# Cardinality Estimation

Cardinality estimation is a classic problem. Pinot solves it with multiple ways each of which has a trade-off between accuracy and latency.

## Exact Results

Functions:

**DistinctCount***(x) -> LONG*

Returns accurate count for all unique values in a column.

The underlying implementation is using a IntOpenHashSet in library: `it.unimi.dsi:fastutil:8.2.3`

to hold all the unique values.

## Approximate Results

It usually takes a lot of resources and time to compute exact results for unique counting on large datasets. In some circumstances, we can tolerate a certain error rate, in which case we can use approximation functions to tackle this problem.

### HyperLogLog

HyperLogLog is an approximation algorithm for unique counting. It uses fixed number of bits to estimate the cardinality of given data set.

Pinot leverages HyperLogLog Class in library `com.clearspring.analytics:stream:2.7.0`

as the data structure to hold intermediate results.

Functions:

**DistinctCountHLL**(x)_ -> LONG_

For column type **INT**/**LONG**/**FLOAT**/**DOUBLE**/**STRING**, Pinot treats each value as an individual entry to add into HyperLogLog Object, and then computes the approximation by calling method _**cardinality()**.

For column type **BYTES**, Pinot treats each value as a serialized HyperLogLog Object with pre-aggregated values inside. The bytes value is generated by `org.apache.pinot.core.common.ObjectSerDeUtils.HYPER_LOG_LOG_SER_DE.serialize(hyperLogLog)`

.

All deserialized HyperLogLog object will be merged into one then calling method **_cardinality()** to get the approximated unique count._

### HyperLogLogPlusPlus

The HyperLogLog++ algorithm proposes several improvements in the HyperLogLog algorithm to reduce memory requirements and increase accuracy in some ranges of cardinalities.

64-bit hash function is used instead of the 32 bits used in the original paper. This reduces the hash collisions for large cardinalities allowing to remove the large range correction.

Some bias is found for small cardinalities when switching from linear counting to the HLL counting. An empirical bias correction is proposed to mitigate the problem.

A sparse representation of the registers is implemented to reduce memory requirements for small cardinalities, which can be later transformed to a dense representation if the cardinality grows.

Pinot leverages HyperLogLogPlus Class in library `com.clearspring.analytics:stream:2.7.0`

as the data structure to hold intermediate results.

Functions:

**DistinctCountHLLPlus**(*<HllPlusColumn>*)_ -> LONG_**DistinctCountHLLPlus**(*<HllPlusColumn>, <p>*)_ -> LONG_**DistinctCountHLLPlus**(*<HllPlusColumn>, <p>, <sp>*)_ -> LONG_

For column type **INT**/**LONG**/**FLOAT**/**DOUBLE**/**STRING** , Pinot treats each value as an individual entry to add into HyperLogLogPlus Object, then compute the approximation by calling method **_cardinality()**.

For column type **BYTES**, Pinot treats each value as a serialized HyperLogLogPlus Object with pre-aggregated values inside. The bytes value is generated by `org.apache.pinot.core.common.ObjectSerDeUtils.HYPER_LOG_LOG_PLUS_SER_DE.serialize(hyperLogLogPlus)`

.

All deserialized HyperLogLogPlus object will be merged into one then calling method **_cardinality()** to get the approximated unique count._

### Theta Sketches

The Theta Sketch framework enables set operations over a stream of data, and can also be used for cardinality estimation. Pinot leverages the Sketch Class and its extensions from the library `org.apache.datasketches:datasketches-java:4.2.0`

to perform distinct counting as well as evaluating set operations.

Functions:

**DistinctCountThetaSketch(**<thetaSketchColumn>, <thetaSketchParams>, predicate1, predicate2..., postAggregationExpressionToEvaluate**) **-> LONGthetaSketchColumn (required): Name of the column to aggregate on.

thetaSketchParams (required): Parameters for constructing the intermediate theta-sketches. Currently, the only supported parameter is

`nominalEntries`

.predicates (optional)_: _ These are individual predicates of form

`lhs <op> rhs`

which are applied on rows selected by the`where`

clause. During intermediate sketch aggregation, sketches from the`thetaSketchColumn`

that satisfies these predicates are unionized individually. For example, all filtered rows that match`country=USA`

are unionized into a single sketch. Complex predicates that are created by combining (AND/OR) of individual predicates is supported.postAggregationExpressionToEvaluate (required)

*:*The set operation to perform on the individual intermediate sketches for each of the predicates. Currently supported operations are`SET_DIFF, SET_UNION, SET_INTERSECT`

, where DIFF requires two arguments and the UNION/INTERSECT allow more than two arguments.

In the example query below, the `where`

clause is responsible for identifying the matching rows. Note, the where clause can be completely independent of the `postAggregationExpression`

. Once matching rows are identified, each server unionizes all the sketches that match the individual predicates, i.e. `country='USA'`

, `device='mobile'`

in this case. Once the broker receives the intermediate sketches for each of these individual predicates from all servers, it performs the final aggregation by evaluating the `postAggregationExpression`

and returns the final cardinality of the resulting sketch.

**DistinctCountRawThetaSketch(**<thetaSketchColumn>, <thetaSketchParams>, predicate1, predicate2..., postAggregationExpressionToEvaluate**)** -> HexEncoded Serialized Sketch Bytes

This is the same as the previous function, except it returns the byte serialized sketch instead of the cardinality sketch. Since Pinot returns responses as JSON strings, bytes are returned as hex encoded strings. The hex encoded string can be deserialized into sketch by using the library `org.apache.commons.codec.binary`

as `Hex.decodeHex(stringValue.toCharArray())`

.

### Tuple Sketches

The Tuple Sketch is an extension of the Theta Sketch. Tuple sketches store an additional summary value with each retained entry which makes the sketch ideal for summarizing attributes such as impressions or clicks. Tuple sketches are interoperable with the Theta Sketch and enable set operations over a stream of data, and can also be used for cardinality estimation.

Functions:

**avgValueIntegerSumTupleSketch(**<tupleSketchColumn>, <tupleSketchLgK>**) -> LongtupleSketchColumn (required): Name of the column to aggregate on.

tupleSketchLgK (optional): lgK which is the the log2 of K, which controls both the size and accuracy of the sketch.

This function can be used to combine the summary values from the random sample stored within the Tuple sketch and formulate an estimate for an average that applies to the entire dataset. The average should be interpreted as applying to each key tracked by the sketch and is rounded to the nearest whole number.

**distinctCountTupleSketch(**<tupleSketchColumn>, <tupleSketchLgK>**) -> LONGtupleSketchColumn (required): Name of the column to aggregate on.

tupleSketchLgK (optional): lgK which is the the log2 of K, which controls both the size and accuracy of the sketch.

This returns the cardinality estimate for a column where the values are already encoded as Tuple sketches, stored as BYTES.

**distinctCountRawIntegerSumTupleSketch(**<tupleSketchColumn>, <tupleSketchLgK>**) -> HexEncoded Serialized Sketch Bytes

This is the same as the previous function, except it returns the byte serialized sketch instead of the cardinality sketch. Since Pinot returns responses as JSON strings, bytes are returned as hex encoded strings. The hex encoded string can be deserialized into sketch by using the library `org.apache.commons.codec.binary`

as `Hex.decodeHex(stringValue.toCharArray())`

.

**sumValuesIntegerSumTupleSketch(**<tupleSketchColumn>, <tupleSketchLgK>**) -> LongtupleSketchColumn (required): Name of the column to aggregate on.

tupleSketchLgK (optional): lgK which is the the log2 of K, which controls both the size and accuracy of the sketch.

This function can be used to combine the summary values (using `sum`

) from the random sample stored within the Tuple sketch and formulate an estimate that applies to the entire dataset. See `avgValueIntegerSumTupleSketch`

for extracting an average for integer summaries. If other merging options are required, it is best to extract the raw sketches directly or to implement a new Pinot aggregation function to support these.

### Compressed Probability Counting (CPC) Sketches

The Compressed Probability Counting(CPC) Sketch enables extremely space-efficient cardinality estimation. The stored CPC sketch can consume about 40% less space than an HLL sketch of comparable accuracy. Pinot can aggregate multiple existing CPC sketches together to get a total distinct count or estimated directly from raw values.

Functions:

**distinctCountCpcSketch(**<cpcSketchColumn>, <cpcSketchLgK>**) -> Long`cpcSketchColumn`

(required): Name of the column to aggregate on.`cpcSketchLgK`

(optional): lgK which is the the log2 of K, which controls both the size and accuracy of the sketch.

This returns the cardinality estimate for a column.

**distinctCountRawCpcSketch(**<cpcSketchColumn>, <cpcSketchLgK>**) -> HexEncoded Serialized Sketch Bytes`cpcSketchColumn`

(required): Name of the column to aggregate on.`cpcSketchLgK`

(optional): lgK which is the the log2 of K, which controls both the size and accuracy of the sketch.

This is the same as the previous function, except it returns the byte serialized sketch instead of the cardinality sketch. Since Pinot returns responses as JSON strings, bytes are returned as hex encoded strings. The hex encoded string can be deserialized into sketch by using the library `org.apache.commons.codec.binary`

as `Hex.decodeHex(stringValue.toCharArray())`

.

### UltraLogLog (ULL) Sketches

The UltraLogLog Sketch from Dynatrace is a variant of *HyperLogLog* and is used for approximate distinct counts. The UltraLogLog sketch shares many of the same properties of a typical HyperLogLog sketch but requires less space and also provides a simpler and faster estimator.

Pinot uses an production-ready Java implementation available in Hash4j available under the Apache license.

Functions:

**distinctCountULL(**<ullSketchColumn>, <ullSketchPrecision>**) -> Long`ullSketchColumn`

(required): Name of the column to aggregate on.`ullSketchPrecision`

(optional): p which is the precision parameter, which controls both the size and accuracy of the sketch.

This returns the cardinality estimate for a column.

**distinctCountRawULL(**<cpcSketchColumn>, <ullSketchPrecision>**) -> HexEncoded Serialized Sketch Bytes`ullSketchColumn`

(required): Name of the column to aggregate on.`ullSketchPrecision`

(optional): p which is the precision parameter, which controls both the size and accuracy of the sketch.

`org.apache.commons.codec.binary`

as `Hex.decodeHex(stringValue.toCharArray())`

.