Sublinear Time Algorithms

https://people.csail.mit.edu/ronitt/sublinear.html

by gone35 • 1 year ago

138 72 1 year ago

people.csail.mit.edu

dataflow9• 1 year ago

There are problems for which deterministic exact sublinear time algorithms are known.

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?

gleenn• 1 year ago

Anything probabilistic? There are so many interesting fields where you can assume the distribution of a dataset and the take a sample of data and assert things about it with a high degree of confidence. All of modern AI is built on so much of this. All the Deep Neural Nets are making grand assumptions about the shape of meaning of data, they literally assume convexity of the space and they have clearly very interesting results despite the imprecision of the model. Anything dealing with finance is also dealing in lack of data. So if you had a list of prices of a stock over time, you could probably start making assumptions exactly like that, tgat the probability that it doubles over a short time is so unlikely so you can subsample the data and have it still be super useful to make assumptions exactly, especially when you have intractably large data.

dataflow9• 1 year ago

> deterministic exact
Anything probabilistic?

Are you sure you're answering the same question I'm asking?

_jab• 1 year ago

Binary search is the obvious example.

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.

dataflow9• 1 year ago

I can't imagine that's what they meant? The text very specifically says: "Indeed, it is hard to imagine doing much better than that, since for any nontrivial problem, it would seem that an algorithm must consider all of the input in order to make a decision." For them to be thinking of binary search, they would have to be effectively saying "it is hard to think of binary search", which would be a rather absurd position from a CS professor, especially given binary search is quite literally the first algorithm every programmer learns.

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.

spoaceman7777• 1 year ago

I mean, yeah, binary search is sublinear, but the data has to be ordered into a binary search tree to be able to use it, which has a much more familiar (non-sub-linear) runtime.

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.

karparov2• 1 year ago

Another obvious example: What's the mean of an unsorted list of integers? If you do a random sample of sqrt(n) values and mean over that, you guess is with high probability pretty good. Or a log(n) sample. (That's how election polling works too which uses even a O(1) sample though not random.)

Edit: Ah, GP asked for deterministic exact.

Aurornis2• 1 year ago

Binary search requires a sorted input, which requires that you first consider all elements of the input data set.

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.

tbrownaw• 1 year ago

Public opinion polling is sub-linear in the size of the population.

dataflow9• 1 year ago

Public opinion polling is sub-linear in the size of the population.

How is public opinion polling deterministic and exact?

wellow• 1 year ago

Lempel-Ziv compression is a good example: https://arxiv.org/abs/2409.12146

latency-guy2• 1 year ago

I wouldn't call that example silly IMO.

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.

z2210558• 1 year ago

Assuming shuffled list: estimate of mean, estimate of cardinality etc etc

munchler• 1 year ago

I think any sort of estimation is ruled out by the word “exact”.

qyph2• 1 year ago

https://en.wikipedia.org/wiki/AKS_primality_test though it's number theory, and concerned with numbers of size n, rather than lists of length n.

Also relevant: https://www.cs.yale.edu/homes/aspnes/pinewiki/Derandomizatio...

dataflow9• 1 year ago

https://en.wikipedia.org/wiki/AKS_primality_test though it's number theory, and concerned with numbers of size n, rather than lists of length n.

They were talking about not reading a lot of the input, so that's not it.

Ar-Curunir• 1 year ago

AKS is not sublinear. It runs in poly(n) time, where n is the number of bits in the input (i.e. input size).

alok-g• 1 year ago

For that case, a better 'n' to use could be the number of digits in the number.

minutillo• 1 year ago

https://en.wikipedia.org/wiki/Boyer%E2%80%93Moore_string-sea...

dataflow9• 1 year ago

Isn't that O(mn) worst-case run time?

BugsJustFindMe2• 1 year ago

Consider that you often need to decide when to stop looking at data before making a decision using what you've seen so far.

https://en.wikipedia.org/wiki/Optimal_stopping

dataflow9• 1 year ago

Cool as that is, I don't think that's a "deterministic exact sublinear time algorithm".

ssivark4• 1 year ago

Think from an information theory perspective. It is rarely true that you cannot say anything more about the data than what is assumed by classical algorithms. We almost always have some more information depending on the specific domain under consideration. Eg: Sorting a list of ages might be very different from sorting a list of account balances.

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.

ssivark4• 1 year ago

As one example of such a more general framework -- (variants of) belief propagation might be a good answer if dataset constraints could be cleanly formulated as distributions to be reasoned with.

an_ko• 1 year ago

Fully dynamic connectivity on general graphs comes to mind. https://en.m.wikipedia.org/wiki/Dynamic_connectivity (Graphs, with operations to connect and disconnect nodes, and to check whether two nodes are connected by some path.)

State of the art there is poly-logarithmic time worst case.

JonChesterfield• 1 year ago

Data structured as trees permit a lot of sublinear operations. Set intersection for example, you traverse the two trees in the same order, and where a node exists in one and not the other, you know nothing under it is in the intersection.

Aurornis2• 1 year ago

In this case, Sublinear Algorithms refers to algorithms that don't consider the entire input set.

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.

indoordin0saur• 1 year ago

I'm surprised this is even a debate on HN. Aren't we mostly computer scientists here? Several examples on wikipedia: https://en.wikipedia.org/wiki/Big_O_notation#Orders_of_commo...

deycallmeajay• 1 year ago

What about GWP-ASan? It basically samples a portion of allocations with ASan looking for memory corruption bugs. If your app is used enough it’ll find the bugs eventually without the performance overhead. https://llvm.org/docs/GwpAsan.html

ice-water• 1 year ago

A round-robin tournament with n players, where you have the results (win/lose) of all the games and you must determine whether there is a player who won all his games.

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).

dooglius2• 1 year ago

Often for problems taking integer input x, formal CS will define the input to be something like 1^x (the character '1' repeated x times, as opposed to binary) so that the time complexity is in terms of x. This class of problems seems amenable to sublinear time since only only needs log(x) steps to determine x.

LPisGood• 1 year ago

Actually if you try to formally define sublinear time algorithms in this manner they all collapse to constant time algorithms.

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.

tromp2• 1 year ago

The notion of sublinear time only makes sense with Random-access Machines [1], not with Turing Machines.

[1] https://en.wikipedia.org/wiki/Random-access_machine

shae• 1 year ago

This sounds like a useful way to mix in statistics and get useful approximations. I'm reading one of the survey links and it's approximately eye opening.

doormatt2• 1 year ago

So like HyperLogLog?

qyph2• 1 year ago

Hyperloglog analyses generally assume access to the full data stream, and so are O(n) at a minimum. Perhaps by running hyperloglog on a sublinear sample of the dataset you'd get an algorithm in this class.

doormatt2• 1 year ago

That makes sense, thanks for explaining!

dataflow9• 1 year ago

HyperLogLog uses sublinear space, not sublinear time.

Sublinear Time Algorithms

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