Sketch of the Day: Probabilistic Counting with Stochastic Averaging (PCSA)

Before there was LogLog, SuperLogLog or HyperLogLog there was Probabilistic Counting with Stochastic Averaging (PCSA) from the seminal work “Probabilistic Counting Algorithms for Data Base Applications” (also known as the “FM Sketches” due to its two authors, Flajolet and Martin). The basis of PCSA matches that of the other Flajolet distinct value (DV) counters: hash values from a collection into binary strings, use patterns in those strings as indicators for the number of distinct values in that collection (bit-pattern observables), then use stochastic averaging to combine m trials into a better estimate. Our HyperLogLog post has more details on these estimators as well as stochastic averaging.


The choice of observable pattern in PCSA comes from the knowledge that in a collection of randomly generated binary strings, the following probabilities occur:

\begin{aligned}  &P( ..... 1) &= 2^{-1} \\  &P( .... 10) &= 2^{-2} \\  &P( ... 100) &= 2^{-3}  \end{aligned}


P(...10^{k-1}) = 2^{-k}

For each value added to the DV counter, a suitable hash is created and the position of the least-significant (right-most) 1 is determined. The corresponding position in a bitmap is updated and stored. I’ve created the simulation below so that you can get a feel for how this plays out.

Bit Pattern Simulation

Click above to run the bit-pattern simulation

(All bit representations in this post are numbered from 0 (the least-significant bit) on the right. This is the opposite of the direction in which they’re represented in the paper.)

Run the simulation a few times and notice how the bitmap is filled in. In particular, notice that it doesn’t necessarily fill in from the right side to the left — there are gaps that exist for a time that eventually get filled in. As the cardinality increases there will be a block of 1s on the right (the high probability slots), a block of 0s on the left (the low probability slots) and a “fringe” (as Flajolet et al. called it) of 1s and 0s in the middle.

I added a small pointer below the bitmap in the simulation to show how the cardinality corresponds to the expected bit position (based on the above probabilities). Notice what Flajolet et al. saw when they ran this same experiment: the least-significant (right-most) 0 is a pretty good estimator for the cardinality! In fact when you run multiple trials you see that this least-significant 0 for a given cardinality has a narrow distribution. When you combine the results with stochastic averaging it leads to a small relative error of 0.78 / \sqrt{m} and estimates the cardinality of the set quite well. You might have also observed that the most-significant (left-most) 1 can also be used for an estimator for the cardinality but it isn’t as clear-cut. This value is exactly the observable used in LogLog, SuperLogLog and HyperLogLog and does in fact lead to the larger relative error of 1.04 / \sqrt{m} (in the case of HLL).


The PCSA algorithm is elegant in its simplicity:

m = 2^b # with b in [4...16]

bitmaps = [[0]*32]*m # initialize m 32bit wide bitmaps to 0s

# Construct the PCSA bitmaps
for h in hashed(data):
    bitmap_index = 1 + get_bitmap_index( h,b ) # binary address of the rightmost b bits
    run_length = run_of_zeros( h,b ) # length of the run of zeroes starting at bit b+1
    bitmaps[ bitmap_index ][ run_length ] = 1 # set the bitmap bit based on the run length observed

# Determine the cardinality
phi = 0.77351
DV = m / phi * 2 ^ (sum( least_sig_bit( bitmap ) ) / m) # the DV estimate

Stochastic averaging is accomplished via the arithmetic mean.

You can see PCSA in action by clicking on the image below.

HyperLogLog Simulation

Click above to run the PCSA simulation

There is one point to note about PCSA and small cardinalities: Flajolet et al. mention that there are “initial nonlinearities” in the algorithm which result in poor estimation at small cardinalities (n/m \approx 10 \, \text{to} \, 20 ) which can be dealt with by introducing corrections but they leave it as an exercise for the reader to determine what those corrections are. Scheuermann et al. did the leg work in “Near-Optimal Compression of Probabilistic Counting Sketches for Networking Applications” and came up with a small correction term (see equation 6). Another approach is to simply use the linear (ball-bin) counting introduced in the HLL paper.

Set Operations

Just like HLL and KMV, unions are trivial to compute and lossless. The PCSA sketch is essentially a “marker” for runs of zeroes, so to perform a union you merely bit-wise OR the two sets of bitmaps together. Folding a PCSA down to a smaller m works the same way as HLL but instead of HLL’s max you bit-wise OR the bitmaps together. Unfortunately for intersections you have the same issue as HLL, you must perform them using the inclusion/exclusion principle. We haven’t done the plots on intersection errors for PCSA but you can imagine they are similar to HLL (and have the benefit of the better relative error 0.78 / \sqrt{m}).


That fact that PCSA has a better relative error than HyperLogLog with the same number of registers (1.04 / 0.78 \approx 1.33 ) is slightly deceiving in that m (the number of stored observations) are different sizes. A better way to look at it is to fix the accuracy of the sketches and see how they compare. If we would like to have the same relative error from both sketches we can see that the relationship between registers is:

\text{PCSA}_{RE} = \text{HLL}_{RE}

\dfrac{0.78}{\sqrt{m_{\scriptscriptstyle PCSA}}} = \dfrac{1.04}{\sqrt{m_{\scriptscriptstyle HLL}}}

m_{\scriptscriptstyle PCSA} = \left( \dfrac{0.78}{1.04} \right)^2 m_{\scriptscriptstyle HLL} \approx 0.563 \ m_{\scriptscriptstyle HLL}

Interestingly, PCSA only needs a little more than half the registers of an HLL to reach the same relative error. But this is also deceiving. What we should be asking is what is the size of each sketch if they provide the same relative error? HLL commonly uses a register width of 5 bits to count to billions whereas PCSA requires 32 bits. That means a PCSA sketch with the same accuracy as an HLL would be:

\begin{aligned}  \text{Size of PCSA} &= 32 \text{bits} \ m_{\scriptscriptstyle PCSA} = 32 \text{bits} \, ( 0.563 \ m_{\scriptscriptstyle HLL} )   \\  \\  \text{Size of HLL} &= 5 \text{bits} \ m_{\scriptscriptstyle HLL}  \end{aligned}


\dfrac{\text{Size of PCSA}} {\text{Size of HLL}} = \dfrac{32 \text{bits} \, ( 0.563 \ m_{\scriptscriptstyle HLL} )}{5 \text{bits} \ m_{\scriptscriptstyle HLL}} \approx 3.6

A PCSA sketch with the same accuracy is 3.6 times larger than HLL!


But what if you could make PCSA smaller by reducing the size of the bitmaps? Near the end of the paper in the Scrolling section, Flajolet et al. bring up the point that you can make the bitmaps take up less space. From the simulation you can observe that with a high probability there is a block of consecutive 1s on the right side of the bitmap and a block of consecutive 0s on the left side of the bitmap with a fringe in between. If one found the “global fringe” — that is the region defined by the left-most 1 and right-most 0 across all bitmaps — then only those bits need to be stored (along with an offset value). The authors theorized that a fringe width of 8 bits would be sufficient (though they fail to mention if there are any dependencies on the number of distinct values counted). We put this to the test.

PCSA with Fringe Cardinality Error

For a plot with larger m values, click here.

In our simulations it appears that a fringe width of 12 bits is necessary to provide an unbiased estimator comparable to full-fringe PCSA (32-bit) for the range of distinct values we analyzed. (Notice the consistent bias of smaller fringe sizes.) There are many interesting reasons that this “fringe” concept can fail. Look at the notes to this post for more. If we take the above math and update 32 to 12 bits per register (and include the 32 bit offset value) we get:

\begin{aligned}  \dfrac{\text{Size of PCSA}} {\text{Size of HLL}}  &= \dfrac{12 \text{bits} \, ( 0.563 \ m_{\scriptscriptstyle HLL} ) + 32\text{bits}} {5 \text{bits} \cdot m_{\scriptscriptstyle HLL}}  \\  \\  &= \dfrac{12 \text{bits} \, ( 0.563 \ m_{\scriptscriptstyle HLL} )} {5 \text{bits} \cdot m_{\scriptscriptstyle HLL}} + \dfrac{32\text{bits}} {5 \text{bits} \cdot m_{\scriptscriptstyle HLL}}  \\  \\  &\approx 1.35 \text{ (for }m\gg64 \text{)}    \end{aligned}

This is getting much closer to HLL! The combination of tighter bounds on the estimate and the fact that the fringe isn’t really that wide in practice result in PCSA being very close to the size of the much lauded HLL. This got us thinking about further compression techniques for PCSA. After all, we only need to get the sketch about 1/3 smaller to be comparable in size to HLL. In a future post we will talk about what happens if you Huffman code the PCSA bitmaps and the tradeoffs you make when you do this.


PCSA provides for all of the goodness of HLL: very fast updates making it suitable for real-time use, small footprint compared to the information that it provides, tunable accuracy and unions. The fact that it has a much better relative error per register than HLL indicates that it should get more credit than it does. Unfortunately, each bitmap in PCSA requires more space than HLL and you still get less accuracy per bit. Look for a future post on how it is possible to use compression (e.g. Huffman encoding) to reduce the number of bits per bitmap, thus reducing the error per bit to match that of HLL, resulting in an approach that matches HLL in size but exceeds its precision!

Notes on the Fringe

While we were putting this post together we discovered many interesting things to look at with respect to fringe optimization. One of the questions we wanted to answer was “How often does the limited size of the fringe muck up a bitmap?” Below is a plot that shows how often any given sketch had a truncation event (that affected the DV estimate) in the fringe of any one of its bitmaps for a given fringe width (i.e. some value could not be stored in the space available).


Note that this is an upper bound on the error that could be generated by truncation. If you compare the number of runs that had a truncation event (almost all of the runs) with the error plot in the post it is quite shocking that the errors are as small as they are.

Since we might not get around to all of the interesting research here, we are calling out to the community to help! Some ideas:

  1. There are likely a few ways to improve the fringe truncation. Since PCSA is so sensitive to the least-significant 1 in each bitmap, it would be very interesting to see how different approaches affect the algorithm. For example, in our algorithm we “left” truncated meaning that all bitmaps had to have a one in the least-significant position of the bitmap in order to move up the offset. It would be interesting to look at “right” truncation. If one bitmap is causing many of the others to not record incoming values perhaps it should be bumped up. Is there some math to back up this intuition?
  2. It is interesting to us that the fringe width truncation events are DV dependent. We struggled with the math on this for a bit before we just stopped. Essentially we want to know what is the width of the theoretical fringe? It obviously appears to be DV dependent and some sort of coupon collector problem with unequal probabilities. Someone with better math skills than us needs to help here.

Closing thoughts

We uncovered PCSA again as a way to go back to first principles and see if there are lessons to be learned that could be applied to HLL to make it even better. For instance, can all of this work on the fringe be applied to HLL to reduce the number of bits per register while still maintaining the same precision? HLL effectively records the “strandline” (what we call the left-most 1s). More research into how this strandline behaves and if it is possible to improve the storage of it through truncation could reduce the standard HLL register width from 5 bits to 4, a huge savings! Obviously, we uncovered a lot of open questions with this research and we feel there are algorithmic improvements to HLL right around the corner. We have done some preliminary tests and the results so far are intriguing. Stay tuned!

Sketch of the Day: HyperLogLog — Cornerstone of a Big Data Infrastructure


In the Zipfian world of AK, the HyperLogLog distinct value (DV) sketch reigns supreme. This DV sketch is the workhorse behind the majority of our DV counters (and we’re not alone) and enables us to have a real time, in memory data store with incredibly high throughput. HLL was conceived of by Flajolet et. al. in the phenomenal paper HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm. This sketch extends upon the earlier Loglog Counting of Large Cardinalities (Durand et. al.) which in turn is based on the seminal AMS work FM-85, Flajolet and Martin’s original work on probabilistic counting. (Many thanks to Jérémie Lumbroso for the correction of the history here. I am very much looking forward to his upcoming introduction to probabilistic counting in Flajolet’s complete works.) UPDATE – Rob has recently published a blog about PCSA, a direct precursor to LogLog counting which is filled with interesting thoughts. There have been a few posts on HLL recently so I thought I would dive into the intuition behind the sketch and into some of the details.

Just like all the other DV sketches, HyperLogLog looks for interesting things in the hashed values of your incoming data.  However, unlike other DV sketches HLL is based on bit pattern observables as opposed to KMV (and others) which are based on order statistics of a stream.  As Flajolet himself states:

Bit-pattern observables: these are based on certain patterns of bits occurring at the beginning of the (binary) S-values. For instance, observing in the stream S at the beginning of a string a bit- pattern O^{\rho-1}1 is more or less a likely indication that the cardinality n of S is at least 2^\rho.

Order statistics observables: these are based on order statistics, like the smallest (real) values, that appear in S. For instance, if X = min(S), we may legitimately hope that n is roughly of the order of 1/X…

In my mind HyperLogLog is really composed of two insights: Lots of crappy things are sometimes better than one really good thing; and bit pattern observables tell you a lot about a stream. We’re going to look at each component in turn.

Bad Estimator

Even though the literature refers to the HyperLogLog sketch as a different family of estimator than KMV I think they are very similar. It’s useful to understand the approach of HLL by reviewing the KMV sketch. Recall that KMV stores the smallest k values that you have seen in a stream. From these k values you get an estimate of the number of distinct elements you have seen so far. HLL also stores something similar to the smallest values ever seen. To see how this works it’s useful to ask “How could we make the KMV sketch smaller?” KMV stores the actual value of the incoming numbers. So you have to store k 64 bit values which is tiny, but not that tiny. What if we just stored the “rank” of the numbers?  Let’s say the number 94103 comes through (I’ll use base 10 here to make things easier). That number is basically 9*10^4 plus some stuff. So, let’s just store the exponent, i.e. 4. In this way I get an approximation of the size of numbers I have seen so far. That turns the original KMV algorithm into only having to store the numbers 1-19 (since 2^{64} \approx 10^{19}) which is a whole lot less than 2^{64} numbers. Of course, this estimate will be much worse than storing the actual values.

Bit Pattern Observables

In actuality HLL, just like all the other DV sketches, uses hashes of the incoming data in base 2. And instead of storing the “rank” of the incoming numbers HLL uses a nice trick of looking for runs of zeroes in the hash values. These runs of zeroes are an example of “bit pattern observables”. This concept is similar to recording the longest run of heads in a series of coin flips and using that to guess the number of times the coin was flipped. For instance, if you told me that you spent some time this afternoon flipping a coin and the longest run of heads you saw was 2 I could guess you didn’t flip the coin very many times. However, if you told me you saw a run of 100 heads in a row I would gather you were flipping the coin for quite a while. This “bit pattern observable”, the run of heads, gives me information about the stream of data it was pulled from. An interesting thing to note is just how probable long runs of heads are. As Mark Shilling points out, you can almost always tell the difference between a human generated set of coin flips and an actual one, due to humans not generating long runs. (The world of coin flipping seems to be a deep and crazy pit.) Disclaimer: The only thing I am trying to motivate here is that by keeping a very small piece of information (the longest run of heads) I can get some understanding of what has happened in a stream. Of course, you could probably guess that even though we have now reduced the storage of our sketch the DV estimate is pretty crummy. But what if we kept more than one of them?

Stochastic Averaging

In order to improve the estimate, the HLL algorithm stores many estimators instead of one and averages the results. However, in order to do this you would have to hash the incoming data through a bunch of independent hash functions. This approach isn’t a very good idea since hashing each value a bunch of times is expensive and finding good independent hash families is quite difficult in practice. The work around for this is to just use one hash function and “split up” the input into m buckets while maintaining the observable (longest run of zeroes) for each bucket. This procedure is called stochastic averaging. You could do this split in KMV as well and it’s easier to visualize. For an m of 3 it would look like:

To break the input into the m buckets, Durand suggests using the first few (k) bits of the hash value as an index into a bucket and compute the longest run of zeroes (R) on what is left over. For example, if your incoming datum looks like 010100000110 and k = 3 you could use the 3 rightmost bits, 110, to tell you which register to update (110_2 = 6) and from the remaining bits, 010100000, you could take the longest run of zeroes (up to some max), which in this case is 5. In order to compute the number of distinct values in the stream you would just take the average of all of the m buckets:

\displaystyle DV_{LL} = \displaystyle\text{constant} * m*2^{\overline{R}}

Here \overline{R} is the average of the values R in all the buckets. The formula above is actually the estimator for the LogLog algorithm, not HyperLogLog. To get HLL, you need one more piece…

Harmonic Mean

A fundamental insight that Flajolet had to improve LogLog into HyperLogLog was that he noticed the distribution of the values in the m registers is skewed to the right, and there can be some dramatic outliers that really mess up the average used in LogLog (see Fig. 1 below). He and Durand knew this when they wrote LogLog and did a bunch of hand-wavey stuff (like cut off the top 30% of the register values) to create what he called the “SuperLogLog”, but in retrospect this seems kind of dumb. He fixed this in HLL by tossing out the odd rules in SuperLogLog and deciding to take the harmonic mean of the DV estimates. The harmonic mean tends to throw out extreme values and behave well in this type of environment. This seems like an obvious thing to do. I’m a bit surprised they didn’t try this in the LogLog paper, but perhaps the math is harder to deal with when using the harmonic mean vs the geometric mean.

Fig. 1:  The theoretical distribution of register values after v distinct values have been run through an HLL.

Throw all these pieces together and you get the HyperLogLog DV estimator:

\displaystyle DV_{HLL} = \displaystyle\text{constant} * m^2 *\left (\sum_{j=1}^m 2^{-R_j} \right )^{-1}

Here R_j is the longest run of zeroes in the i^{th} bucket.

Putting it All Together

Even with the harmonic mean Flajolet still has to introduce a few “corrections” to the algorithm. When the HLL begins counting, most of the registers are empty and it takes a while to fill them up. In this range he introduces a “small range correction”. The other correction is when the HLL gets full. If a lot of distinct values have been run through an HLL the odds of collisions in your hash space increases. To correct for hash collisions Flajolet introduces the “large range collection”. The final algorithm looks like (it might be easier for some of you to just look at the source in the JavaScript HLL simulation):

m = 2^b #with b in [4...16]

if m == 16:
    alpha = 0.673
elif m == 32:
    alpha = 0.697
elif m == 64:
    alpha = 0.709
    alpha = 0.7213/(1 + 1.079/m)

registers = [0]*m # initialize m registers to 0

# Construct the HLL structure
for h in hashed(data):
    register_index = 1 + get_register_index( h,b ) # binary address of the rightmost b bits
    run_length = run_of_zeros( h,b ) # length of the run of zeroes starting at bit b+1
    registers[ register_index ] = max( registers[ register_index ], run_length )

# Determine the cardinality
DV_est = alpha * m^2 * 1/sum( 2^ -register )  # the DV estimate

if DV_est < 5/2 * m: # small range correction
    V = count_of_zero_registers( registers ) # the number of registers equal to zero
    if V == 0:  # if none of the registers are empty, use the HLL estimate
          DV = DV_est
          DV = m * log(m/V)  # i.e. balls and bins correction

if DV_est <= ( 1/30 * 2^32 ):  # intermediate range, no correction
     DV = DV_est
if DV_est > ( 1/30 * 2^32 ):  # large range correction
     DV = -2^32 * log( 1 - DV_est/2^32)

Rob wrote up an awesome HLL simulation for this post. You can get a real sense of how this thing works by playing around with different values and just watching how it grows over time. Click below to see how this all fits together.

HyperLogLog Simulation

Click above to run the HyperLogLog simulation


Unions are very straightforward to compute in HLL and, like KMV, are lossless. All you need to do to combine the register values of the 2 (or n) HLL sketches is take the max of the 2 (or n) register values and assign that to the union HLL. With a little thought you should realize that this is the same thing as if you had fed in the union stream to begin with. A nice side effect about lossless unions is that HLL sketches are trivially parallelizable. This is great if, like us, you are trying to digest a firehose of data and need multiple boxes to do summarization. So, you have:

for i in range(0, len(R_1)):
     R_new[i] = max( R_1[i], R_2[i] )

To combine HLL sketches that have differing sizes read Chris’s blog post about it.

Wrapping Up

In our research, and as the literature says, the HyperLogLog algorithm gives you the biggest bang for the buck for DV counting. It has the best accuracy per storage of all the DV counters to date. The biggest drawbacks we have seen are around intersections. Unlike KMV, there is no explicit intersection logic, you have to use the inclusion/exclusion principle and this gets really annoying for anything more than 3 sets. Aside from that, we’ve been tickled pink using HLL for our production reporting. We have even written a PostgreSQL HLL data type that supports cardinality, union, and intersection. This has enabled all kinds of efficiencies for our analytics teams as the round trips to Hadoop are less and most of the analysis can be done in SQL. We have seen a massive increase in the types of analytics that go on at AK since we have adopted a sketching infrastructure and I don’t think I’m crazy saying that many big data platforms will be built this way in the future.

P.S.  Sadly, Philippe Flajolet passed away in March 2011. It was actually a very sad day for us at Aggregate Knowledge because we were so deep in our HLL research at the time and would have loved to reach out to him, he seems like he would have been happy to see his theory put to practice. Based on all I’ve read about him I’m very sorry to have not met him in person. I’m sure his work will live on but we have definitely lost a great mind both in industry and academia. Keep counting Philippe!

Photo courtesy of