Author: Øyvind Langsrud

# The problem of singletons and zeros in ‘easySdcTable’

#### Øyvind Langsrud

## Introduction

This document discuss background for changes made in easySdcTable after parameter `threshold` was introduced as a new possibility in sdcTable.

    library(easySdcTable)

### Note added in version 1.1.1

The terminology in this vignette is somewhat misleading. According to Langsrud (2024), “ones and zeros” should have been used instead of “singletons and zeros.”

Langsrud, Ø. (2024): Secondary Cell Suppression by Gaussian Elimination: An Algorithm Suitable for Handling Issues with Zeros and Singletons, Published in the Springer LNCS proceedings of the conference Privacy in Statistical Databases.

## Example datasets

Below are four two-way example datasets. The data is organized here in wide format so that the frequencies are in several columns. It is thus one row variable and one column variable. The dataset, data1b, comes from Kristian Lønø. He used this to point out a problem that has led to changes in the latest version of r-package sdcTable. The details are below in this document. The other datasets are modified variants.

    data1a = data.frame(row = c("r1","r2"), A=c(0,2), B=c(1,0), H=c(7,0), M=c(1,2), W=c(0,8))
    data1b = data.frame(row = c("r1","r2"), A=c(1,1), B=c(1,0), H=c(7,0), M=c(1,2), W=c(0,8))
    data0a = data.frame(row = c("r1","r2"), A=c(5,5), B=c(0,9), H=c(7,9), M=c(0,5), W=c(9,8))
    data0b = data.frame(row = c("r1","r2"), A=c(0,0), B=c(0,9), H=c(7,9), M=c(0,2), W=c(9,8))

data1a
row | A | B | H | M | W
--- | --- | --- | --- | --- | ---
r1 | 0 | 1 | 7 | 1 | 0
r2 | 2 | 0 | 0 | 2 | 8

data1b
row | A | B | H | M | W
--- | --- | --- | --- | --- | ---
r1 | 1 | 1 | 7 | 1 | 0
r2 | 1 | 0 | 0 | 2 | 8

data0a
row | A | B | H | M | W
--- | --- | --- | --- | --- | ---
r1 | 5 | 0 | 7 | 0 | 9
r2 | 5 | 9 | 9 | 5 | 8

data0b
row | A | B | H | M | W
--- | --- | --- | --- | --- | ---
r1 | 0 | 0 | 7 | 0 | 9
r2 | 0 | 9 | 9 | 2 | 8

## The singleton problem

In the first run of the first dataset, we use `protectZeros = FALSE`. This means that 0s are not suppressed. All 0s are shown and none of them are secondary suppressed. We use the (previously) usual method, `"SIMPLEHEURISTIC_OLD"`.

    s1a = ProtectTable(data1a, 1, 2:6, protectZeros = FALSE, method = "SIMPLEHEURISTIC_OLD",
        suppression = ".")$suppressed

s1a
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | 0 | . | 7 | . | 0 | 9
r2 | . | 0 | 0 | . | 8 | 12
Total | . | . | 7 | . | 8 | 21

Here it is easy to reveal that both the suppressed numbers in the first row must be 1 since the sum should be 9 and neither number can be 0. This is called the singleton problem. In the underlying function of sdcTable there is a parameter, `detectSingletons` (default is `FALSE`), which is intended to handle this problem. Such parameters in sdcTable can also be used as input to ProtectTable.

    s1aSingle = ProtectTable(data1a, 1, 2:6, protectZeros = FALSE, method = "SIMPLEHEURISTIC_OLD",
        detectSingletons = TRUE, suppression = ".")$suppressed

s1aSingle
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | 0 | . | . | . | 0 | 9
r2 | . | 0 | 0 | . | 8 | 12
Total | . | . | . | . | 8 | 21

Now it is sufficiently suppressed so that the values can no longer be revealed. In the next dataset it will be different.

    s1bSingle = ProtectTable(data1b, 1, 2:6, protectZeros = FALSE, method = "SIMPLEHEURISTIC_OLD",
        detectSingletons = TRUE, suppression = ".")$suppressed

s1bSingle
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | 7 | . | 0 | 10
r2 | . | 0 | 0 | . | . | 11
Total | . | . | 7 | . | . | 21

We can reveal that the suppressed numbers in the first row must be 1. This problem has led to changes in the latest version of sdcTable. A new parameter, `threshold`, is introduced.

## Solving the singleton problem by using `threshold`

The new parameter, `threshold`, is a number that can be specified. The parameter means that the sum of the suppressed cells is required to be at least threshold. This means that `threshold = 3` will solve problems in a similar way to `detectSingletons = TRUE`. In the case of data1b, then the problem is not solved since the sum is already 3. But the problem can be solved by setting `threshold = 4`.

    s1bThreshold4 = ProtectTable(data1b, 1, 2:6, protectZeros = FALSE, method = "SIMPLEHEURISTIC_OLD",
        threshold = 4, suppression = ".")$suppressed

s1bThreshold4
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | . | . | 0 | 10
r2 | . | 0 | 0 | . | . | 11
Total | . | . | . | . | . | 21

Now it has been suppressed sufficiently. But a problem is that one cannot know, without examining the data, what threshold is needed. It is not difficult to create example data where `threshold = 4` is not enough. One could imagine a very large value of `threshold`. The threshold parameter affects not only 1s but also other suppressed numbers. Above, not only were the 7s in the first row removed, but also the 8s in the second row. It might not be required. It is possible to know that there must be 1 and 2 in the row, but not complete disclosure.

In an imagined example where 4 is secondary suppressed to protect 2, extra cells will be suppressed if `threshold = 7`. So, the parameter threshold does not solve the singleton problem in an optimal way. But as shown below, this looks better in the case where zeros are suppressed.

## The problem of zeros

Now we consider data0a use `protectZeros = TRUE`. This means that 0s are primary suppressed.

    s0a = ProtectTable(data0a, 1, 2:6, protectZeros = TRUE, method = "SIMPLEHEURISTIC_OLD",
        suppression = ".")$suppressed

s0a
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | 5 | . | 7 | . | 9 | 21
r2 | 5 | . | 9 | . | 8 | 36
Total | 10 | 9 | 16 | 5 | 17 | 57

Here it is easy to reveal that both the suppressed numbers in the first row must be 0 since the sum of the numbers shown is already 21. This problem is similar to the problem with 1s, but it is not called the singleton problem. It doesn’t help using `detectSingletons = TRUE`. The answer will be the same.

In the next data set (data0b) there are three 0’s and the problem is the same.

    s0b = ProtectTable(data0b, 1, 2:6, protectZeros = TRUE, method = "SIMPLEHEURISTIC_OLD",
        suppression = ".")$suppressed

s0b
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | 7 | . | 9 | 16
r2 | . | . | 9 | . | 8 | 28
Total | . | 9 | 16 | . | 17 | 44

## Solving the problem of zeros by using `threshold=1`

The threshold parameter solves the above problem (data0a). It is sufficient to set `threshold = 1` to prevent only 0s being suppressed.

    s0aThreshold1 = ProtectTable(data0a, 1, 2:6, protectZeros = TRUE, method = "SIMPLEHEURISTIC_OLD",
        threshold = 1, suppression = ".")$suppressed

s0aThreshold1
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | 7 | . | 9 | 21
r2 | . | . | 9 | . | 8 | 36
Total | 10 | 9 | 16 | 5 | 17 | 57

When there are three (as below, data0b) or more zeros, the problem is also solved. It is suppressed extra to avoid disclosure.

    s0bThreshold1 = ProtectTable(data0b, 1, 2:6, protectZeros = TRUE, method = "SIMPLEHEURISTIC_OLD",
        threshold = 1, suppression = ".")$suppressed

s0bThreshold1
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | . | . | 9 | 16
r2 | . | . | . | . | 8 | 28
Total | . | 9 | 16 | . | 17 | 44

## Discussion and new default

The new parameter threshold is not an optimal solution to the singleton problem (1s). Users must consider what value to use. What is great is that the threshold parameter solves problems with 0’s, ie when `protectZeros = TRUE`.

Note also that the threshold parameter can be used to increase the degree of protection in general, even without 0s or 1s.

In easySdcTable, `protectZeros = TRUE` is the default. It is not in sdcTable. The parameter is also renamed. The method `"SIMPLEHEURISTIC"` which is default in sdcTable, has also been default easySdcTable. This is now changed to `"SimpleSingle"` whos new definition is:

- When `protectZeros=FALSE`: `"SIMPLEHEURISTIC"` with `detectSingletons=TRUE`.
- when `protectZeros=TRUE`: `"SIMPLEHEURISTIC"` with `threshold=1` (can be overridden by input).

The problem of zeros is solved. Otherwise, the data is protected the old way using `detectSingletons`. In addition, it is possible to manually set the parameter `threshold` to provide better protection. If this is done, the parameter `detectSingletons` will not be used.

Note that parameters `detectSingletons` and `threshold` increase the computing time.

## Note after easySdcTable version 0.8.0

Method `"Gauss"` made default (See NEWS).

For all the examples to still be relevant, `"SIMPLEHEURISTIC_OLD"` is used instead of `"SIMPLEHEURISTIC"`. In the solution after `threshold=1`, more cells than earlier (more than needed) are suppressed.

Methodology to handle the problem of singletons and zeros are also included in “Gauss” . Below is output:

    s1aGauss = ProtectTable(data1a, 1, 2:6, protectZeros = FALSE, suppression = ".")$suppressed

s1aGauss
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | 0 | . | . | . | 0 | 9
r2 | . | 0 | 0 | . | 8 | 12
Total | . | . | . | . | 8 | 21

    s1bGauss = ProtectTable(data1b, 1, 2:6, protectZeros = FALSE, suppression = ".")$suppressed

s1bGauss
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | . | . | 0 | 10
r2 | . | 0 | 0 | . | 8 | 11
Total | . | . | . | . | 8 | 21

    s0aGauss = ProtectTable(data0a, 1, 2:6, protectZeros = TRUE, suppression = ".")$suppressed

s0aGauss
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | 7 | . | 9 | 21
r2 | . | . | 9 | . | 8 | 36
Total | 10 | 9 | 16 | 5 | 17 | 57

    s0bGauss = ProtectTable(data0b, 1, 2:6, protectZeros = TRUE, suppression = ".")$suppressed

s0bGauss
row | A | B | H | M | W | Total
--- | --- | --- | --- | --- | --- | ---
r1 | . | . | 7 | . | . | 16
r2 | . | . | 9 | . | . | 28
Total | . | 9 | 16 | . | 17 | 44