Socks, birthdays and hash collisions

Socks, birthdays and hash collisions

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Nath knits awesome socks Suppose you’ve got a huge mixed-up pile of white, black, green and red socks, with roughly equal numbers of each. You randomly choose two of them. What is the probability that they are a matched pair?

There are sixteen ways of choosing a pair of socks: WW, WB, WG, WR, BW, BB, … Of those sixteen pairs, four of them are matched pairs. So chances are 25% that you get a matched pair.

Suppose you choose three of them. What is the probability that amongst the socks you chose, there exists at least one matched pair?

Well, we already know that chances are 25% after you pick out just the first two. If you get a matched pair right off, great. If you don’t, then there are two colours in hand you might match. So the odds are going to be a lot better.

There are 64 ways of choosing three socks: WWW, WWB, … and so on. Of those 64 possible combinations, 40 of them have at least one matched pair, so that’s about a 63% chance.

Suppose you choose four. There are 256 possible combinations, 232 of which have at least one matched pairs, so that’s a 91% chance.

Of course by the time we get to five socks, we have a 100% chance of getting a pair; five socks, four colours, there have got to be two alike.

It might appear that we’ve slightly messed up the probabilities here because once you choose one white sock, odds are slightly better that the next sock you pick will not be white, since there are now fewer white socks in the pile. But if the pile is big enough then we can neglect this minor problem.

From now on we’ll call getting a matched pair a “collision”.

It seems clear that as we increase the number of possible sock colours, we decrease the probability of getting a collision in some sample size. And as we increase the size of the sample, we increase the probability of the sample containing a collision.

Suppose you have 365 different colours of socks - perhaps each sock has a number on it giving its colour number, so that we can tell them apart - and a pile of about six billion socks, with roughly equal numbers of each sock colour. What is the probability that we’ll get a collision if we pull out two socks at random? One in 365, clearly. Three socks? A little bit better than double that.  And so on. To work out the exact probabilities we’d work out the number of possible combinations, and the number of those combinations that contain at least one collision.

Turns out that the point where you have a better than 50% chance of having a collision is 23 socks. This is the famous “birthday paradox”; if instead of 365 colours of socks we have 365 possible birthdays (ignoring leap years, the fact that more people are born on certain days than others, and so on) and we have a large group of people to choose from at random, then once you get to 23 people the odds are about fifty-fifty that two of them have the same birthday. By 50 people, chances are about 97% that two have the same birthday.

Which is maybe a nice party trick next time you’re at a party with 30 to 50 people – if you go around the room and ask everyone to say their birthday, odds are very good that two people will say the same day. But what’s my point?

Suppose you have just over four billion possible sock colours and a truly enormous supply of socks of each colour, such that each one is about equally likely. You start pulling socks out of the pile. What is the probability that you get a collision based on the number of socks you pull out? Four billion is an awfully big number compared to 4 or 365. What’s your intuition about the likelihood of a collision? How long until you have to start worrying about it?

Not nearly as long as you might think. I’ve worked out the math and summarized it in this handy log-log chart:


Man, is there anything better than getting a straight line on a log-log chart?

Anyway, you end up with a 1% chance of a collision after about 9300 tries, and a 50% chance after only 77000 tries. By the time you get into the mid six-digit numbers chances are for practical purposes 100% that there is a collision in there somewhere.

This is why it is a really bad idea to use 32 bit hash codes as “unique” identifiers. Hash values aren't random per se, but if they're well-distributed then they might as well be for our purposes. You might think “well, sure, obviously they are not truly unique since there are more than four billion possible values, but only four billion hash codes available. But there are so many possible hash values, odds are really good that I’m going to get unique values for my hashes”. But are the chances really that good? 9300 objects is not that many and 1% is a pretty high probability of collision.

  • Eric wrote:

    "As noted below, this kind of thing comes up frequently on StackOverflow. Also, as one of my technical interview questions I pose a problem which requires the generation of a unique 32 bit integer; many people immediately go to random numbers or hash codes and are then unable to compute the probability of collision."

    But wouldn't it be difficult to compute the probability of a collision on a random number if the person doesn't know the algorithm used to generate that number?

    Exactly, yes. A truly random number should have the probability of collision outlined in this article. If its not truly random, then yes, the probabily of collision depends strongly upon the details of the algorithm used. -- Eric

    Yes, more bits buys you a lowered risk of avoiding a collision, but I suppose to some extent it depends what one is going to do with that "unique integer". If it is going to be used for cryptography purposes, it may work in the short term, but in the long term with the increase in computing power that is available all it probably really buys you is more time.

    Absolutely. One of the reasons I ask this as part of an interview question is I get to see whether the candidate will ask clarifying questions about how the code is being used. What you want to know is (1) what is a reasonably expected time to first collision? If the candidate can ask questions about how many of these "unique" numbers need to be generated per second then they can work out whether there's going to be a collision every single time the program runs, or if we could go for millions of years with only a tiny chance of a collision. And (2) what are the negative consequences of a collision? If a collision means crashing the program or sending sensitive user data to an untrusted caller, then a solution which produces a collision in any reasonable amount of time is likely to be unacceptable. If the negative consequences are merely that a hash algorithm becomes a little sub-optimal, then no big deal. Good candidates clarify these things before they try to write code. -- Eric


    Thanks for the chart/explanation...very informative.

  • Based on your previous commented reply, Eric, are you implying that when using a HashTable, with its algorithm for collisions, this is not a serious issue, even with a million or so entries, updated every couple of seconds? Even if the ID of the entry is based on a string value (say, a GUID), in the hash table this ID will still be hashed first, so what have I gained from using a string value?

    Also, as mentioned in @ShuggyCoUk's comment, what is a perfect hash? I have recently been involved with Java development, and there the direction is to always override the 'hashCode' and 'equals' methods, with the hash-code usually involving prime-numbers and bitwise operations on 64-bit values.

  • > Also, as mentioned in @ShuggyCoUk's comment, what is a perfect hash?

    A perfect hash is a function hash(x) such that, for any x != y, hash(x) != hash(y). Simply put, it's a hash which has no collisions.

    >  I have recently been involved with Java development, and there the direction is to always override the 'hashCode' and 'equals' methods, with the hash-code usually involving prime-numbers and bitwise operations on 64-bit values.

    It is certainly not common practice to override equals/hashCode in Java (nor Equals/GetHashCode in .NET) for every random class. It is normally done for those classes which either 1) have value semantics (e.g. String), or 2) have their own identity distinct from runtime object identity (e.g. ORM entities with explicit primary key fields - as seen in Hibernate etc).

    If your entity class has a unique ID which is within the domain of hash values (i.e. 32-bit integer for hashCode & GetHashCode), then that ID is the perfect hash for that class. Otherwise the usual dance with "prime numbers and bitwise operations" is done on the primary key to trim it to desired size in a more-or-less evenly distributed way, but then the result is not a perfect hash anymore.

  • I had a friend whose wife tied his pairs of socks together so that he'd have 100% probability of getting a matched pair on the first try.  This was because he would stop after picking the second sock whether it matched or not, no matter how many were in the drawer.

    Hey, if I turned on the lights to get a matched pair of socks, I'd wake her up. Oh, wait, you weren't talking about me and my wife... as you were. -- Eric

  • I just want to clarify is it ok to use GetHashCode for a Quick Check of the content of a String for example, and if you do get a match couple that with a Real Check?

  • @Paul, My intuitive answer is no, it's not ok. Intuitively, I think it could take longer (having to generate a hash code). The lazy programmer in me would stick with String.Equals and it's plethora culture-aware overrides. If it is actually faster, I reckon string.Equals already does it for me.

    But what do your benchmarks say? Does your profiler indicate you're spending 80% of your time in String.Equals()? What about case-(in)sensitive and/or culture (in)sensitive comparisons?

  • Phew, looks like buying all those GUIDs on Ebay was a deal after all.

  • "Man, is there anything better than getting a straight line on a log-log chart?"

    Two very big thumbs up.

    Great article, thanks.

  • @csharptest: I am guilty! I have used the hashcode as an identifier!

    Well, I need to weaken this statement, let me explain: In very seldom events, our sever may get some kind of hiccup when dealing with a request: The exception's hashcode will also be written into the log file. This id can be picked up at the client which made the request.

    My reason of thought was that the event of an unexpected exception is seldom (the chart appears to prove me right) and that we do actually have additional correlation clues, like the day and possibly even the time. Plus, if there would be a collision it would be far from the end of the world. So far, this lil' correlation id has served us well.

  • jsrfc58 said: To complicate matters, there are also other types of socks that are not worn--wind socks, drift socks, etc.

    Eric said: Clearly more research needs to be done on this problem

    I've been trying to think of a way of punning on garter and Gartner for too many minutes now. It just ain't gonna happen.

  • There are also multicolored socks. For example, former-president Clinton's cat.

    The former president's former cat. Socks died in 2009. -- Eric

  • As a quick first-order approximation, cut the number of bits in half to get a 50% chance of a collision and shift right 3 bits to get a 1% chance. E.g. if you have 2^32 sock colors, choose approximately 2^16 socks to get a 50% chance of a duplicate or 2^13 socks to get about a 1% chance of a duplicate color.

  • Everyone talks about version 1 GUIDs (time & mac address), but almost all software generates version 4 GUIDs (random numbers).  I've experienced at LEAST 10 duplicate GUIDs in the last 8 years.

  • Its useful to point out that 23 = sqrt(365), and that the point at which it is likely (>=50%) that you will have a collision is at the squareroot of the size of your keyspace.

    Following from this, you can trivially calculate the tipping point for any hash given its length, as it is 2**(hashlength/2)

    Futhermore, you chances are that good if you assume random uniform distribution. When you're hashing a bunch of items, they probably have things in common that make them less than perfectly random.

    Moral of the story: Be careful using hashes as UIDs, but if you're using a 128bit hash you will *probably* be fine.

  • >> Everyone talks about version 1 GUIDs (time & mac address), but almost all software generates version 4 GUIDs (random numbers).  I've experienced at LEAST 10 duplicate GUIDs in the last 8 years.

    That is statistically very unlikely (even if it's not impossible) unless they're being generated in a particularly poor manner.

    note that using System.Random or C's rand() function or anything else that only starts with a 32-bit seed qualifies - you've basically got a 32-bit random* number (the original seed) that's been stretched out to 122 bits without adding any more real randomness, since all your bits are dependent on that original seed so you can only generate 2^32 possible GUIDs. You'll have that same high probability of collisions with semi-random GUIDs generated in the same way by the same implementation, and an unknown (somewhere between that high probability and zero) probability between it and different implementations.

    *or not even really so, since the default seed for System.Random is time-dependent... and rand()'s _default_ seed is fixed, but its _typical_ seed is going to also be time-dependent

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