# Theta Sketch and Tuple Sketch Set Operation Corner Cases

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 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 real number between 0 and 1 inclusive. In the code it is expressed as a double-precision (64-bit) floating-point value. 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:

• By definition, if empty = true, the number of retained entries must be zero. The value of theta is irrelevant in this case, and can be assumed to be 1.0.
• If empty = false, the retained entries can be zero or greater than zero, and theta can be 1.0 or less than 1.0.

We have developed a shorthand 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

• theta can be 1.0 or <1.0
• retained entries can be either 0 or >0
• empty can be either T or F

Each of the above three states can be represented as a 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 2016, pages 6:1–6:17, 2016.

## Valid States of a Sketch

Of the eight possible combinations of the three boolean variables and using the above notation, there are four valid states of a Theta sketch.

### Empty{1.0, 0, T}

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 empty sketch that has been untouched and unaffected by the input arguments to the set operation.

### Exact{1.0, >0, F}

All of the Theta sketches have an internal buffer that is effectively a list of hash values of the items received by the sketch. If the number of distinct input items 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 distinct values are in the buffer. retained entries is the count of those values in the buffer, and the sketch is not empty.

### Estimation{<1.0, >0, F}

Here, the number of distinct inputs to the sketch have exceeded the size of the 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.

### Degenerate{<1.0, 0, F}2

This requires some explanation.

Imagine we have two large data sets, A and B, with only a few items in common. The exact intersection of these two sets, A∩B would result in those few common items.

Now suppose we compute Sketch(A) and Sketch(B). Because sketches are approximate and the items from each set are chosen at random, there is some probability that one of the sketches may not contain any of the common items. As a result, the sketch intersection of these two sets, Sketch(A)∩Sketch(B), which is also approximate, might contain zero retained entries. Even though the retained entries are zero, the upper bound of the estimated number of distinct values from the input domain is clearly greater than zero, but missed by the sketch intersection. This upper bound can be computed statistically. It is too complex to discuss further here, but the sketch code actually performs this estimation.

Where both input sketches are non-empty, there is a non-zero probability that the intersection will have zero entries, yet the statistics tell us that the result may not be really empty, we may have been just unlucky.
We indicate this by setting the result empty = F, and retained entries = 0. The resulting theta = min(thetaA, thetaB). Calling getUpperBound(…) on the resulting intersection will reveal the best estimate of how many values might exist in the intersection of the raw data. The getLowerBound(…) will be zero because it is also possible that the two sets, A and B, were exactly disjoint.

2Note that this degenerate state can also result from an AnotB operation or the Union operation, which will be discussed below.

### Summary Table of the Valid States of a Sketch

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

• theta = 1.0 := 4, else 0
• retained entries >0 := 2, else 0
• empty = true := 1, else 0

The octal digit ID = ((theta == 1.0) ? 4 : 0) | ((retainedEntries > 0) ? 2 : 0) | (empty ? 1 : 0);

Shorthand
Notation
Theta Retained
Entries
Empty Has Seen
Data
Empty
{1.0,0,T}
1.0 0 T F 5 Empty Sketch
Exact
{1.0,>0,F}
1.0 >0 F T 6 Exact Mode Sketch
Estimation
{<1.0,>0,F}
<1.0 >0 F T 2 Estimation Mode Sketch
Degenerate
{<1.0,0,F}3
<1.0 0 F T 0 Degenerate and valid
Intersect or AnotB result

3 Degenerate: This can occur as an estimating result of a an Intersection of two disjoint sets, an AnotB of two identical sets, or the Union of two Degenerate sets.

## Invalid States of a Sketch

The remaining four combinations of the variables are invalid 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 | F | T | If it has seen data, Empty = F.4
∴ Theta cannot be = 1.0 AND Entries = 0 | | 1.0 | >0 | T | F | If it has not seen data, Empty = T.
∴ Entries cannot be > 0 | | <1.0 | >0 | T | F | If it has not seen data, Empty = T.
∴ Theta cannot be < 1.0 OR Entries > 0 | | <1.0 | 0 | T | F | If it has not seen data, Empty = T.5
∴ Theta cannot be < 1.0 | — 4This can occur internally as the result from an intersection of two exact, disjoint sets, or AnotB of two exact, identical sets. There is no probability distribution, so this is converted internally to EMPTY {1.0, 0, T}. A Union cannot produce this result.

5This can occur internally as the initial state of an UpdateSketch if p was set to less than 1.0 by the user and the sketch has not seen any data. There is no probability distribution because the sketch has not been offered any data, so this is converted internally to EMPTY {1.0, 0, T}.

## State Combinations of Two Sketches and Set Operation Results

Each sketch can have four valid states, which means we can have 16 combinations of states of two sketches as expanded in the following table.

Sketch A
State
Sketch B
State
Pair
ID
Intersection
Action
AnotB
Action
Union
Action
Action IDs
Empty
{1.0,0,T}
Empty
{1.0,0,T}
55 Empty
{1.0,0,T}=A=B
Empty
{1.0,0,T}=A
Empty
{1.0,0,T}=A=B
E,E,E
Empty
{1.0,0,T}
Exact
{1.0,>0,F}
56 Empty
{1.0,0,T}=A
Empty
{1.0,0,T}=A
Sketch B E,E,B
Empty
{1.0,0,T}
Estimation
{<1.0,>0,F}
52 Empty
{1.0,0,T}=A
Empty
{1.0,0,T}=A
Sketch B E,E,B
Empty
{1.0,0,T}
Degenerate
{<1.0,0,F}
50 Empty
{1.0,0,T}=A
Empty
{1.0,0,T}=A
Degenerate
{ThetaB,0,F}=B
E,E,DB
Exact
{1.0,>0,F}
Empty
{1.0,0,T}
65 Empty
{1.0,0,T}=B
Sketch A Sketch A E,A,A
Exact
{1.0,>0,F}
Exact
{1.0,>0,F}
66 Full Intersect Full AnotB Full Union I,N,U
Exact
{1.0,>0,F}
Estimation
{<1.0,>0,F}
62 Full Intersect Full AnotB Full Union I,N,U
Exact
{1.0,>0,F}
Degenerate
{<1.0,0,F}
60 Degenerate
{ThetaB,0,F}=B
Trim A
by minTheta
Trim A
by minTheta
D,TA,TA
Estimation
{<1.0,>0,F}
Empty
{1.0,0,T}
25 Empty
{1.0,0,T}=B
Sketch A Sketch A E,A,A
Estimation
{<1.0,>0,F}
Exact
{1.0,>0,F}
26 Full Intersect Full AnotB Full Union I,N,U
Estimation
{<1.0,>0,F}
Estimation
{<1.0,>0,F}
22 Full Intersect Full AnotB Full Union I,N,U
Estimation
{<1.0,>0,F}
Degenerate
{<1.0,0,F}
20 Degenerate
{minTheta,0,F}
Trim A
by minTheta
Trim A
by minTheta
D,TA,TA
Degenerate
{<1.0,0,F}
Empty
{1.0,0,T}
05 Empty
{1.0,0,T}=B
Degenerate
{ThetaA,0,F}=A
Degenerate
{ThetaA,0,F}=A
E,DA,DA
Degenerate
{<1.0,0,F}
Exact
{1.0,>0,F}
06 Degenerate
{ThetaA,0,F}=A
Degenerate
{ThetaA,0,F}=A
Trim B
by minTheta
DA,DA,TB
Degenerate
{<1.0,0,F}
Estimation
{<1.0,>0,F}
02 Degenerate
{minTheta,0,F}
Degenerate
{minTheta,0,F}
Trim B
by minTheta
D,D,TB
Degenerate
{<1.0,0,F}
Degenerate
{<1.0,0,F}
00 Degenerate
{minTheta,0,F}
Degenerate
{minTheta,0,F}
Degenerate
{minTheta,0,F}
D,D,D

Column Descriptions:

• Pair ID: two octal digits, the first digit represents the state of Sketch A, the second digit represents the state of Sketch B.
• Sketch A State
• Sketch B State
• Intersection Action
• AnotB Action
• Union Action
• Action Codes: Intersection, AnotB, Union.

The action IDs are given by the following table along with description and where used:

Action ID Action
Description
Intersection AnotB Union
A Sketch A
TA Trim Sketch A
by minTheta

B Sketch B
TB Trim Sketch B
by minTheta

D Degenerate
{minTheta,0,F}
DA Degenerate
{ThetaA,0,F}
(optional)

DB Degenerate
{ThetaB,0,F}
(optional)

E Empty
{1.0,0,T}
I Full Intersect
N Full AnotB
U Full Union

Note that the results of Full Intersect, Full AnotB, or Full Union actions will require further interpretation of the resulting state. For example:

• If the resulting sketch is {1.0,0,?}, then an Empty{1.0,0,T} is returned.
• If the resulting sketch is {<1.0,0,?} then a Degenerate{<1.0,0,F} is returned.
• Otherwise, the sketch returned will be an estimating {minTheta, >0, F}, or exact {1.0, >0, F}.

## Testing

The above information is encoded as a model into the special class org.apache.datasketches.SetOperationsCornerCases. 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 tests for the Theta Sketch can be found in the class org.apache.datasketches.theta.CornerCaseThetaSetOperationsTest

The tests for the Tuple Sketch can be found in the class org.apache.datasketches.tuple.aninteger.CornerCaseTupleSetOperationsTest

The details of how this model is used in run-time code can be found in the class org.apache.datasketches.tuple.AnotB.java.