Nearly All Binary Searches and Mergesorts are Broken (2006)

http://googleresearch.blogspot.co.uk/2006/06/extra-extra-read-all-about-it-nearly.html

by prajjwal • 13 years ago

66 46 13 years ago

googleresearch.blogspot.co.uk

I would guess that nearly all code, period, is vulnerable to integer overflow issues. I don't think it makes sense to worry about this except in very special cases.

AndrewBissell2• 13 years ago

Having an array with more than about 1.2 billion elements is all that it would have taken to break the old binary search, and that's not all that uncommon anymore.

Pure JavaScript code isn't vulnerable to integer overflow by virtue of not having any integer types, and I believe errors like this were the reason for leaving them out of the language.

vbit• 13 years ago

JavaScript numbers are just floats and susceptible to precision loss as well as overflow to Infinity, aren't they?

In fact, if incremented repeatedly, 64-bit floats lose precision before 64-bit ints overflow.

dsego• 13 years ago

That wasn't the reason, the reason was to make the language simpler, with only one number type.

yetanotherphd4• 13 years ago

But any system with such a large array would almost certainly be 64 bit by now.

zvrba• 13 years ago

I would guess that nearly all code, period, is vulnerable to integer overflow issues.

Lack of checked integer arithmetic in some form is what I think is Java's biggest failure. Next to it is a lack of unsigned less/greater-than operator; it cannot be emulated easily with signed arithmetic. [There is unsigned right shift.]

(.Net _does_ have checked arithmetic; at least as an option.)

jadyoyster• 13 years ago

How so? From [1] it would seem that Lisps (idiomatic ones at least), Python, Perl, Haskell and Ruby are all free from the possibility of integer overflow. That definitely doesn't sound like "nearly all code".

[1] https://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic

Perseids2• 13 years ago

With Haskell I would not be so sure. There is the 'Integer' arbitrary size data type but what is mostly used is 'Int' for efficiency reasons afaik. Compiled as a 64 bit executable this should be a 64 bit integer though.

yetanotherphd4• 13 years ago

In that case the language fixes the issue so the code is irrelevant.

In languages where integer overflow is possible, my point still holds.

simula67• 13 years ago

Previous discussions :

https://news.ycombinator.com/item?id=1130463

https://news.ycombinator.com/item?id=621557

jgoodwin• 13 years ago

A bit off topic, but most of statistics also breaks.

If you go back to _Mathematical Statistics_ by RA Fisher, early in the last century, and look at his arguments about binning 'big data' into histograms, he has a nice little construction that uses the notion of an 'angle' running through the data set, does a Fourier Series expansion, keeps the 'DC' term from the cosine series, and waves his hand about second order effects. He does estimate them for the sine-like series, and finds for a data set of size N=1 Trillion it might be a 10% effect.

The only remnant of this whole proceeding in modern lore (and even Ph.D. statisticians may not have heard of it) is Sheppard's correction for equal class-interval histograms:

http://mathworld.wolfram.com/SheppardsCorrection.html

But of course when your datasets start to be 1 billion rows routinely, 10% effects a mere 3 orders of magnitude away in the size of the dataset should start to make you nervous.

Moral: once you get a billion data points of anything or so, it's time to redo the Maths, very very carefully.

yetanotherphd4• 13 years ago

I am a PhD statistician and I can't understand your comment.

What statistic is being calculated for the N=1 Trillion dataset? And what is the way of calculating that would be off by 10%?

kagebe2• 13 years ago

In my opinion, the main take-away here is that the proofs were, obviously, no proofs. If you program with modulo arithmetic you have to do your proofs with modulo arithmetic. If you use IEEE floating point, say goodbye to your theorems about real arithmetic.

If you forget/omit a single fact about your target platform/machine/api (whatever axioms you found your reasoning on) in your proof, it may be worth nothing.

koverstreet• 13 years ago

Yup.

More annoyingly, for a binary search over an array (i.e. something that can fit in memory) their code is still wrong - they should be using size_t.

smsm42• 13 years ago

That's what I immediately wondered - why it's int and not size_t? Well, I guess in Java engine there could be technical reasons why it is int, but in general case of implementation you'd probably assume it's size_t and then only one of the fixes works.

MaulingMonkey• 13 years ago

On top of that, this 'fix' from the article, even if it were corrected to use size_t instead of unsigned int...

  In C and C++ (where you don't have the >>> operator), you can do this:
  6:             mid = ((unsigned int)low + (unsigned int)high)) >> 1;

Has the same bug as the original snippet, just in unsigned space. Now, granted, binary searching a >2GB byte array in a 32-bit process is an unlikely use case... but it is technically feasible!

byerley• 13 years ago

As a theory guy I'd argue you're looking at it backwards.

The proofs are fine in that the correctness logically follows given the appropriate assumptions. In this particular example, it's implicitly assumed that integer overflow isn't a concern. Ideally, something like that should be explicitly stated, but let's be honest, Algorithms and Math in general have to be shorthand heavy - the alternative is painful and often not human readable.

It's always been the responsibility of the implementer to dive deep enough into the theory to grasp all the omitted assumptions. A less gratuitous headline might read - "Common implementation pitfalls: Integer Overflow"

kagebe2• 13 years ago

Well, then I restate: "Those proofs were not proofs of the theorems they/Bloch wanted to prove." The point being: Proving is not just about your proof, but also about writing meaningful Theorems, or as it is more commonly called when programming: specification.

The implementer is not free of proving her implementation correct, given that the original algorithm was proven correct on some theoretical computational model. Bloch argues that, since the original proof is not sufficient, the implementer needs to test. I'd add, that if the implementer would adapt/redevelop the proof for her implementation, she might forgo testing. Of course, this is practically impossible for many execution environments.

gkhnarik• 13 years ago

I don't understand why such an old article came here again.

VladRussian2• 13 years ago

because Java arrays length is int, 32bit limited. I think we'll see these articles more and more :)

greenlakejake• 13 years ago

Off topic(since it's not about C/Java), but Python has a long integer type which won't over/underflow unless you run out of memory.

% python

Python 2.7.5 (default, Aug 25 2013, 00:04:04)

[GCC 4.2.1 Compatible Apple LLVM 5.0 (clang-500.0.68)] on darwin

Type "help", "copyright", "credits" or "license" for more information.

>> i = long(9999999999999999999999999999999)
>> i

9999999999999999999999999999999L

>>

masklinn2• 13 years ago

Python has a long integer type which won't over/underflow unless you run out of memory.

IIRC, so do Erlang, Ruby or Haskell (when using `Integer`), FWIW. And Java has BigInteger (though that one's a pain to use).

But there's a cost to their existence (they need to check for overflow at every operation), and a cost to going above machine word size. Also, now you've got "integers" which can take arbitrary amounts of memory and integer operations in O(n)

Still, definitely a plus on the correctness side.

There's also the option of type-encoded value ranges as in Pascal or Ada.

crntaylor2• 13 years ago

I don't agree with this statement

  "It is not sufficient merely to prove a program correct;
   you have to test it too."

It is sufficient to prove a program correct - as long as your proof is not faulty! The problem in this case was not that the program had a bug despite being proved correct. The problem was that the 'proof' was not a proof at all.

Machine ints are not mathematical integers. Floats are not real numbers. You can't prove things about programs that use ints/floats without taking these things into account.

Of course, the question of how one knows that a proof is correct is still left open - but that's a metatheoretical argument that it might be best to leave aside. I suspect that most faulty proofs are faulty for pedestrian reasons (incorrect type assumptions, failing to deal with null/NaN etc) rather than high-falutin' concerns about the validity of first-order logic.

magicalist• 13 years ago

Not that it detracts from your point, but the phrase is a reference to a famous Knuth quote:

http://www-cs-faculty.stanford.edu/~knuth/faq.html (see the last question)

stiff2• 13 years ago

Computers are now complicated enough for Computer Science to have turned to some extent into an empirical science - it is impossible for a single person to have in their head everything that goes in a typical computer, operating system, compiler and so forth, so one is often forced to resort to experiment to find things out, it's no longer a theory where you can just reason things out, maybe it never was one in fact, because many simple imperative programs are so complex to reason about.

In empirical sciences, you not only have to have a mathematically valid theory, but you also have to check if the theory fits reality by making predictions and experimentally checking them with the real world. It's the same now in Computer Science, there are so many places where theoretical assumptions might deviate from reality that having a proof is not enough. In fact, you have to have those assumptions to make things mathematically tractable. Imagine mathematicians or computer scientists re-proving real analysis theorems using floating point arithmetic...

In other words, your vision of proofs being enough as long as all the assumptions are part of the theory, seems utopian to me. In fact, even some of the most devoted advocates of correctness proofs have admitted this:

http://www.gwern.net/docs/1996-hoare.pdf

crntaylor2• 13 years ago

Imagine mathematicians or computer scientists re-proving real analysis theorems using floating point arithmetic...

Mathematicians and computer scientists do prove theorems about floating point arithmetic! For example, the most widely-cited floating point reference contains no fewer than fifteen theorems about floating point:

http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.ht...

Or here's a presentation about representing functions with their Taylor expansions using floating point arithmetic, providing strong error bounds on the result of adding or multiplying two functions represented in this way:

http://perso.ens-lyon.fr/nathalie.revol/talks/ICIAM07.pdf

I agree that many proofs in computer science are harder than proofs in mathematics, because you can't deal with idealizations - you always have to think about the machine. But unlike empirical sciences, we have access to the design of the machine. We know what many of the axioms are. Formal reasoning is valid for a far larger part of computer science than for the natural sciences.

I'm not going to argue that proofs are a panacea for every situation. But I also don't categorically reject them in the domains where they can be usefully applied.

chrismonsanto• 13 years ago

I do agree with the quoted statement. I have a lot of experience with machine-verified proofs (in a language called Coq). Even though Coq guarantees that an accepted proof is correct, you still can't necessarily trust it, because your formal model of the program may be not accurate enough, or your statement of correctness is improperly stated.

In this case, it seems like their proof was correct with respect to their model, but their model did not match up with reality. No amount of formalization will ever be able to solve this problem completely. I think the role of testing in this context is to make sure your formalization of the problem says what you meant to say about the world that you meant to refer to. Input/output examples (aka, tests) are a great way of convincing yourself of this.

skrebbel• 13 years ago

This hinges on the definition of "program". The OP did not say "algorithm", which is often assumed to be something on paper rather than a piece of working code in $LANGUAGE. I believe that nearly anyone will assume that a "program" is something tangible, something that a computer can run.

So, the OP did not prove the program correct, but he did prove the underlying algorithm correct.

chattoraj• 13 years ago

It is sufficient to prove a program correct - as long as your proof is not faulty!

This is unquestionable truth. Proof:

Proposition A(X): X is true in theory

Proposition B : For all X such that A(X), X is true in practice

Theoretically, there is no difference between theory and practice. ... (1)

Theoretically, statement B is true. [using (1)] ... (2)

Therefore, B is true in practice. [using (1) and (2)]

QED.

rytis• 13 years ago

theory is very often contrasted to "practice" [...] a Greek term for "doing", which is opposed to theory because pure theory involves no doing apart from itself. [1]

[1] http://en.wikipedia.org/wiki/Theory

phelmig• 13 years ago

I would change it to "It is not sufficient merely to prove AN ALGORITHM IS correct; You have to test your implementation as well" ...

edanm• 13 years ago

One of the uncommon times a sensational headline is actually correct!

I love this bug - it's been my go-to example of how there are bugs in every piece of code, no matter how supposedly common. A bug in Java's implementatino of Binary Search - one of the most popular languages, and one of the most used algorithms - and still a bug managed to lay in wait for 9 years.

SideburnsOfDoom• 13 years ago

I see that as a limitation. "works only with data sizes up to half of MAXINT" is not a very surprising limitation of

     int mid = (low + high) / 2;

kokey• 13 years ago

I was also thinking this was a stereotypical HN sensational headline blog post. Apart from being strangely correct, it somehow makes me want to code something in Java today.

Perseids2• 13 years ago

I understand why people actually using these huge data sets see this as a bug but personally I don't agree. Consider how hard programming becomes when you have to take into account integer overflows even for a /single addition/ of array indices. Our mainstream programming languages are build around the assumption that indices don't overflow and I would rather use the binary sort code as argument to support this than to support the argument that bug free code is really hard.

The real lesson in my opinion is to stop using 32 bit integers in big data applications. Which unfortunately doesn't seem trivial in Java.

bproctor• 13 years ago

I don't believe this is necessarily a bug. All code will break under some extremes and nowhere are the requirements specified for what this code is supposed to be able to handle. Without requirements, you can't "prove" it correct (or incorrect for that matter).

eximius• 13 years ago

Well, that was an overly melodramatic title.

graycat• 13 years ago

I read far enough into that to see the code that could cause overflow and quit reading.

I've programmed various cases of binary search, merge sort, etc. for decades and in all cases wrote code with indexing that could never overflow. If some silly academic make-work, prof-scam, busy-work, nonsense code could overflow, so be it.

Heck, I even worry ahout

do i = 1 to n

wondering if the code is written to exit the loop when i = n + 1 tests larger than n, which in case n is the largest integer could never happen.

The title here "nearly all" is to me total BS. Take your insult of my code and stuff it. Capiche?

mtp0101• 13 years ago

Good thing I learned about this in 15-122 freshman year

Nearly All Binary Searches and Mergesorts are Broken (2006)

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