5 ms·

Nearly All Binary Searches and Mergesorts are Broken (2006)

66 points by prajjwal 13y ago

jgoodwin 13y 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 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.
- yetanotherphd 13y 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%?
kagebe 13y 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 13y 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 13y 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.
    - chj 13y ago
      Yes, best fix: int mid = low + (high - low) / 2;
  - MaulingMonkey 13y 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!
    - PhantomGremlin 13y ago
      Yeah. That "fix" immediately jumped out at me in the sense of WTF??? All he did was double the max size of the array before his code fails again! That so-called "fixed" code just couldn't do anything useful in a 32-bit address space. A billion 32-bit ints completely fills up the address space by itself. As you note, perhaps it could "technically" be possible to search a billion 8-bit bytes, but that's not what's being passed in to the function. And if he's running in a 64-bit address space (otherwise how could he pass in an array of a billion ints), then he should be using 64-bit integer arithmetic. (I don't know enough about Java to know how feasible it is to do that).
gkhnarik 13y ago
I don't understand why such an old article came here again.
- VladRussian2 13y ago
  because Java arrays length is int, 32bit limited. I think we'll see these articles more and more :)
graycat 13y 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?
yetanotherphd 13y ago
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.
- zvrba 13y 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.)
- AndrewBissell 13y 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 13y 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.
    - AndrewBissell 13y ago
      Oh yes, floating point numbers have ALL SORTS of other issues. I was just stating the JS rationale for leaving ints out of the language, not saying I find it particularly convincing at all.
    - masklinn 13y ago
      > 64-bit floats lose precision before 64-bit ints overflow. More precisely, a 64 bit IEEE 754 ("double precision") has 53 bits worth of "integer"[0], which allows for 15 digits (and almost, but not quite, 16: it can encode 15.95 decimal digits) [0] even though only 52 bits are allocated to the fraction, because the fraction part has an implicit 53rd bit set to 1 outside of special values
simula67 13y ago
Previous discussions : https://news.ycombinator.com/item?id=1130463 https://news.ycombinator.com/item?id=1130463 https://news.ycombinator.com/item?id=621557 https://news.ycombinator.com/item?id=621557
greenlakejake 13y 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 >>>
- masklinn 13y 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.
crntaylor 13y 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 13y 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 http://www-cs-faculty.stanford.edu/~knuth/faq.html (see the last question)
- stiff 13y 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 http://www.gwern.net/docs/1996-hoare.pdf
edanm 13y 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 13y 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 13y 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.
Perseids 13y 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 13y 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 13y ago
Well, that was an overly melodramatic title.
mtp0101 13y ago
Good thing I learned about this in 15-122 freshman year

66 points by prajjwal 13y 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 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.

yetanotherphd 13y 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%?

kagebe 13y 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 13y 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 13y 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.
  - chj 13y ago
    Yes, best fix: int mid = low + (high - low) / 2;
- MaulingMonkey 13y 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!
  - PhantomGremlin 13y ago
    Yeah. That "fix" immediately jumped out at me in the sense of WTF??? All he did was double the max size of the array before his code fails again! That so-called "fixed" code just couldn't do anything useful in a 32-bit address space. A billion 32-bit ints completely fills up the address space by itself. As you note, perhaps it could "technically" be possible to search a billion 8-bit bytes, but that's not what's being passed in to the function. And if he's running in a 64-bit address space (otherwise how could he pass in an array of a billion ints), then he should be using 64-bit integer arithmetic. (I don't know enough about Java to know how feasible it is to do that).

gkhnarik 13y ago

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

VladRussian2 13y ago
because Java arrays length is int, 32bit limited. I think we'll see these articles more and more :)

graycat 13y 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?

yetanotherphd 13y ago

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.

zvrba 13y 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.)
AndrewBissell 13y 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 13y 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.
  - AndrewBissell 13y ago
    Oh yes, floating point numbers have ALL SORTS of other issues. I was just stating the JS rationale for leaving ints out of the language, not saying I find it particularly convincing at all.
  - masklinn 13y ago
    > 64-bit floats lose precision before 64-bit ints overflow. More precisely, a 64 bit IEEE 754 ("double precision") has 53 bits worth of "integer"[0], which allows for 15 digits (and almost, but not quite, 16: it can encode 15.95 decimal digits) [0] even though only 52 bits are allocated to the fraction, because the fraction part has an implicit 53rd bit set to 1 outside of special values

simula67 13y ago

Previous discussions : https://news.ycombinator.com/item?id=1130463 https://news.ycombinator.com/item?id=1130463 https://news.ycombinator.com/item?id=621557 https://news.ycombinator.com/item?id=621557

greenlakejake 13y 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 >>>

masklinn 13y 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.

crntaylor 13y 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 13y 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 http://www-cs-faculty.stanford.edu/~knuth/faq.html (see the last question)
stiff 13y 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 http://www.gwern.net/docs/1996-hoare.pdf

edanm 13y 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 13y 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 13y 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.

Perseids 13y 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 13y 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 13y ago

Well, that was an overly melodramatic title.

mtp0101 13y ago

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