Sublinear Time Algorithms
people.csail.mit.edu
people.csail.mit.edu
I can imagine silly examples (like "find the minimum element in this list under the assumption that no more than O(sqrt(n)) elements exceed the minimum"...), but what's an interesting example of this?
> Anything probabilistic?
Are you sure you're answering the same question I'm asking?
What it and your example have in common is that a significant constraint exists on the input. I can't imagine how a deterministic algorithm with unconstrained input can process that input in sublinear time, but I would love to learn otherwise.
So I took it to mean there's something interesting here where the inputs could literally be anything, not heavily constrained. But I can't imagine what that might be.
I get what you're saying, and it doesn't change your point, but: no _way_ is binary search the first algorithm people learn. For binary search to even be intelligible you already have to know, at a minumum linear search and the concept of sorting (you probably know a sorting algorithm first). You also learn dozens of other really simple algorithms first in practice.
But, giving them binary search before sorting kinda works. It is motivating. If you do sorting first, it just seems like a weird high-effort excursion into a niche bookkeeping thing about lists. Once they see how fast binary search is (just give them pre-sorted lists to run it on), sorting becomes interesting, right?
But it does need to be covered.
Unfortunately.
(Mergesort is best).
Even worse is the fact for naive implementations (such as students might come up with) the worst case behaviour occurs in very common cases such as sorting already sorted lists or reverse-sorted lists.
It's based on a very simple but powerful idea/strategy (divide & conquer). Its flexibility means it can be adapted to partially sort, find top-N, find the median or any other rank, all optimally.
And it's so much faster in practice than everything else (why?), that even after mitigating its worse case, it often comes out ahead.
Also, it is relevant/necessary to teach the concept of average, best and worse cases in complexity analysis. What best way to do it than “the best sorting algorithm is terrible for some inputs”?
You can also use it to teach/learn adaptive algorithms (you're almost expected too): switch to something else on the base case; or on the worst case; can we do better for low cardinality; etc.
So, of course it needs to be covered. There's more to learn from 200 lines of Quicksort than from Mergesort: https://github.com/ncruces/sort/blob/main/quick/quick.go
radix sort
If you’re forced to use comparative sorting and write it by hand and it’s near random etc then Quicksort isn’t that bad but even then there’s better options.
And it is hard to beat, which is why the standard libraries of languages like C++, Rust, Go, C#, Java (etc) use some quicksort variant for their unstable sorts (and some mergesort variant when they need stability).
All of the above aren't forced to use comparison sorts, they just do; no other caveats required.
You mentioned it was a comparisons sort, but not that you were only comparing it with other comparison sorts.
There’s also a huge caveat, libraries know less about your data than you do. Thus different default choices become optimal but any decent library will give you many options for very good reasons.
Quicksort is relatively terrible for partially ordered lists like appended timestamps from multiple processes etc etc. It’s only ok at a very narrow range of problems, and the important bit isn’t the implementation but where those borders are.
It's not, not really: pdqsort handles those organically.
I'm sorry if "best sort", "so much faster" and "very hard to beat" felt like baits. But "huge caveat", "relatively terrible" and being "forced to use comparative sorting" are not fair descriptions of quicksort or why it's used and chosen by standard libraries.
Regardless, my point was that there's a lot to learn from quicksort. It's not “a shame” that it must be taught, and mergesort is not “best.”
pdqsort fails to detect many partially ordered lists.
So you might argue about the scale of “Huge caveat” but trying to discover information about a list takes computing cycles. Even just pdqsort vs pdqsort_branchless exists due to that exact caveat, it’s fundamental to the nature of the problem. There’s infinitely ways data could have an inherent pattern which would speed up the process of sorting it and no way to algorithmically notice all patterns in such a way as to universally speed up sorting.
As to what there is to learn from Quicksort. I think it’s a poor introduction to algorithms not because it’s of the nature as an algorithm, but early on its many pitfalls draw attention away from more useful topics. Later on it’s simply not complex enough to be worth much attention. So sure in an academic context it looks really appealing, yet when you dig into what the point of teaching algorithms it’s much harder to justify. It’s covered so frequently you’re not even going to see it on interviews.
Quicksort has all sorts of nonsense bouncing around and picking pivots. Eww. Terrible.
I will admit that I could have been more clear that I was evaluating algorithms based on beauty, but given that info I’m sure the conclusion is obvious.
Just the other day I sped up a process by an order of magnitude because someone was sorting to do weighted median selection. Which is easy to build on top of std::nth_element. Top-k is another common need.
Quicksort is still worthy today also because of this.
Almost every 6-10 year old kid who had to use physical dictionaries intuitively learned (probably even discovered by themselves) something like binary search. It's a different matter whether they could formalize that into an algorithm and write code to handle all the edge cases. But the basic idea is very intuitive. Kids can also pick up the intuition to incorporate improvements even beyond balanced binary search eg. there might be a lot of words starting with "S" so split into two groups at a little less than the middle, etc.
If you are asking which is the "first algorithm" a human learns in their life then it's likely more related to movement (crawl? walk? move food towards mouth?) or selection (which item can I eat? who are my parents?) rather than a physical dictionary. Even considering that it's been a while since kids encountered a physical dictionary.
If you are asking about formal algorithms then we're talking about the beginning of a programmers or computer scientists education and then it's usually some form of O(n^2) sort that they will encounter first, if we don't count things like "how to add two multi-digit integers" which is typically an algorithm every kid learns in primary school.
Binary search tends to be one of the first recursive algorithms that are taught which is another level entirely regarding intellectual development.
Looking over your comment again, I also don't dispute that linear search and sorting are simpler -- even toddlers learn these.
I find this highly unlikely. It might be true for those children who grow up to study computer science, though.
It legit was the first one we learned, the first algorithm written on the blackboard by the professor (this was in the 2000s but the first algorithm lessons were on blackboard and paper!)
Probably because something simpler linear like "find the minimum value in a list" is too dull as an algorithm example
It's an unusually intuitive algorithm, so it wouldn't surprise me if it were one many people learn first.
And your first quote is followed by "However, for most natural problems", which further indicates that the known exact algorithms are for trivial problems.
> there are classical optimization problems whose values can be approximated in sublinear time
This can actually be quite useful if the approximation is guaranteed, or even if it isn't, as long as it works well in practice.
https://www.dcs.warwick.ac.uk/~czumaj/PUBLICATIONS/DRAFTS/Dr...
Searching in sorted lists is the first example, although they acknowledge that "the assumption that the input array is sorted is not natural in typical applications." They then give an non-trivial variant of this problem, where the sorted list is stored as a linked list, so that you cannot directly jump to some element at position i.
Another example for a sublinear algorithm they give is to check whether 2 convex polygons intersect, using the algorithm by Chazelle and Dobkin.
I have to assume the reason for the article wasn't to talk about the runtime of algorithms that operate on data that's already in a partially solved state.
Edit: Ah, GP asked for deterministic exact.
The sublinear algorithms this page is discussing require that the algorithm not consider all elements of the data set and you're not allowed to pre-process it with an O(n) or greater algorithm.
So no trees, no binary search. It's a different set of algorithms.
How is public opinion polling deterministic and exact?
I'd consider all the varieties of B-Tree to be real example, which goes to any DBMS. You can extend this out to any direction you want like logging for concrete examples.
GIS/Mapping/computer vision has tons of algorithms and data structures that all needed to do better than linear time as well.
Stream processing in general is another, but that ends up being probabilistic more often than not, so weak punt into that direction.
If you expand the use case out to sublinear space as well, I'd argue for compression of all kinds.
Also relevant: https://www.cs.yale.edu/homes/aspnes/pinewiki/Derandomizatio...
They were talking about not reading a lot of the input, so that's not it.
Anyway, the point is that it's O(n/m) in the usual case. Which remains technically linear, not sub-linear; but at least it's O(n) with a constant factor smaller than 1.
It's not "maimed", it's literally the original algorithm. And what the article was specifically analyzing. And exactly what the parent was citing. "No" here makes no sense, unless your goal was just to write "no" to someone on the internet.
> That's a special usage of the phrase "worst case"! In the absolute worst case, your implementation could have a bug and never terminate at all!
Wikipedia is describing that algorithm, not a different broken one. If your code is buggy then you're not implementing that algorithm, you're implementing a different one that happens to be buggy. It's completely absurd to suggest "the absolute worst case" of an algorithm could include that of a different algorithm. Whether the latter is correct or buggy.
> Anyway, the point is that it's O(n/m) in the usual case.
Sure, and the halting problem is O(1) in the best case.
> Which remains technically linear, not sub-linear
So it's neither an example of what the page was talking about (sublinear) nor an answer to my question (interesting sublinear).
> but at least it's O(n) with a constant factor smaller than 1.
If "usually only reads a fraction of the input" was what I was looking for, I would've realized String.indexOf(char) or Array.find(element) is an answer, and not needed to ask a question here.
Any time I have information that reduces the entropy of the dataset, I want to be able to leverage that into runtime improvements of algorithms for pertinent questions. And it would be great to develop a structured framework for that instead of handling special cases in an ad-hoc manner.
Do you have any illustrative example so I can understand better what you are hinting at?
Cheers
State of the art there is poly-logarithmic time worst case.
A B-Tree would not qualify because it must first consider the entire input set. Only later operations can be less than O(n) because you've already done O(n) or greater work on the data set.
The input is the n(n-1)/2 bits indicating the results, but the existence of a winner can be determined in O(n) steps (fun exercise).
To see why, realize that to determine x, the TM needs to look at x many bits. If this algorithm only needed, say, log(x) many bits to produce output then all input values with more than log(x) bits are indistinguishable by this TM.