The TupleSketch is an extension of the ThetaSketch and both are part of the Theta Sketch Framework1. In this document, the term Theta (upper case) when referencing sketches will refer to both the ThetaSketch and the TupleSketch. This is not to be confused with the term theta (lower case), which refers to the sketch variable that tracks the sampling probability of the sketch.
Because Theta sketches provide the set operations of intersection and difference (A and not B or just A not B), a number of interesting corner cases arise that require some analysis to determine how the code should handle them.
Theta sketches track three key variables in addition to retained data:
theta: This is the current sampling probability of the sketch and mathematically expressed as a 64-bit, double floating value between 0.0 and 1.0. However, internally in the sketch, this value is expressed as a 64-bit, signed, long integer (usually identified as thetaLong in the code), where the maximum positive value (Long.MAX_VALUE) is interpreted as the double 1.0. In this document we will only refer to the mathematical quantity theta.
retained entries or count: This is the number of hash values currently retained in the sketch. It can never be less than zero.
empty:
We have developed a short hand notation for these three variables to record their state as {theta, retained entries, empty}. When analyzing the corner cases of the set operations, we only need to know whether theta is 1.0 or less than 1.0, retained entries is zero or greater than zero, and empty is true or false. These are further abbreviated as
Each of the above three variables can be represented as boolean variable. Thus, there are 8 possible combinations of the three variables.
1 Anirban Dasgupta, Kevin J. Lang, Lee Rhodes, and Justin Thaler. A framework for estimating stream expression cardinalities. In *EDBT/ICDT Proceedings ‘16 *, pages 6:1–6:17, 2016.
Of the eight possible combinations of the three variables and using the above notation, there are five valid states of a Theta sketch.
When a new sketch is created, theta is set to 1.0, retained entries is set to zero, and empty is true. This state can also occur as the result of a set operation, where the operation creates a new sketch to potentially load result data into the sketch but there is no data to load into the sketch. So it effectively returns a new sketch that has been untouched and unaffected by the input arguments to the set operation.
All of the Theta sketches have an input buffer that is effectively a list of items received by the sketch. If the number of unique input values does not exceed the size of that buffer, the sketch is in exact mode. There is no probabilistic estimation involved so theta = 1.0, which indicates that all unique values presented to the sketch are in the buffer. retained entries is the count of those values in the buffer, and the sketch is clearly not empty.
Here, the number of inputs to the sketch have exceeded the size of the input buffer, so the sketch must start choosing what values to retain in the sketch and starts reducing the value of theta accordingly. theta < 1.0, retained entries > 0, and empty = F.
This is a new sketch where the user has set the sampling probability, p < 1.0 and the sketch has not been presented any data. Internally at initialization, theta is set to p, so if p = 0.5, theta will be set to 0.5. Since the sketch has not seen any data, retained entries = 0 and empty = T. This is degenerative form of a new sketch, thus its name.
This requires some explanation. Imagine the intersection of two estimating sketches where the values retained in the two sketches are disjoint (i.e, no overlap). Since the two sketches chose their internal values at random, there remains some probability that there could be common values in an exactly computed intersection, but it just so happens that one of the two sketches did not select any of them in the random sampling process. Therefore, the retained entries = 0. The value 1.0 - theta represents the probability that there could be intersecting values in the exact distribution. Since there is a positive probability of an intersection, empty = F. This is also a degenerative case in the sense that theta < 1.0 and empty = F like an estimating sketch, except that no actual values were found in the operation, so retained entries = 0.
The Has Seen Data column is not an independent variable, but helps with the interpretation of the state.
We can assign a single octal digit ID to each state where
| Shorthand Notation | theta | retained entries | empty | Has Seen Data | ID | Comments |
|---|---|---|---|---|---|---|
| New {1.0,0,T} | 1.0 | 0 | T | F | 5 | New Sketch, p=1.0 (default) |
| Exact {1.0,>0,F} | 1.0 | >0 | F | T | 6 | Exact Mode |
| Estimation {<1.0,>0,F} | <1.0 | >0 | F | T | 2 | Estimation Mode |
| NewDegen {<1.0,0,T}2 | <1.0 | 0 | T | F | 1 | New Sketch, user sets p<1.0 |
| ResultDegen {<1.0,0,F}3 | <1.0 | 0 | F | T | 0 | Valid Intersect or AnotB result |
2 New Degenerate: New Empty Sketch where the user sets p < 1.0. This can be safely reinterpreted as {1.0,0,T} because it has not seen any data.
3 Result Degenerate: Can appear as a result of a an Intersection or AnotB of certain combination of sketches.
The remaining three combinations of the variables represent internal errors and should not occur. The Has Seen Data column is not an independent variable, but helps with the interpretation of the state.
| Theta | Retained Entries | Empty Flag | Has Seen Data | Comments |
|---|---|---|---|---|
| 1.0 | 0 | T | T | If it has seen data, Theta != 1.0 AND Entries = 0. |
| 1.0 | >0 | F | F | If it has not seen data, Entries !> 0. |
| <1.0 | >0 | F | F | If it has not seen data, Entries !> 0. |
Each sketch can have 5 valid states, which means we can have 25 combinations of states of two sketches as expanded in the following table.
| ID | Sketch A | Sketch B | Intersection Result | AnotB Result | Result Actions |
|---|---|---|---|---|---|
| 00 | ResultDegen {<1.0,0,F} | ResultDegen {<1.0,0,F} | New {minTheta,0,F} | New {minTheta,0,F} | 2,2 |
| 01 | ResultDegen {<1.0,0,F} | NewDegen {<1.0,0,T} | New {1.0,0,T} | New {ThetaA,0,F} | 1,3 |
| 02 | ResultDegen {<1.0,0,F} | Estimation {<1.0,>0,F} | New {minTheta,0,F} | New {minTheta,0,F} | 2,2 |
| 05 | ResultDegen {<1.0,0,F} | New {1.0,0,T} | New {1.0,0,T} | New {ThetaA,0,F} | 1,3 |
| 06 | ResultDegen {<1.0,0,F} | Exact {1.0,>0,F} | New {minTheta,0,F} | New {ThetaA,0,F} | 2,3 |
| 10 | NewDegen {<1.0,0,T} | ResultDegen {<1.0,0,F} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 11 | NewDegen {<1.0,0,T} | NewDegen {<1.0,0,T} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 12 | NewDegen {<1.0,0,T} | Estimation {<1.0,>0,F} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 15 | NewDegen {<1.0,0,T} | New {1.0,0,T} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 16 | NewDegen {<1.0,0,T} | Exact {1.0,>0,F} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 20 | Estimation {<1.0,>0,F} | ResultDegen {<1.0,0,F} | New {minTheta,0,F} | Trim A by minTheta | 2,4 |
| 21 | Estimation {<1.0,>0,F} | NewDegen {<1.0,0,T} | New {1.0,0,T} | Sketch A | 1,5 |
| 22 | Estimation {<1.0,>0,F} | Estimation {<1.0,>0,F} | Full Intersect | Full AnotB | 6,7 |
| 25 | Estimation {<1.0,>0,F} | New {1.0,0,T} | New {1.0,0,T} | Sketch A | 1,5 |
| 26 | Estimation {<1.0,>0,F} | Exact {1.0,>0,F} | Full Intersect | Full AnotB | 6,7 |
| 50 | New {1.0,0,T} | ResultDegen {<1.0,0,F} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 51 | New {1.0,0,T} | NewDegen {<1.0,0,T} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 52 | New {1.0,0,T} | Estimation {<1.0,>0,F} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 55 | New {1.0,0,T} | New {1.0,0,T} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 56 | New {1.0,0,T} | Exact {1.0,>0,F} | New {1.0,0,T} | New {1.0,0,T} | 1,1 |
| 60 | Exact {1.0,>0,F} | ResultDegen {<1.0,0,F} | New {minTheta,0,F} | Trim A by minTheta | 2,4 |
| 61 | Exact {1.0,>0,F} | NewDegen {<1.0,0,T} | New {1.0,0,T} | Sketch A | 1,5 |
| 62 | Exact {1.0,>0,F} | Estimation {<1.0,>0,F} | Full Intersect | Full AnotB | 6,7 |
| 65 | Exact {1.0,>0,F} | New {1.0,0,T} | New {1.0,0,T} | Sketch A | 1,5 |
| 66 | Exact {1.0,>0,F} | Exact {1.0,>0,F} | Full Intersect | Full AnotB | 6,7 |
The description of each column:
| Result Action | Result Code | Description |
|---|---|---|
| New{1.0,0,T} | 1 | New empty sketch |
| New{min,0,F} | 2 | Min=min(thetaA,thetaB) |
| New{thA,0,F} | 3 | thA=theta of A |
| SkA Min | 4 | Trim A by minTheta |
| Sketch A | 5 | Sketch A exactly |
| Full Inter | 6 | Full intersect |
| Full AnotB | 7 | Full AnotB |
Note that the results of a Full Intersect or a Full AnotB will require further interpretation of the resulting state. For example, if the resulting sketch is {1.0,0,?}, then a New{1.0,0,T} is returned. If the resulting sketch is {<1.0,0,?} then a ResultDegen{<1.0,0,F} is returned.
Otherwise, the sketch returned will be an estimating or exact {theta, >0, F}.
The above information is encoded as a model into the special class org.apache.datasketches.SetOperationCornerCases.java. This class is made up of enums and static methods to quickly determine for a sketch what actions to take based on the state of the input arguments. This model is independent of the implementation of the Theta Sketch, whether the set operation is performed as a Theta Sketch, or a Tuple Sketch and when translated can be used in other languages as well.
Before this model was put to use an extensive set of tests was designed to test any potential implementation against this model. These tests are slightly different for the Tuple Sketch than the Theta Sketch because the Tuple Sketch has more combinations to test, but the model is the same.
The details of how this mode is used in run-time code can be found in the class org.apache.datasketches.tuple.AnotB.java.