries' cache behavior is predictable in a way that hash maps' isn't.

This is even more relevant for branch prediction: hash functions (by design) are harder to predict, giving the lookup algorithm of a trie an advantage.

A cache aware design like "adaptive radix trees" delivers really good performance in practice, comparable or better than hash maps while also supporting fast iteration and certain queries (like iterating over all keys with a given prefix).

Take a look at the adaptive radix tree paper[1]: the idea is very accessible, and the resulting performance very impressive. They also generalize the system to work with different types of keys through a systematic method for turning values into binary strings.

[1]: http://www3.informatik.tu-muenchen.de/~leis/papers/ART.pdf

[bane]

I don't know why the complexity estimates I find for Hashes are always so bad. They never account for growth (or depending on the implementation shrinking on delete), never account for the hash function, etc.

By the time you hash a key, you could have likely already inserted it into a trie.

Lookups on hashes are also not O(1) for similar reasons. You have to hash the search string, then compare the value at whatever location it hashed to (usually a string-comparison operation which aren't O(1)) and depending on the collision strategy, do more things if it doesn't match, but isn't an empty value.

[chrisseaton]

When you say O(something), the something has a unit.

Hash table lookups and insertions take time O(1), when talking about number of items already in the hash table - and that 'when talking about' is implicit and doesn't have to be said as anyone talking about the complexity of a hash table knows that, or would state otherwise as it would be an exception.

Talking about the length of strings used as keys in the hash table is therefore nonsense when we have already agreed that we're talking about the number of elements in the hash table, as length of the strings isn't a parameter in that function.

And they never account for growth - yeah, it's amortised isn't it? That's what we wanted to do when we do an O().

I mean what you are proposing would be an interesting study - the complexity of hash tables parameterised for all sorts of different properties such as the size increase function or the load factor or whatever, but it's not what anyone means when they talk about complexity of a hash table unless they state otherwise.

So nobody's getting it wrong or making a mistake. They're using some assumptions that everyone's already agreed on and know about. If you want to talk about the complexity of something else you need to clarify.

[rnovak]

I have two kind of nits with this logic, but I could totally be wrong, and you should feel absolutely free to correct me.

I'm fairly positive that a unit of measure should _never_ be variable, otherwise it's fairly pointless. And if you don't think hashing a 1TB string takes significantly longer than a 100byte string ...

Futher more, big O notation is supposed to be a wide upper bound, but I don't think that works well if it's not actually an upper bound. If you were to tell your bosses something took constant time, and it took an hour for string A (1TB) and 100ms for string B (10B), I'm pretty sure your opinion wouldn't mean much after that.

The hashmap should absolutely be considered in terms of the hash function, because it's the longest running part of the algorighthm. To do otherwise is disingenous.

Using that logic, I could call any iterative algorithm O(1) since the top level function only gets called once.

[chrisseaton]

I think the problem you would have with your boss would be that your boss asked you 'how long will this program run', and if you told them O(1), the question you are really answering is 'what is the time complexity of this algorithm, parameterised by the number of entries, as the number of entries tends towards infinity'.

If your boss really wanted O(), then they wouldn't care that hashing one key takes a day and another a second, because they're thinking in terms of a hash with infinite entries, so the difference between a day and a second to hash is irrelevant.

[rnovak]

If you released software that was exponential, but your QA department only ever tested small inputs, I think it would be negligent to omit to your boss and|or clients the rate at which run-time could expand.

and I think it would be a horrible manager to not care about the difference between a day and a second.

[chrisseaton]

Yeah I agree but my point is none of what you are talking about has anything to do with O(), which doesn't attempt or claim to do what you want to do.

It's like someone gave you a hammer and you're saying it's broken because it doesn't cut wood very well. It's not designed for that.

[chinpokomon]

To implement the hash table you wouldn't have to hash the whole string... of course this will depend on the data that you are trying to store. Assuming that the data is random, 100 bytes vs. 100 terabytes, you only need to figure out what bucket the data is saved. You could still base it on this concept if sightly modified

[rnovak]

Yes. A valid hash function certainly can be defined as a F(n): N x N -> n % 100

But that certainly cannot be considered a reasonable hash function. A string is basically an array of bytes (or code-points, in case of UTF-8).

To have any decent property (like, producing different outputs for miniscule changes in the input), you have to touch every element in the array.

For custom objects, yes, you don't have to hash every property, but for strings, yeah, the hash function will almost always depend on the length of the string.

[bane]

Part of the problem when estimating hash complexity is that what's usually considered is something that's basically memory + offset. Basically a few mov's and an add.

However, this is absolutely dominated by complexity of the hashing function, growth (which can be amortized, but is not O(1)) deletion (also not O(1)) and comparison functions (which are usually O(mn) or O(n) (or some similar depending)).

We end up measuring the things that are the most minimal in hashing and pretending like the expensive operations, which aren't O(1), don't exist. This is wrong and dishonest when considering algorithms. We know for example that string comparison is not O(1), and it's usually a part of most hash table algorithms, and yet it magically disappears when analyzing hash table complexity and everything is somehow supposed to be considered as a mem+offset which is stupid.

Hash-tables also usually have exponential memory growth complexity which nobody ever pays attention to. Of course there are versions which don't do this and/or have fixed memory sizes, but the random kind you find in most languages grow O(n^2) or similar. And this is also ignored and we pretend it doesn't happen and resizing magically becomes part of the "1" in O(1)....even though copying arrays isn't free and as resizes happen arrays get bigger and big-O should get worse.

Hash-table operations can be pretty expensive and faulty big-O analysis doesn't help. So sure, for a static, fixed size, hash table, with no growth, instantaneous hash functions that use temporal oracles, don't need to deal with collisions, O(1) is correct. But these hash functions pretty much don't exit in nature, and most programmers won't be working with them. The standard libraries for most languages sure as hell don't implement such ideal hash tables.

O(n) is at least an honest approximation. I honestly have never seen a well considered analysis of hash function complexity but I know it grows super-linearly from just using them a lot. This kind of fanciful analysis doesn't help anybody.

So yes, O(something) where something has a unit. But the unit sure as heck isn't whatever "1" is supposed to represent. It's probably closer to string length or array length (or some combination of the two), but a single "add" it is most definitely not.

[Cushman]

All of what you're saying is true, about the speed of real algorithms running on real hardware with real problem sizes.

But if you're talking about big-O, you are explicitly not talking about that. You're talking about how the speed of the algorithm hypothetically scales as some parameter tends to infinity.

To wit, O(n) doesn't mean "this algorithm takes kn time to run for a given n", it means "this algorithm's runtime for all n > c for is bounded above by nk for some c and k".

Sound like a analytic club that's rarely accurate to real-world performance? Yup, that's big-O :)

[bane]

Right, so I'm asserting that the parameter usually called for in big-O analysis of hash table operations is the wrong one since it measures the lookup complexity and not the hashing complexity, which is usually O(n). And this results in meaningless complexity analysis which gives you things like "Insert is O(1) on Average but O(n) Worst Case or Θ(n)".

This analysis is using the hash table length as the parameter under consideration, but that's silly, because most of the complexity of hash-tables is in the hashing which (depending on the hash function) usually has a known complexity of O(n). Where n is the length of the input string.

This is far more important in hash table complexity analysis than table size because this part of the operation dominates runtime.

You can also do multi-paramter big-O analysis. O(mn) is a thing for example, with two differently sized parameters, both of which contribute to the complexity of the algorithm and can't be dismissed.

So charitably, if you need to provide a big-O for say, hash table inserts, it's reasonable to say O(mn) where m is the length of the table and n is the length of the input string, but it's not necessary since table length usually has little contribution to the complexity of the algorithm...hence why people keep saying inserts are O(1) on average, because table length isn't much of a determinant in hash table operation complexity. Just like big-O hides constants, we can hide parameters that are dominated by another. O(1) is doing this backwards.

My guess is that O(1) is some weird cargo-culted remnant from some early work done on Hash tables as a generalization of Arrays, likely from when the hash functions were just modulus operations on integer keys or some other simple, fixed length hashing method that was easy to ignore (like the universal hash in Corman's "Introduction to Algorithms". But modern hash-tables can't make these assumptions and I, and quite a few other folks, think that this needs to be rethought.

Some examples (I don't agree with all of these, but I think it makes the point):

https://stackoverflow.com/questions/2771368/can-hash-tables-...

http://lemire.me/blog/archives/2009/08/18/do-hash-tables-wor...

> Sound like a analytic club that's rarely accurate to real-world performance? Yup, that's big-O

If your complexity analysis isn't measurable by real-world performance, it's likely that you aren't analyzing the correct parameters.

[Cushman]

> And this results in meaningless complexity analysis which gives you things like "Insert is O(1) on Average but O(n) Worst Case or Θ(n)".

I dunno, that just sounds like a time complexity to me. Quick, what's the time complexity of quicksort?

> This analysis is using the hash table length as the parameter under consideration, but that's silly

I think you're just talking about something else, then? Sure, the analysis generally assumes a finite key size, and looks at performance as the table size increases. That's just pragmatism; people generally have a bounded key size and an unbounded amount of data.

If your complaint is that treating the key length as finite results in a complexity of O(1), then... that's the point. Treating the key length as finite results in a complexity of O(1).

Table size isn't much of a determinant. It isn't any of a determinant, on average. Only the key length matters. That conclusion is the whole point of this analysis.

> it's reasonable to say O(mn)

I'm confused if this is what you meant to write-- this is not an accurate complexity for a hash table insert because, as you have pointed out, a hash table insert doesn't depend on the table size. There should be no factor of m. Edit: Er, technically O(n) is in O(mn)? Is that your point? But O(mn) doesn't simplify to O(n) unless m is constant, which I don't think you're saying.

With respect to key size n and table size m, the average complexity should be O(n). If we let key size be finite relative to the size of the table, that gives us O(1). But if you don't like that, you can let key size grow to infinity and you're right back to O(n).

None of this is going to tell you if it's the right data structure to use, though.

> If your complexity analysis isn't measurable by real-world performance, it's likely that you aren't analyzing the correct parameters.

No, in this case you are looking at the correct parameters but using the wrong model. At least, I think; it's still not clear what you're trying to get done here where big-O is letting you down.

[miscfuck]

On a hash with a million elements, the O(n) hashing of a 10-char key is negligible.

Also, you have to make apples to apples comparisons. In this case, you're comparing the time to search against the number of elements, and that's it. If you want the time to hash AND search - as a function of n - then your analysis holds, but you can't then compare that against other datastructures that do not have an equivalent hash step.

[sorokod]

> computing the hash of a string key is typically an O(n) operation.

When strings are immutable, the hash values can be cached so the computation is amortized (this is how it works in Java)

[fiatmoney]

Additionally, it's provable that in the context of a physical computer, no data structure actually has O(1) read / write / anything performance over an arbitrary amount of data.

Consider that the universe enforces both a maximum information density per cubic meter, and a maximum speed at accessing a given physical location.

[the8472]

> computing the hash of a string key is typically an O(n) operation.

The n in operation time on data structures usually refers to the number of members in the data structure. String length would be a different variable.

For example worst case for finding a string member in a linked list would be O(n * k) where n is the elements in the linked list and k is the string length. I.e. the worst case here assumes lots of members with shared prefixes.

So the O(1) in hash tables actually is O(1 * k).

This distinction often is important because the length of strings and the number of members are independent variables.

And for many use-cases (e.g. member names in classes or text-protocol keywords) k is essentially fixed, e.g. a maximum member name length of 255 characters or a similar limitation. And for big O notation that means O(1 * 1), since it's a constant.

[masklinn]

> Linked lists don't suffer from this problem, but at the cost of 1-2 pointers per item.

And decreased memory locality (more cache misses) and increased number of allocations.

[LgWoodenBadger]

Worst case performance of a Map is O(n).