# Parallel Computation of Rough Set Approximations in Information Systems with Missing Decision Data

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Basic Concepts

#### 3.1. Rough Set

#### 3.2. MapReduce Model

**Map**takes an input pair ($<{K}_{1},{V}_{1}>$) and produces a set of intermediate key/value pairs ($[<{K}_{2},{V}_{2}>]$). The MapReduce library groups all intermediate values associated with the intermediate key ${K}_{2}$, shuffles, sorts, and sends them to the Reduce function.

**Reduce**accepts an intermediate key (${K}_{2}$) and a set of values for that key ($\left[{V}_{2}\right]$). It merges these values together to create a possibly smaller set of values, and finally produces $<{K}_{3},{V}_{3}>$ pairs as output.

#### 3.3. The Usage of MapReduce to Rough Set Processing

## 4. Twofold Rough Approximations for IDS

**Example**

**1.**

## 5. Computing Rough Approximations in IDS

#### 5.1. Sequential Algorithm

Algorithm 1 Sequential algorithm to calculate twofold rough approximations |

#### 5.2. MapReduce Based Algorithms

#### 5.2.1. Computing Equivalence Classes in Parallel (EC)

**Proposition**

**1.**

**Proof.**

**Example**

**2.**

Algorithm 2 function EC Map |

Algorithm 3 function EC Reduce |

**Example**

**3.**

#### 5.2.2. Computing Possible and Certain Equivalence Classes in Parallel (PEC)

**Proposition**

**2.**

**Proof.**

Algorithm 4 function PEC Map |

Algorithm 5 function PEC Reduce |

**Example**

**4.**

#### 5.2.3. Aggregating Possible and Certain Equivalence Classes in Parallel (AP)

**Proposition**

**3.**

**Proposition**

**4.**

**Proof.**

Algorithm 6 function AP Map |

Algorithm 7 function AP Reduce |

**Example**

**5.**

#### 5.2.4. Computing Rough Approximations in Parallel (RA)

**Proposition**

**5.**

**Proof.**

Algorithm 8 function RA Map |

Algorithm 9 function RA Reduce |

**Example**

**6.**

**Proposition**

**6.**

**Proof.**

#### 5.3. Evaluation Test

- Execution time increases when the volume of data increases in both sequential and parallel algorithms.
- The most intensive step is RA step, and the least intensive step is EC, PEC step. The AP step takes less time than the RA step in our experiments because we aggregate very few condition attributes. The more attributes we aggregate, the more time the AP step will take.
- In AP and RA steps, the parallel algorithms outperform the sequential algorithm. To the dataset Kdd100, the former performs 25 times faster than the latter at the AP step, and four times faster at the RA step, respectively. This is important since these are the most intensive computational steps. For the EC, PEC step, the parallel algorithm costs more time. This is because we divide this step into two separate MapReduce jobs: EC and PEC. Since each job requires a certain time to start up its mappers and reducers, the time consumed by both jobs becomes larger than the one of the sequential algorithm, especially when the input data is small. Notice that the time difference becomes smaller when the input data is larger (63 s in case of Kdd50, and 32 s in case of Kdd100). It is intuitive that the parallel algorithm is more efficient if we input larger datasets. In addition, since this step costs the least amount of time, it will not impact the total execution time of both algorithms.
- As the size of the input data increases, the parallel algorithm outperforms the sequential algorithm. We can verify this through the total execution time. The parallel algorithm is around four times less than the sequential algorithm in datasets Kdd50 and Kdd100. This proves the efficiency of our proposed parallel algorithm.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Compare execution time of each step between sequential and parallel algorithms: (

**a**) Step 1 of the sequential algorithm and EC, PEC steps of the parallel algorithm, (

**b**) Step 2 of the sequential algorithm and AP step of the parallel algorithm, (

**c**) Steps 3,4 of the sequential algorithm and RA step of the parallel algorithm, and (

**d**) total steps of the sequential and parallel algorithms.

${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | d | |
---|---|---|---|

${x}_{1}$ | 1 | 2 | 0 |

${x}_{2}$ | * | 1 | 1 |

${x}_{3}$ | 2 | * | 2 |

${x}_{4}$ | 1 | 2 | * |

${x}_{5}$ | 1 | 1 | 0 |

${x}_{6}$ | 2 | * | 2 |

${x}_{7}$ | * | 2 | 0 |

${x}_{8}$ | 3 | 1 | 3 |

a | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ |
---|---|---|

${U}_{a\ne \ast}/a$ | $\begin{array}{c}\hfill c\{{x}_{1},{x}_{4},{x}_{5}\},\{{x}_{3},{x}_{6}\},\left\{{x}_{8}\right\}\end{array}$ | $\{{x}_{2},{x}_{5},{x}_{8}\},\{{x}_{1},{x}_{4},{x}_{7}\}$ |

${U}_{a=\ast}$ | $\{{x}_{2},{x}_{7}\}$ | $\{{x}_{3},{x}_{6}\}$ |

$Cer(U/a)$ | $\begin{array}{c}\hfill c\{{x}_{1},{x}_{4},{x}_{5}\},\{{x}_{3},{x}_{6}\},\left\{{x}_{8}\right\},\varnothing \end{array}$ | $\{{x}_{2},{x}_{5},{x}_{8}\},\{{x}_{1},{x}_{4},{x}_{7}\},\varnothing $ |

$Pos(U/a)$ | $\begin{array}{c}\hfill c\{{x}_{2},{x}_{7}\},\{{x}_{1},{x}_{2},{x}_{4},{x}_{5},{x}_{7}\},\{{x}_{2},{x}_{3},{x}_{6},{x}_{7}\},\{{x}_{2},{x}_{7},{x}_{8}\}\end{array}$ | $\begin{array}{c}\hfill c\{{x}_{3},{x}_{6}\},\{{x}_{2},{x}_{3},{x}_{5},{x}_{6},{x}_{8}\},\{{x}_{1},{x}_{3},{x}_{4},{x}_{6},{x}_{7}\}\end{array}$ |

**Table 3.**Calculate ${\underline{r}}_{A}^{Cer}\left(x\right)$ and ${\overline{r}}_{A}^{Cer}\left(x\right)$.

$\mathit{x}\in {\mathit{X}}_{\mathit{C}\mathit{e}\mathit{r}}$ | ${\underline{\mathit{r}}}_{\mathit{A}}^{\mathit{C}\mathit{e}\mathit{r}}\left(\mathit{x}\right)$ | ${\overline{\mathit{r}}}_{\mathit{A}}^{\mathit{C}\mathit{e}\mathit{r}}\left(\mathit{x}\right)$ |
---|---|---|

$\{{x}_{1},{x}_{5},{x}_{7}\}$ | ∅ | $\left\{{x}_{5}\right\},\{{x}_{1},{x}_{4}\}$ |

$\left\{{x}_{2}\right\}$ | ∅ | ∅ |

$\{{x}_{3},{x}_{6}\}$ | ∅ | ∅ |

$\left\{{x}_{8}\right\}$ | ∅ | $\left\{{x}_{8}\right\}$ |

**Table 4.**Calculate ${\underline{r}}_{A}^{Pos}\left(x\right)$ and ${\overline{r}}_{A}^{Pos}\left(x\right)$.

$\mathit{x}\in {\mathit{X}}_{\mathit{P}\mathit{o}\mathit{s}}$ | ${\underline{\mathit{r}}}_{\mathit{A}}^{\mathit{P}\mathit{o}\mathit{s}}\left(\mathit{x}\right)$ | ${\overline{\mathit{r}}}_{\mathit{A}}^{\mathit{P}\mathit{o}\mathit{s}}\left(\mathit{x}\right)$ |
---|---|---|

$\{{x}_{1},{x}_{4},{x}_{5},{x}_{7}\}$ | $\left\{{x}_{5}\right\},\left\{{x}_{7}\right\},\{{x}_{1},{x}_{4},{x}_{7}\}$ | $\left\{{x}_{7}\right\},\{{x}_{2},{x}_{5}\},\{{x}_{3},{x}_{6},{x}_{7}\},\{{x}_{1},{x}_{4},{x}_{7}\}$ |

$\{{x}_{2},{x}_{4}\}$ | $\left\{{x}_{2}\right\},\left\{{x}_{4}\right\}$ | $\left\{{x}_{2}\right\},\{{x}_{2},{x}_{8}\},\{{x}_{2},{x}_{5}\},\{{x}_{2},{x}_{3},{x}_{6}\},\{{x}_{1},{x}_{4},{x}_{7}\}$ |

$\{{x}_{3},{x}_{4},{x}_{6}\}$ | $\{{x}_{3},{x}_{6}\},\left\{{x}_{4}\right\}$ | $\begin{array}{c}\hfill c\{{x}_{3},{x}_{6}\},\{{x}_{2},{x}_{3},{x}_{6}\},\{{x}_{3},{x}_{6},{x}_{7}\},\{{x}_{1},{x}_{4},{x}_{7}\}\end{array}$ |

$\{{x}_{4},{x}_{8}\}$ | $\left\{{x}_{8}\right\},\left\{{x}_{4}\right\}$ | $\{{x}_{2},{x}_{8}\},\{{x}_{1},{x}_{4},{x}_{7}\}$ |

$\left\{{x}_{4}\right\}$ | $\left\{{x}_{4}\right\}$ | $\{{x}_{1},{x}_{4},{x}_{7}\}$ |

${\mathit{U}}^{1}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | d | ${\mathit{U}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | d |
---|---|---|---|---|---|---|---|

${x}_{1}$ | 1 | 2 | 0 | ${x}_{5}$ | 1 | 1 | 0 |

${x}_{2}$ | * | 1 | 1 | ${x}_{6}$ | 2 | * | 2 |

${x}_{3}$ | 2 | * | 2 | ${x}_{7}$ | * | 2 | 1 |

${x}_{4}$ | 1 | 2 | * | ${x}_{8}$ | 3 | 1 | 3 |

Records | Size (MB) | |
---|---|---|

Kdd10 | 489,844 | 44 |

Kdd50 | 2,449,217 | 219 |

Kdd10 | 4,898,431 | 402 |

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**MDPI and ACS Style**

Cao, T.; Yamada, K.; Unehara, M.; Suzuki, I.; Nguyen, D.V.
Parallel Computation of Rough Set Approximations in Information Systems with Missing Decision Data. *Computers* **2018**, *7*, 44.
https://doi.org/10.3390/computers7030044

**AMA Style**

Cao T, Yamada K, Unehara M, Suzuki I, Nguyen DV.
Parallel Computation of Rough Set Approximations in Information Systems with Missing Decision Data. *Computers*. 2018; 7(3):44.
https://doi.org/10.3390/computers7030044

**Chicago/Turabian Style**

Cao, Thinh, Koichi Yamada, Muneyuki Unehara, Izumi Suzuki, and Do Van Nguyen.
2018. "Parallel Computation of Rough Set Approximations in Information Systems with Missing Decision Data" *Computers* 7, no. 3: 44.
https://doi.org/10.3390/computers7030044