Have you ever found yourself in a situation where you needed to keep track of whether or not you’ve seen an item, but the number of items you have to keep track of is either gigantic or completely unbounded?

This comes up a lot in “massive data” situations (like internet search engines), but it can also come up in games sometimes – like maybe having a monster in an MMO game remember which players have tried to attack it before so it can be hostile when it sees them again.

One solution to the game example could be to keep a list of all players the monster had been attacked by, but that will get huge and eat up a lot of memory and take time to search the list for a specific player.

You might instead decide you only want to keep the last 20 players the monster had been attacked by, but that still uses up a good chunk of memory if you have a lot of different monsters tracking this information for themselves, and 20 may be so low, that the monster will forget people who commonly attack it by the time it sees them again.

Enter The Bloom Filter

There is actually an interesting solution to this problem, using a probabilistic data structure called a bloom filter – and before you ask, no, it has nothing to do with graphics. It was invented by a guy named Burton Howard Bloom in 1970.

A bloom filter basically has M number of bits as storage and K number of hashes.

To illustrate an example, let’s say that we have 10 bits and only 1 hash.

When we insert an item, what we want to do is hash that item and then modulus that has value against 10 to get a pseudo-random (but deterministic) value 0-9. That value is the index of the bit we want to set to true. After we set that bit to true, we can consider that item is inserted into the set.

When we want to ask if an item exists in a given bloom filter, what we do is again hash the item and modulus it against 10 to get the 0-9 value again. If that bit is set to false, we can say that item is NOT in the set with certainty, but if it’s true we can only say it MIGHT be in the set.

If the bit is true, we can’t say that for sure it’s part of the set, because something else may have hashed to the same bit and set it, so we have to consider it a maybe, not a definite yes. In fact, with only 10 bits and 1 hash function, with a good hash function, there is a 10% chance that any item we insert will result in the same bit being set.

To be a little more sure of our “maybe” being a yes, and not a false positive, we can do two things… we can increase the number of bits we use for the set (which will reduce collisions), or we can increase the number of hashes we do (although there comes a point where adding more hashes starts decreasing accuracy again).

If we add a second hash, all that means is that we will do a second hash, get a second 0-9 value, and write a one to that bit AS WELL when we insert an item. When checking if an item exists in a set, we also READ that bit to see if it’s true.

For N hashes, when inserting an item into a bloom filter, you get (up to) N different bits that you need to set to 1.

When checking if an item exists in a bloom filter, you get (up to) N different bits, all of which need to be 1 to consider the item in the set.

There are a few mathematical formulas you can use to figure out how many bits and hashes you would want to use to get a desired reliability that your maybes are in fact yeses, and also there are some formulas for figuring out the probability that your maybe is in fact a yes for any given bloom filter in a runtime state.

Depending on your needs and usage case, you may treat your “maybe” as a “yes” and move on, or you could treat a “maybe” as a time when you need to do a more expensive test (like, a disk read?). In those situations, the “no” is the valuable answer, since it means you can skip the more expensive test.

Estimating Item Count In Bloom Filters

Besides being able to test an item for membership, you can also estimate how many items are in any given bloom filter:

EstimatedCount = -(NumBits * ln(1 – BitsOn / NumBits)) / NumHashes

Where BitsOn is the count of how many bits are set to 1 and ln is natural log.

Set Operations On Bloom Filters

If you have two bloom filters, you can do some interesting set operations on them.

Union

One operation you can do with two bloom filters is union them, which means that if you have a bloom filter A and bloom filter B, you end up with a third bloom filter C that contains all the unique items from both A and B.

Besides being able to test this third bloom filter to see if it contains specific items, you can also ask it for an estimated count of objects it contains, which is useful for helping figure out how similar the sets are.

Essentially, if A estimates having 50 items and B estimates having 50 items, and their union, C, estimates having 51 items, that means that the items in A and B were almost all the same (probably).

How you union two bloom filters is you just do a bitwise OR on every bit in A and B to get the bits for C. Simple and fast to calculate.

Intersection

Another operation you can do with two bloom filters is to calculate their intersection. The intersection of two sets is just the items that the two sets have in common.

Again, besides being able to test this third bloom filter for membership of items, you can also use it to get an estimated count of objects to help figure out how similar the sets are.

Similary to the union, you can reason that if the intersection count is small compared to the counts of the individual bloom filters that went into it, that they were probably not very similar.

There are two ways you can calculate the intersection of two bloom filters.

The first way is to do a bitwise AND on every bit in A and B to get the bits for C. Again, super simple and faster to calculate.

The other way just involves some simple math:

Count(Intersection(A,B)) = (Count(A) + Count(B)) – Count(Union(A,B))

Whichever method is better depends on your needs and your usage case.

Jaccard Index

Just like I talked about in the KMV post two posts ago, you can also calculate an estimated Jaccard Index for two bloom filters, which will give you a value between 0 and 1 that tells you how similar two sets are.

If the Jaccard index is 1, that means the sets are the same and contain all the same items. If the Jaccard index is 0, that means the sets are completely different. If the Jaccard index is 0.5 that means that they have half of their items in common.

To calculate the estimated Jaccard Index, you just use this simple formula:

Jaccard Index = Count(Intersection(A,B)) / Count(Union(A,B))

Estimating False Positive Rate

The more items you insert into a bloom filter the higher the false positive rate gets, until it gets to 100% which means all the bits are set and if you ask if it contains an item, it will never say no.

To combat this, you may want to calculate your error rate at runtime and maybe spit out a warning if the error rate starts getting too high, so that you know you need to adjust the number of bits or number of hashes, or at very least you can alert the user that the reliability of the answers has degraded.

Here is the formula to calculate the false positive rate of a bloom filter at runtime:

ErrorRate = (1 – e^(-NumHashes * NumItems / NumBits))^NumHashes

NumItems is the number of unique items that have been inserted into the bloom filter. If you know that exact value somehow you can use that real value, but in the more likely case, you won’t know the exact value, so you can use the estimated unique item count formula described above to get an estimated unique count.

Managing the Error Rate of False Positives

As I mentioned above, there are formulas to figure out how many bits and how many hashes you need, to store a certain number of items with a specific maximum error rate.

You can work this out in advance by figuring out about how many items you expect to see.

First, you calculate the ideal bit count:

NumBits = – (NumItems * ln(DesiredFalsePositiveProbability)) / (ln(2)^2)

Where NumItems is the number of items you expect to put into the bloom filter, and DesiredFalsePositiveProbability is the error rate you want when the bloom filter has the expected number of items in it. The error rate will be lower until the item count reaches NumItems.

Next, you calculate the ideal number of hashes:

NumHashes = NumBits / NumItems * ln(2)

Then, you just create your bloom filter, using the specified number of bits and hashes.

Example Code

Here is some example bloom filter c++ code with all the bells and whistles. Instead of using multiple hashes, I just grabbed some random numbers and xor them against the hash of each item to make more deterministic but pseudo-random numbers (that’s what a hash does too afterall). If you want to use a bloom filter in a more serious usage case, you may want to actually use multiple hashes, and you probably want to use a better hashing algorithm than std::hash.

#include 
#include 
#include 
#include 
#include 
#include 
#include 
 
// If you get a compile error here, remove the "class" from this enum definition.
// It just means your compiler doesn't support enum classes yet.
enum class EHasItem
{
    e_no,
    e_maybe
};
 
// The CBloomFilter class
template <typename T, unsigned int NUMBYTES, unsigned int NUMHASHES, typename HASHER = std::hash>
class CBloomFilter
{
public:
    // constants
    static const unsigned int c_numHashes = NUMHASHES;
    static const unsigned int c_numBits = NUMBYTES*8;
    static const unsigned int c_numBytes = NUMBYTES;
 
    // friends
    template 
    friend float UnionCountEstimate(const CBloomFilter& left, const CBloomFilter& right);
	template 
	friend float IntersectionCountEstimate (const CBloomFilter& left, const CBloomFilter& right);
 
    // constructor
    CBloomFilter (const std::array& randomSalt)
        : m_storage()              // initialize bits to zero
        , m_randomSalt(randomSalt) // store off the random salt to use for the hashes
    { }
 
    // interface
    void AddItem (const T& item)
    {
        const size_t rawItemHash = HASHER()(item);
        for (unsigned int index = 0; index < c_numHashes; ++index)
        {
            const size_t bitIndex = (rawItemHash ^ m_randomSalt[index])%c_numBits;
            SetBitOn(bitIndex);
        }
    }
 
    EHasItem HasItem (const T& item) const
    {
        const size_t rawItemHash = HASHER()(item);
        for (unsigned int index = 0; index < c_numHashes; ++index)
        {
            const size_t bitIndex = (rawItemHash ^ m_randomSalt[index])%c_numBits;
            if (!IsBitOn(bitIndex))
                return EHasItem::e_no;
        }
        return EHasItem::e_maybe;
    }
 
    // probabilistic interface
    float CountEstimate () const
    {
        // estimates how many items are actually stored in this set, based on how many
        // bits are set to true, how many bits there are total, and how many hashes
        // are done for each item.
        return -((float)c_numBits * std::log(1.0f - ((float)CountBitsOn() / (float)c_numBits))) / (float)c_numHashes;
    }
 
    float FalsePositiveProbability (size_t numItems = -1) const
    {
        // calculates the expected error.  Since this relies on how many items are in
        // the set, you can either pass in the number of items, or use the default
        // argument value, which means to use the estimated item count
        float numItemsf = numItems == -1 ? CountEstimate() : (float)numItems;
        return pow(1.0f - std::exp(-(float)c_numHashes * numItemsf / (float)c_numBits),(float)c_numHashes);
    }
     
private:
 
    inline void SetBitOn (size_t bitIndex)
    {
        const size_t byteIndex = bitIndex / 8;
        const uint8_t byteValue = 1 << (bitIndex%8);
        m_storage[byteIndex] |= byteValue;
    }
 
    inline bool IsBitOn (size_t bitIndex) const
    {
        const size_t byteIndex = bitIndex / 8;
        const uint8_t byteValue = 1 << (bitIndex%8);
        return ((m_storage[byteIndex] & byteValue) != 0);
    }
 
    size_t CountBitsOn () const
    {
        // likely a more efficient way to do this but ::shrug::
        size_t count = 0;
        for (size_t index = 0; index < c_numBits; ++index)
        {
            if (IsBitOn(index))
                ++count;
        }
        return count;
    }
     
    // storage of bits
    std::array m_storage;
 
    // Storage of random salt values
    // It could be neat to use constexpr and __TIME__ to make compile time random numbers.
    // That isn't available til like c++17 or something though sadly.
    const std::array& m_randomSalt;
};
 
// helper functions
template 
float UnionCountEstimate (const CBloomFilter& left, const CBloomFilter& right)
{
    // returns an estimated count of the unique items if both lists were combined
    // example: (1,2,3) union (2,3,4) = (1,2,3,4) which has a union count of 4
    CBloomFilter temp(left.m_randomSalt);
    for (unsigned int index = 0; index < left.c_numBytes; ++index)
        temp.m_storage[index] = left.m_storage[index] | right.m_storage[index];
     
    return temp.CountEstimate();
}
 
template 
float IntersectionCountEstimate (const CBloomFilter& left, const CBloomFilter& right)
{
    // returns an estimated count of the number of unique items that are shared in both sets
    // example: (1,2,3) intersection (2,3,4) = (2,3) which has an intersection count of 2
    CBloomFilter temp(left.m_randomSalt);
    for (unsigned int index = 0; index < left.c_numBytes; ++index)
        temp.m_storage[index] = left.m_storage[index] & right.m_storage[index];
     
    return temp.CountEstimate();
}
 
float IdealBitCount (unsigned int numItemsInserted, float desiredFalsePositiveProbability)
{
    // given how many items you plan to insert, and a target false positive probability at that count, this returns how many bits
    // of flags you should use.
    return (float)-(((float)numItemsInserted*log(desiredFalsePositiveProbability)) / (log(2)*log(2)));
}
 
float IdealHashCount (unsigned int numBits, unsigned int numItemsInserted)
{
    // given how many bits you are using for storage, and how many items you plan to insert, this is the optimal number of hashes to use
    return ((float)numBits / (float)numItemsInserted) * (float)log(2.0f);
}
 
// random numbers from random.org
// https://www.random.org/cgi-bin/randbyte?nbytes=40&format=h
// in 64 bit mode, size_t is 64 bits, not 32.  The random numbers below will be all zero in the upper 32 bits!
static const std::array s_randomSalt =
{
    0x6ff3f8ef,
    0x9b565007,
    0xea709ce4,
    0xf7d5cbc7, 
    0xcb7e38e1,
    0xd54b5323,
    0xbf679080,
    0x7fb78dee,
    0x540c9e8a,
    0x89369800
};
 
// data for adding and testing in our list
static const char *s_dataList1[] =
{
    "hello!",
    "blah!",
    "moo",
    nullptr
};
 
static const char *s_dataList2[] =
{
    "moo",
    "hello!",
    "mooz",
    "kitty",
    "here is a longer string just cause",
    nullptr
};
 
static const char *s_askList[] =
{
    "moo",
    "hello!",
    "unf",
    "boom",
    "kitty",
    "mooz",
    "blah!",
    nullptr
};
 
// driver program
void WaitForEnter ()
{
    printf("nPress Enter to quit");
    fflush(stdin);
    getchar();
}
 
void main(void)
{
    CBloomFilter set1(s_randomSalt);
    CBloomFilter set2(s_randomSalt);
	std::set actualSet1;
	std::set actualSet2;
 
    printf("Creating 2 bloom filter sets with %u bytes of flags (%u bits), and %u hashes.nn", set1.c_numBytes, set1.c_numBits, set1.c_numHashes);
 
	// create data set 1
    unsigned int index = 0;
    while (s_dataList1[index] != nullptr)
    {
        printf("Adding to set 1: "%s"n", s_dataList1[index]);
        set1.AddItem(s_dataList1[index]);
		actualSet1.insert(s_dataList1[index]);
        index++;
    }
 
	// create data set 2
    printf("n");
    index = 0;
    while (s_dataList2[index] != nullptr)
    {
        printf("Adding to set 2: "%s"n", s_dataList2[index]);
        set2.AddItem(s_dataList2[index]);
		actualSet2.insert(s_dataList2[index]);
        index++;
    }
 
	// query each set to see if they think that they contain various items
    printf("n");
    index = 0;
    while (s_askList[index] != nullptr)
    {
        printf("Exists: "%s"? %s & %s (actually %s & %s)n",
            s_askList[index],
            set1.HasItem(s_askList[index]) == EHasItem::e_maybe ? "maybe" : "no",
            set2.HasItem(s_askList[index]) == EHasItem::e_maybe ? "maybe" : "no",
			actualSet1.find(s_askList[index]) != actualSet1.end() ? "yes" : "no",
			actualSet2.find(s_askList[index]) != actualSet2.end() ? "yes" : "no");
        index++;
    }

	// show false positive rates
    printf ("nFalse postive probability = %0.2f%% & %0.2f%%n", set1.FalsePositiveProbability()*100.0f, set2.FalsePositiveProbability()*100.0f);
    printf ("False postive probability at 10 items = %0.2f%%n", set1.FalsePositiveProbability(10)*100.0f);
    printf ("False postive probability at 25 items = %0.2f%%n", set1.FalsePositiveProbability(25)*100.0f);
    printf ("False postive probability at 50 items = %0.2f%%n", set1.FalsePositiveProbability(50)*100.0f);
    printf ("False postive probability at 100 items = %0.2f%%n", set1.FalsePositiveProbability(100)*100.0f);

	// show ideal bit counts and hashes.
    const unsigned int itemsInserted = 10;
    const float desiredFalsePositiveProbability = 0.05f;
    const float idealBitCount = IdealBitCount(itemsInserted, desiredFalsePositiveProbability);
    const float idealHashCount = IdealHashCount((unsigned int)idealBitCount, itemsInserted);
    printf("nFor %u items inserted and a desired false probability of %0.2f%%nYou should use %0.2f bits of storage and %0.2f hashesn",
        itemsInserted, desiredFalsePositiveProbability*100.0f, idealBitCount, idealHashCount);

	// get the actual union
	std::set actualUnion;
	std::for_each(actualSet1.begin(), actualSet1.end(), [&actualUnion] (const std::string& s) {
		actualUnion.insert(s);
	});
	std::for_each(actualSet2.begin(), actualSet2.end(), [&actualUnion] (const std::string& s) {
		actualUnion.insert(s);
	});

	// get the actual intersection
	std::set actualIntersection;
	std::for_each(actualSet1.begin(), actualSet1.end(), [&actualIntersection,&actualSet2] (const std::string& s) {
		if (actualSet2.find(s) != actualSet2.end())
			actualIntersection.insert(s);
	});

	// caclulate actual jaccard index
	float actualJaccardIndex = (float)actualIntersection.size() / (float)actualUnion.size();

	// show the estimated and actual counts, and error of estimations
	printf("nSet1: %0.2f estimated, %u actual.  Error: %0.2f%%n",
		set1.CountEstimate(),
		actualSet1.size(),
		100.0f * ((float)set1.CountEstimate() - (float)actualSet1.size()) / (float)actualSet1.size()
	);
	printf("Set2: %0.2f estimated, %u actual.  Error: %0.2f%%n",
		set2.CountEstimate(),
		actualSet2.size(),
		100.0f * ((float)set2.CountEstimate() - (float)actualSet2.size()) / (float)actualSet2.size()
	);

	float estimatedUnion = UnionCountEstimate(set1, set2);
	float estimatedIntersection = IntersectionCountEstimate(set1, set2);
	float estimatedJaccardIndex = estimatedIntersection / estimatedUnion;
	printf("Union: %0.2f estimated, %u actual.  Error: %0.2f%%n",
		estimatedUnion,
		actualUnion.size(),
		100.0f * (estimatedUnion - (float)actualUnion.size()) / (float)actualUnion.size()
	);
	printf("Intersection: %0.2f estimated, %u actual.  Error: %0.2f%%n",
		estimatedIntersection,
		actualIntersection.size(),
		100.0f * (estimatedIntersection - (float)actualIntersection.size()) / (float)actualIntersection.size()
	);
	printf("Jaccard Index: %0.2f estimated, %0.2f actual.  Error: %0.2f%%n",
		estimatedJaccardIndex,
		actualJaccardIndex,
		100.0f * (estimatedJaccardIndex - actualJaccardIndex) / actualJaccardIndex
	);
 
    WaitForEnter();
}

And here is the output of that program:

In the output, you can see that all the existence checks were correct. All the no’s were actually no’s like they should be, but also, all the maybe’s were actually present, so there were no false positives.

The estimated counts were a little off but were fairly close. The first list was estimated at 2.4 items, when it actually had 3. The second list was estimated at 4.44 items when it actually had 5 items.

It’s reporting a very low false positive rate, which falls in line with the fact that we didn’t see any false positives. The projected false positive rates at 10, 25, 50 and 100 items show us that the set doesn’t have a whole lot more capacity if we want to keep the error rate low.

The union, intersection and jaccard index error rate was pretty low, but the error rate was definitely larger than the false positive rate.

Interestingly, if you look at the part which reports the ideal bit and hash count for 10 items, it says that we should actually use FEWER hashes than we do and a couple less bits. You can actually experiment by changing the number of hashes to 4 and seeing that the error rate goes down. In the example code we are actually using TOO MANY hashes, and it’s hurting our probability rate, for the number of items we plan on inserting.

Interesting Idea

I was chatting with a friend Dave, who said he was using a bloom filter like structure to try and help make sure he didn’t try the same genetic algorithm permutation more than once. An issue with that is that hash collisions could thwart the ability for evolution to happen correctly by incorrectly disallowing a wanted permutation from happening just because it happened to hash to the same value as another permutation already tried. To help this situation, he just biased against permutations found in the data structure, instead of completely blocking them out. Basically, if the permutation was marked as “maybe seen” in the data structure, he’d give it some % chance that it would allow it to happen “again” anyways.

Unfortunately, the idea in general turned out to be impractical. He had about 40 bits of genetic information which is about 1 trillion unique items (2^40).

for being able to store only 1 billion items – which is 1000 times smaller – with 5% false positive rate, that would require about 750MB of storage of bits.

Dropping the requirement to being 25% false positive rate, it still requires 350MB, and to 75% requires 70MB. Even at 99% false positive rate allowed, it requires 2.5MB, and we are still 1000 times too small.

So, for 1 trillion permutations, the size of the bloom filter is unfortunately far too large and he ended up going a different route.

The technique of rolling a random number when you get a maybe is pretty neat though, so wanted to mention it (:

Next Up

We’ve now talked about a probabilistic unique value counter (KMV) that can count the number of unique objects seen.

We then talked about a probabilistic set structure (Bloom Filter) that can keep track of set membership of objects.

How about being able to probabilistically store a list of non unique items?

One way to do this would be to have a count with each item seen instead of only having a bit of whether it’s present or not. When you have a count per item, this is known as a multiset, or the more familiar term: histogram.

If you change a bloom filter to have a count instead of a single bit, that’s called a counting bloom filter and completely works.

There’s a better technique for doing this though called count min sketch. Look for a post soon!

Links

Wikipedia: Bloom Filter
Interactive Bloom Filter Demo

Check this out, it’s a physical implementation of a bloom filter (WAT?!)
Wikipedia: Superimposed code

A few months ago I saw what I’m about to show you for the first time, and my jaw dropped. I’m hoping to share that experience with you right now, as a I share a quick intro to probabilistic algorithms, starting with KMV, otherwise known as K-Minimum Values, which is a Distinct Value (DV) Sketch, or a DV estimator.

Thanks to Ben Deane for exposure to this really interesting area of computer science.

The Setup

Let’s say that you needed a program to count the number of unique somethings that you’ve encountered. The unique somethings could be unique visitors on a web page, unique logins into a server within a given time period, unique credit card numbers, or anything else that has a way for you to get some unique identifier.

One way to do this would be to keep a list of all the things you’ve encountered before, only inserting a new item if it isn’t already in the list, and then count the items in the list when you want a count of unique items. The obvious downside here is that if the items are large and/or there are a lot of them, your list can get pretty large. In fact, it can take an unbounded amount of memory to store this list, which is no good.

A better solution may be instead of storing the entire item in the list, maybe instead you make a 32 bit hash of the item and then add that hash to a list if it isn’t already in the list, knowing that you have at most 2^32 (4,294,967,296) items in the list. It’s a bounded amount of memory, which is an improvement, but it’s still pretty large. The maximum memory requirement there is 2^32 * 4 which is 17,179,869,184 bytes or 16GB.

You may even do one step better and just decide to have 2^32 bits of storage, and when you encounter a specific 32 bit hash, use that as an index and set that specific bit to 1. You could then count the number of bits set and use that as your answer. That only takes 536,870,912 bytes, or 512MB. A pretty big improvement over 16GB but still pretty big. Also, try counting the number of bits set in a 512MB region of memory. That isn’t the fastest operation around 😛

We made some progress by moving to hashes, which put a maximum size amount of memory required, but that came at the cost of us introducing some error. Hashes can have collisions, and when that occurs in our scenarios above, we have no way of knowing that two items hashing to the same value are not the same item.

Sometimes it’s ok though that we only have an approximate answer, because getting an exact answer may require resources we just don’t have – like infinite memory to hold an infinitely large list, and then infinite computing time to count how many items there are. Also notice that the error rate is tuneable if we want to spend more memory. If we used a 64 bit hash instead of a 32 bit hash, our hash collisions would decrease, and our result would be more accurate, at the cost of more memory used.

The Basic Idea

Let’s try something different, let’s hash every item we see, but only store the lowest valued hash we encounter. We can then use that smallest hash value to estimate how many unique items we saw. That’s not immediately intuitive, so here’s the how and the why…

When we put something into a hash function, what we are doing really is getting a deterministic pseudo random number to represent the item. If we put the same item in, we’ll get the same number out every time, and if we put in a different item, we should get a different number out, even if the second item is really similar to the first item. Interestingly the numbers we get out should have no bearing on the nature of the items we put into the hash. Even if the items we put in are similar, the output should be just as different (on average) as if we put in items that were completely different.

This is one of the properties of a good hash function, that it’s output is (usually) evenly distributed, no matter the nature of the input. What that means is that if we put N items into a hash function, those items are on average going to be evenly spaced in the output of the hash, regardless of how similar or different they were before going into the hash function.

Using this property, the distance from zero to the smallest hash we’ve ever seen can be treated as a representative estimate of the average distance between each item that we hashed. So, to get the number of items in the hash, we convert the smallest hash into a percentage from 0 to 1 of the total hash space (for uint32, convert to float and divide by (float)2^32), and then we can use this formula:

numItems = (1 / percentage) – 1

To understand why we subtract the 1, imagine that our minimum hash value is 0.5. If that is an accurate representation of the space between values, that means that we only have 1 value, right in the middle. But, if we didvide 1 by 0.5 we get 2. We have 2 REGIONS, but we have only 1 item in the list, so we need to subtract 1.

As another example imagine that our minimum hash is 0.3333 and that it is an accurate representation of the space between values. If we divide 1 by 0.3333, we get 3. We do have 3 regions if we cut the whole space into 3 parts, but we only have 2 items (we made 2 cuts to make 3 regions).

Reducing Error

This technique doesn’t suffer too much from hash collisions (so long as your hash function is a decent hash function), but it does have a different sort of problem.

As you might guess, sometimes the hash function might not play nice, and you could hash only a single item, get a hash of 1 and ruin the count estimation. So, there is possibility for error in this technique.

To help reduce error, you ultimately need information about more ranges so that you can combine the multiple pieces of information together to get a more accurate estimate.

Here are some ways to gather information about more ranges:

• Keep the lowest and highest hash seen instead of just the lowest. This doubles the range information you have since you’d know the range 0-min and max-1.
• Keep the lowest N hashes seen instead of only the lowest. This multiplies the amount of range information you have by N.
• Instead of keeping the lowest value from a single hash, perform N hashes instead and keep the lowest value seen from each hash. Again, this multiplies the amount of range information you have by N.
• Alternately, just salt the same hash N different ways (with high quality random numbers) instead of using N different hash functions, but still keep the lowest seen from each “value channel”. Multiplies info by N again.
• Also a possibility, instead of doing N hashes or N salts, do a single hash, and xor that hash against N different high quality random numbers to come up with N deterministic pseudorandom numbers per item. Once again, still keep the lowest hash value seen from each “value channel”. Multiplies info by N again
• Mix the above however you see fit!

Whatever route you go, the ultimate goal is to just get information about more ranges to be able to combine them together.

In vanilla KMV, the second option is used, which is to keep the N lowest hashes seen, instead of just a single lowest hash seen. Thus the full name for KMV: K-Minimum Values.

Combining Info From Multiple Ranges

When you have the info from multiple ranges and need to combine that info, it turns out that using the harmonic mean is the way to go because it’s great at filtering out large values in data that don’t fit with the rest of the data.

Since we are using division to turn the hash value into an estimate (1/percentValue-1), unusually small hashes will result in exponentially larger values, while unusually large hashes will not affect the math as much, but also likely will be thrown out before we ever see them since they will likely not be the minimum hash that we’ve seen.

I don’t have supporting info handy, but from what I’ve been told, the harmonic mean is provably better than both the geometric mean and the regular plain old vanilla average (arithmetic mean) in this situation.

So, to combine information from the multiple ranges you’ve gathered, you turn each range into a distinct value estimate (by calculating 1/percentValue-1) and then putting all those values through the mean equation of your choice (which ought to be harmonic mean, but doesn’t strictly have to be!). The result will be your final answer.

Set Operations

Even though KMV is just a distinct value estimator that estimates a count, there are some interesting probabilistic set operations that you can do with it as well. I’ll be talking about using the k min value technique for gathering information from multiple ranges, but if you use some logic you should be able to figure out how to make it work when you use any of the other techniques.

Jaccard Index

Talking about set operations, I want to start with a concept called the Jaccard index (sometimes called the Jaccard similarity coefficient).

If you have 2 sets, the Jaccard index is calculated by:

Jaccard Index = count(intersection(A,B)) / count(union(A,B))

Since the union of A and B is the combined list of all items in those sets, and the intersection of A and B is the items that they have in common, you can see that if the sets have all items in common, the Jaccard index will be 1 and if the sets have no items in common, the Jaccard index will be 0. If you have some items in common it will be somewhere between 0 and 1. So, the Jaccard Index is just a measurement of how similar two sets are.

Union

If you have the information for two KMV objects, you can get an estimate to the number of unique items there if you were to union them together, even though you don’t have much of the info about the items that are in the set.

To do a union, you just combine the minimum value lists, and remove the K largest ones, so that you are left with the K minimum values from both sets.

You then do business as usual to estimate how many items are in that resulting set.

If you think about it, this makes sense, because if you tracked the items from both lists in a third KMV object, you’d end up with the same K minimums as if you took the K smallest values from both sets individually.

Note that if the two KMV objects are of different size, due to K being different sizes, or because either one isn’t completely filled with K minimum values, you should use the smaller value of K as your union set K size.

Intersection

Finding an estimated count of intersections between two sets is a little bit different.

Basically, having the K minimum hashes seen from both lists, you can think of that as a kind of random sample of a range in both lists. You can then calculate the Jaccard index for that range you have for the two sets (by dividing the size of the intersection by the size of the union), and then use that Jaccard index to estimate an intersection count for the entire set based on the union count estimate.

You can do some algebra to the Jaccard index formula above to get this:

count(intersection(A,B)) = Jaccard Index * count(union(A,B))

Just like with union, if the two KMV objects are of different size, due to K being different sizes, or because either one isn’t completely filled with K minimum values, you should use the smaller value of K.

Sample Code

#include 
#include 
#include 
#include 
#include 
#include 

// The CKMVCounter class
template <typename T, unsigned int NUMHASHES, typename HASHER = std::hash>
class CKMVUniqueCounter
{
public:
	// constants
	static const unsigned int c_numHashes = NUMHASHES;
	static const size_t c_invalidHash = (size_t)-1;

	// constructor
	CKMVUniqueCounter ()
	{
		// fill our minimum hash values with the maximum possible value
		m_minHashes.fill(c_invalidHash);
		m_largestHashIndex = 0;
	}

	// interface
	void AddItem (const T& item)
	{
		// if the new hash isn't smaller than our current largest hash, do nothing
		const size_t newHash = HASHER()(item);
		if (m_minHashes[m_largestHashIndex] <= newHash)
			return;

		// if the new hash is already in the list, don't add it again
		for (unsigned int index = 0; index < c_numHashes; ++index)
		{
			if (m_minHashes[index] == newHash)
				return;
		}

		// otherwise, replace the largest hash
		m_minHashes[m_largestHashIndex] = newHash;

		// and find the new largest hash
		m_largestHashIndex = 0;
		for (unsigned int index = 1; index  m_minHashes[m_largestHashIndex])
				m_largestHashIndex = index;
		}
	}

	// probabilistic interface
	void UniqueCountEstimates (float &arithmeticMeanCount, float &geometricMeanCount, float &harmonicMeanCount)
	{
		// calculate the means of the count estimates.  Note that if there we didn't get enough items
		// to fill our m_minHashes array, we are just ignoring the unfilled entries.  In production
		// code, you would probably just want to return the number of items that were filled since that
		// is likely to be a much better estimate.
		// Also, we need to sort the hashes before calculating uniques so that we can get the ranges by
		// using [i]-[i-1] instead of having to search for the next largest item to subtract out
		SortHashes();
		arithmeticMeanCount = 0.0f;
		geometricMeanCount = 1.0f;
		harmonicMeanCount = 0.0f;
		int numHashes = 0;
		for (unsigned int index = 0; index < c_numHashes; ++index)
		{
			if (m_minHashes[index] == c_invalidHash)
				continue;
			numHashes++;
			float countEstimate = CountEstimate(index);
			arithmeticMeanCount += countEstimate;
			geometricMeanCount *= countEstimate;
			harmonicMeanCount += 1.0f / countEstimate;
		}
		arithmeticMeanCount = arithmeticMeanCount / (float)numHashes;
		geometricMeanCount = pow(geometricMeanCount, 1.0f / (float)numHashes);
		harmonicMeanCount /= (float)numHashes;
		harmonicMeanCount = 1.0f / harmonicMeanCount;
	}

	// friends
	template 
	friend CKMVUniqueCounter KMVUnion (
		const CKMVUniqueCounter& a,
		const CKMVUniqueCounter& b
	);

	template 
	friend float KMVJaccardIndex (
		const CKMVUniqueCounter& a,
		const CKMVUniqueCounter& b
	);

private:

	unsigned int NumHashesSet () const
	{
		unsigned int ret = 0;
		for (unsigned int index = 0; index  0 ? m_minHashes[hashIndex- 1] : 0;
		const float percent = (float)(currentHash-lastHash) / (float)((size_t)-1);
		return (1.0f / percent) - 1.0f;
	}
	
	// the minimum hash values
	std::array m_minHashes;
	size_t m_largestHashIndex;
};

// Set interface
template 
CKMVUniqueCounter KMVUnion (
	const CKMVUniqueCounter& a,
	const CKMVUniqueCounter& b
)
{
	// gather the K smallest hashes seen, where K is the smaller, removing duplicates
	std::set setMinHashes;
	std::for_each(a.m_minHashes.begin(), a.m_minHashes.end(), [&setMinHashes](size_t v) {setMinHashes.insert(v); });
	std::for_each(b.m_minHashes.begin(), b.m_minHashes.end(), [&setMinHashes](size_t v) {setMinHashes.insert(v); });
	std::vector minHashes(setMinHashes.begin(), setMinHashes.end());
	std::sort(minHashes.begin(),minHashes.end());
	minHashes.resize(std::min(a.NumHashesSet(), b.NumHashesSet()));

	// create and return the new KMV union object
	CKMVUniqueCounter ret;
	for (unsigned int index = 0; index < minHashes.size(); ++index)
		ret.m_minHashes[index] = minHashes[index];
	ret.m_largestHashIndex = ret.c_numHashes - 1;
	return ret;
}

template 
float KMVJaccardIndex (
	const CKMVUniqueCounter& a,
	const CKMVUniqueCounter& b
)
{
	size_t smallerK = std::min(a.NumHashesSet(), b.NumHashesSet());

	size_t matches = 0;
	for (size_t ia = 0; ia < smallerK; ++ia)
	{
		for (size_t ib = 0; ib < smallerK; ++ib)
		{
			if (a.m_minHashes[ia] == b.m_minHashes[ib])
			{
				matches++;
				break;
			}
		}
	}

	return (float)matches / (float)smallerK;
}

// data to add to the lists
const char *s_boyNames[] =
{
	"Loki",
	"Alan",
	"Paul",
	"Stripes",
	"Shelby",
	"Ike",
	"Rafael",
	"Sonny",
	"Luciano",
	"Jason",
	"Brent",
	"Jed",
	"Lesley",
	"Randolph",
	"Isreal",
	"Charley",
	"Valentin",
	"Dewayne",
	"Trent",
	"Abdul",
	"Craig",
	"Andre",
	"Brady",
	"Markus",
	"Randolph",
	"Isreal",
	"Charley",
	"Brenton",
	"Herbert",
	"Rafael",
	"Sonny",
	"Luciano",
	"Joshua",
	"Ramiro",
	"Osvaldo",
	"Monty",
	"Mckinley",
	"Colin",
	"Hyman",
	"Scottie",
	"Tommy",
	"Modesto",
	"Reginald",
	"Lindsay",
	"Alec",
	"Marco",
	"Dee",
	"Randy",
	"Arthur",
	"Hosea",
	"Laverne",
	"Bobbie",
	"Damon",
	"Les",
	"Cleo",
	"Robt",
	"Rick",
	"Alonso",
	"Teodoro",
	"Rodolfo",
	"Ryann",
	"Miki",
	"Astrid",
	"Monty",
	"Mckinley",
	"Colin",
	nullptr
};

const char *s_girlNames[] =
{
	"Chanel",
	"Colleen",
	"Scorch",
	"Grub",
	"Anh",
	"Kenya",
	"Georgeann",
	"Anne",
	"Inge",
	"Georgeann",
	"Anne",
	"Inge",
	"Analisa",
	"Ligia",
	"Chasidy",
	"Marylee",
	"Lashandra",
	"Frida",
	"Katie",
	"Alene",
	"Brunilda",
	"Zoe",
	"Shavon",
	"Anjanette",
	"Daine",
	"Sheron",
	"Hilary",
	"Felicitas",
	"Cristin",
	"Ressie",
	"Tynisha",
	"Annie",
	"Sharilyn",
	"Astrid",
	"Charise",
	"Gregoria",
	"Angelic",
	"Lesley",
	"Mckinley",
	"Lindsay",
	"Shanelle",
	"Karyl",
	"Trudi",
	"Shaniqua",
	"Trinh",
	"Ardell",
	"Doreen",
	"Leanna",
	"Chrystal",
	"Treasa",
	"Dorris",
	"Rosalind",
	"Lenore",
	"Mari",
	"Kasie",
	"Ann",
	"Ryann",
	"Miki",
	"Lasonya",
	"Olimpia",
	"Shelby",
	"Lesley",
	"Mckinley",
	"Lindsay",
	"Dee",
	"Bobbie",
	"Cleo",
	"Leanna",
	"Chrystal",
	"Treasa",
	nullptr
};

// driver program
void WaitForEnter ()
{
	printf("nPress Enter to quit");
	fflush(stdin);
	getchar();
}

void main(void)
{
	// how many min values all these KMV objects keep around
	static const int c_numMinValues = 15;

	printf("Using %u minimum valuesnn", c_numMinValues);

	// =====================  Boy Names  =====================
	// put our data into the KVM counter
	CKMVUniqueCounter boyCounter;
	unsigned int index = 0;
	while (s_boyNames[index] != nullptr)
	{
		boyCounter.AddItem(s_boyNames[index]);
		index++;
	}

	// get our count estimates
	float arithmeticMeanCount, geometricMeanCount, harmonicMeanCount;
	boyCounter.UniqueCountEstimates(arithmeticMeanCount, geometricMeanCount, harmonicMeanCount);

	// get our actual unique count
	std::set actualBoyUniques;
	index = 0;
	while (s_boyNames[index] != nullptr)
	{
		actualBoyUniques.insert(s_boyNames[index]);
		index++;
	}

	// print the results!
	printf("Boy Names:n%u actual uniquesn", actualBoyUniques.size());
	float actualCount = (float)actualBoyUniques.size();
	printf("Estimated counts and percent error:n  Arithmetic Mean: %0.2ft%0.2f%%n"
		"  Geometric Mean : %0.2ft%0.2f%%n  Harmonic Mean  : %0.2ft%0.2f%%n",
		arithmeticMeanCount, 100.0f * (arithmeticMeanCount - actualCount) / actualCount,
		geometricMeanCount, 100.0f * (geometricMeanCount - actualCount) / actualCount,
		harmonicMeanCount, 100.0f * (harmonicMeanCount - actualCount) / actualCount);

	// =====================  Girl Names  =====================
	// put our data into the KVM counter
	CKMVUniqueCounter girlCounter;
	index = 0;
	while (s_girlNames[index] != nullptr)
	{
		girlCounter.AddItem(s_girlNames[index]);
		index++;
	}

	// get our count estimates
	girlCounter.UniqueCountEstimates(arithmeticMeanCount, geometricMeanCount, harmonicMeanCount);

	// get our actual unique count
	std::set actualGirlUniques;
	index = 0;
	while (s_girlNames[index] != nullptr)
	{
		actualGirlUniques.insert(s_girlNames[index]);
		index++;
	}

	// print the results!
	printf("nGirl Names:n%u actual uniquesn", actualGirlUniques.size());
	actualCount = (float)actualGirlUniques.size();
	printf("Estimated counts and percent error:n  Arithmetic Mean: %0.2ft%0.2f%%n"
		"  Geometric Mean : %0.2ft%0.2f%%n  Harmonic Mean  : %0.2ft%0.2f%%n",
		arithmeticMeanCount, 100.0f * (arithmeticMeanCount - actualCount) / actualCount,
		geometricMeanCount, 100.0f * (geometricMeanCount - actualCount) / actualCount,
		harmonicMeanCount, 100.0f * (harmonicMeanCount - actualCount) / actualCount);

	// =====================  Set Operations  =====================

	// make the KMV union and get our count estimates
	CKMVUniqueCounter boyGirlUnion = KMVUnion(boyCounter, girlCounter);
	boyGirlUnion.UniqueCountEstimates(arithmeticMeanCount, geometricMeanCount, harmonicMeanCount);

	// make the actual union
	std::set actualBoyGirlUnion;
	std::for_each(actualBoyUniques.begin(), actualBoyUniques.end(),
		[&actualBoyGirlUnion](const std::string& s)
		{
			actualBoyGirlUnion.insert(s);
		}
	);
	std::for_each(actualGirlUniques.begin(), actualGirlUniques.end(),
		[&actualBoyGirlUnion](const std::string& s)
		{
			actualBoyGirlUnion.insert(s);
		}
	);

	// print the results!
	printf("nUnion:n%u actual uniques in unionn", actualBoyGirlUnion.size());
	actualCount = (float)actualBoyGirlUnion.size();
	printf("Estimated counts and percent error:n  Arithmetic Mean: %0.2ft%0.2f%%n"
		"  Geometric Mean : %0.2ft%0.2f%%n  Harmonic Mean  : %0.2ft%0.2f%%n",
		arithmeticMeanCount, 100.0f * (arithmeticMeanCount - actualCount) / actualCount,
		geometricMeanCount, 100.0f * (geometricMeanCount - actualCount) / actualCount,
		harmonicMeanCount, 100.0f * (harmonicMeanCount - actualCount) / actualCount);

	// calculate estimated jaccard index
	float estimatedJaccardIndex = KMVJaccardIndex(boyCounter, girlCounter);

	// calculate actual jaccard index and actual intersection
	size_t actualIntersection = 0;
	std::for_each(actualBoyUniques.begin(), actualBoyUniques.end(),
		[&actualGirlUniques, &actualIntersection] (const std::string &s)
		{
			if (actualGirlUniques.find(s) != actualGirlUniques.end())
				actualIntersection++;
		}
	);
	float actualJaccardIndex = (float)actualIntersection / (float)actualBoyGirlUnion.size();

	// calculate estimated intersection
	float estimatedIntersection = estimatedJaccardIndex * (float)actualBoyGirlUnion.size();

	// print the intersection and jaccard index information
	printf("nIntersection:n%0.2f estimated, %u actual.  Error: %0.2f%%n",
		estimatedIntersection,
		actualIntersection,
		100.0f * (estimatedIntersection - (float)actualIntersection) / (float)actualIntersection);

	printf("nJaccard Index:n%0.2f estimated, %0.2f actual.  Error: %0.2f%%n",
		estimatedJaccardIndex,
		actualJaccardIndex,
		100.0f * (estimatedJaccardIndex-actualJaccardIndex) / actualJaccardIndex
	);

	WaitForEnter();
}

Here’s the output of the program:

Upgrading from Set to Multi Set

Interestingly, if you keep a count of how many times you’ve seen each hash in the k min hash, you can upgrade this algorithm from being a set algorithm to a multi set algorithm and get some other interesting information from it.

Where a set is basically a list of unique items, a multi set is a set of unique items that also have a count associated with them. In this way, you can think of a multiset as just a list of items which may appear more than once.

Upgrading KMV to a multi set algorithm lets you do some new and interesting things where instead of getting information only about unique counts, you can get information about non unique counts too. But to re-iterate, you still keep the ability to get unique information as well, so it is kind of like an upgrade, if you are interested in multiset information.

Links

Want more info about this technique?

Sketch of the Day: K-Minimum Values

Sketch of the Day: Interactive KMV web demo

K-Minimum Values: Sketching Error, Hash Functions, and You

Wikipedia: MinHash – Jaccard similarity and minimum hash values

All Done

I wanted to mention that even though this algorithm is a great, intuitive introduction into probabilistic algorithms, there are actually much better distinct value estimators in use today, such as one called HyperLogLog which seems to be the current winner. Look for a post on HyperLogLog soon 😛

KMV is better than other algorithms at a few things though. Specifically, from what I’ve read, that it can extend to multiset makes it very useful, and also it is much easier to calculate intersections versus other algorithms.

I also wanted to mention that there are interesting usage cases for this type of algorithms in game development, but where these probabilistic algorithms really shine is in massive data situations like google, amazon, netflix etc. If you go out searching for more info on this stuff, you’ll probably be led down many big data / web dev rabbit holes, because that’s where the best information about this stuff resides.

Lastly, I wanted to mention that I’m using the C++ std::hash built in hash function. I haven’t done a lot of research to see how it compares to other hashes, but I’m sure, much like rand(), the built in functionality leaves much to be desired for “power user” situations. In other words, if you are going to use this in a realistic situation, you probably are better off using a better hashing algorithm. If you need a fast one, you might look into the latest murmurhash variant!

More probabilistic algorithm posts coming soon, so keep an eye out!