# Neighbor Discovery Optimization for Big Data Analysis in Low-Power, Low-Cost Communication Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

^{2}slots when a prime number p is selected for neighbor discovery. One of valuable contributions of U-Connect is the performance metric of neighbor discovery protocols called the power-latency (PL) product. U-Connect offers an alternative approach to Disco by using one prime number; however, it still demonstrates a weakness where two nodes are not likely to meet each other quickly in case of a mismatch during the early time slots of a neighbor discovery schedule. This shortcoming may result in a significant increase in the worst-case discovery latency.

## 3. Problem Statement

_{i}= 1. In this situation, they can communicate with each other if they are within the same communication range; thus, it can be noted that nodes u and v identify their neighbors. Conversely, it may be difficult for them to talk to each other when u is awake and v sleeps at slot i, or vice versa. In the latter scenario, an awakening node cannot avoid wasting its energy and the neighboring nodes u and v are unable to identify each other. For neighbor discovery optimization to achieve low-power, lost-cost communication networks, an ultimate goal is to minimize the number of wasted awakening slots as much as possible during the neighbor discovery process.

_{i}= 1. If we apply a bitwise XOR operation (^) to ${\mathcal{S}}_{u}\left(x\right)$ and ${\mathcal{S}}_{v}\left(x\right)$, then it would be possible to determine the number of wasted awakening slots that exist between ${\mathcal{S}}_{u}\left(x\right)$ and ${\mathcal{S}}_{v}\left(x\right)$. In addition, we can calculate the number of overlapping awakening slots that occur between ${\mathcal{S}}_{u}\left(x\right)$ and ${\mathcal{S}}_{v}\left(x\right)$ if we assign a bitwise AND operation (&) to ${\mathcal{S}}_{u}\left(x\right)$ and ${\mathcal{S}}_{v}\left(x\right)$. Let ${\mathcal{S}}_{u^v}\left(1\right)$ be the total number of wasted awakening slots and ${\mathcal{S}}_{u\&v}\left(1\right)$ be the total number of overlapping awakening slots between ${\mathcal{S}}_{u}\left(x\right)$ and ${\mathcal{S}}_{v}\left(x\right)$. Therefore, we define a neighbor discovery optimization problem as follows:

## 4. Block Design for Neighbor Discovery

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (1)
- X is a set of elements (points), and
- (2)
- A is a collection of non-empty subsets of X (blocks).

**Definition**

**5.**

- (1)
- |X| = v,
- (2)
- Each block has exactly k points, and
- (3)
- Every pair of distinct points is included in exactly λ blocks.

A = {{1,2,4}, {2,3,5}, {3,4,6},

{4,5,7}, {5,6,1}, {6,7,2}, {7,1,3}}.

**Theorem**

**1.**

**Theorem**

**2.**

_{i}and S

_{j}, are applied to the (X, A) symmetric-BIBD with parameters (v, k, λ), then the two DSs, S

_{i}and S

_{j}, have λ overlapping awakening slots.

**Proof:**

_{1}, p

_{2}, …, p

_{v}}, and

_{i}and S

_{j}, can be connected to two distinct blocks in A, respectively. Let two blocks in A be B

_{i}and B

_{j}. According to Theorem 1, there exists λ points in B

_{i}and B

_{j}. Consequently, it is possible to mention that | B

_{i}∩ B

_{j}| = λ. It shows that S

_{i}and S

_{j}, which are applied to B

_{i}and B

_{j}, respectively, have λ overlapped active slots. □

^{2}+ k + 1, k + 1, 1)-BIBD for λ = 1; further, this kind of BIBD is a symmetric-BIBD [22]. (7, 3, 1)-BIBD is a case where k = 2. By choosing an appropriate power of the prime k, we would be able to create several DSs with different duty cycles. Although this idea seems as a possible method for designing DSs, there is a practical challenge in adopting a (v, k, 1)-BIBD directly.

^{2}+ k + 1, k + 1, 1)-BIBD.

## 5. Block Construction Mechanism

**Definition**

**6.**

- (1)
- |X| = v,
- (2)
- Each block has exactly k awakening slots, and
- (3)
- Every pair of different blocks includes at least λ common awakening slots.

**Definition**

**7.**

_{a}, k

_{a}, λ

_{a})-BIBD and B = (v

_{b}, k

_{b}, λ

_{b})-BIBD. (4,3,2)- and (3,2,1)-designs were created to demonstrate the operation of the proposed combining process. The (4,3,2)- and (3,2,1)-designs are shown in Figure 3. We defined the (4, 3, 2)-design as base and the (3,2,1)-design as replacement.

_{a}× v

_{b}, k

_{a}× k

_{b}, λ

_{a}× λ

_{b})-NDD by implementing steps 1 and 2 in the final step. A new block design, (12,6,2)-NDD, is shown in Figure 4. It is finally produced by combining the (4,3,2) and (3,2,1)-designs.

**Definition**

**8.**

_{1}, k

_{1}, 1)-BIBD and $\mathcal{R}$ is declared as a (v

_{2}, k

_{2}, 1)-BIBD. F $\otimes \mathcal{R}$ represents that all the awakening slots in F are transformed to $\mathcal{R}$ and all the sleeping slots are changed into a v

_{2}$\times $v

_{2}size matrix of the sleep schedule.

**Theorem**

**3.**

_{1}, k

_{1}, 1)-BIBD and $\mathcal{R}$ is a (v

_{2}, k

_{2}, 1)-BIBD. F $\otimes $$\mathcal{R}$ results in a (v

_{1}× v

_{2}, k

_{1}× k

_{2}, 1)-NDD.

**Proof:**

_{i}| T

_{i}is a schedule}. We assumed that $\mathsf{\Psi}$ = F $\otimes $ $\mathcal{R}$. We know that $\mathsf{\Psi}$ = {${\mathsf{\Psi}}_{i}$ | 1 $\le $ | ${\mathsf{\Psi}}_{i}$ | $\le $ v

_{1}v

_{2}} is a (v

_{3}, k

_{3}, 1)-NDD, where v

_{3}= v

_{1}v

_{2}and k

_{3}= k

_{1}k

_{2}. Thus, ${\mathsf{\Psi}}_{j}$ $\in $ $\mathsf{\Psi},$ where i $\ne $ j, $\exists $ 1 $\le $ h $\le $ v

_{1}v

_{2}such that $\omega $

_{ih}= $\omega $

_{jh}= 1 for any pair of schedules ${\mathsf{\Psi}}_{i}$. Every awakening slot in $\mathrm{F}$ is transformed to $\mathcal{R}$ by $\mathrm{F}$ ⊗ $\mathcal{R}$. In addition, every sleeping slot in $\mathrm{F}$ is changed into a v

_{2}× v

_{2}sized matrix of all zeros. Hence, it is possible to express that the total number of points in $\mathsf{\Psi}$ is v

_{1}× v

_{2}and each block in $\mathsf{\Psi}$ has exactly k

_{1}× k

_{2}awakening slots. These two chrematistics perfectly correspond to the first and second properties in Definition 10. Consequently, we concluded that every pair of differing two blocks has at least λ common awakening slots. $\mathsf{\Psi}$ can be illustrated with the notation of the matrix as follows:

_{1}v

_{2}× v

_{1}v

_{2}sized matrix and each component of $\mathsf{\Psi}$ denotes an awakening or a sleeping slot. Schedules ${\mathsf{\Psi}}_{i}$ and ${\mathsf{\Psi}}_{j}$ represent one of the rows in $\mathsf{\Psi}$. $\mathrm{F}$ is a BIBD. Therefore, $\mathrm{F}$ has a schedule ${\mathrm{F}}_{i}$. Each ${\mathrm{F}}_{i}$ has at least one awakening slot, s

_{im}, where 1 ≤ m ≤ v

_{1}v

_{2}. According to Definition 8, s

_{im}is changed into $\mathcal{R}$.

_{1}. ${\mathsf{\Psi}}_{i}$ and ${\mathsf{\Psi}}_{j}$ contain at least one common awakening slot such that $\psi $

_{ih}= $\psi $

_{jh}= 1 by $\mathcal{R}$.

_{1}, ${\mathsf{\Psi}}_{i}$ and ${\mathsf{\Psi}}_{j}$ include at least one common awakening slot such that $\psi $

_{ih}= $\psi $

_{jh}= 1 by F. According to Cases I and II, schedules ${\mathsf{\Psi}}_{i}$ and ${\mathsf{\Psi}}_{j}$ in $\mathsf{\Psi}$ always contain at least λ common awakening slots. Finally, $\mathsf{\Psi}$ is a (v

_{1}× v

_{2}, k

_{1}× k

_{2}, 1)-NDD. □

## 6. Asymmetric Neighbor Discovery Algorithm

## 7. Numerical Analysis

## 8. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Protocol | DC | L | Active slots |
---|---|---|---|

Disco [12] | $\frac{{p}_{1}+{p}_{2}}{{p}_{1}\xb7{p}_{2}}$ | ${p}_{1}\xb7{p}_{2}$ | ${p}_{1}+{p}_{2}$ |

U-Connect [13] | $\frac{p+1}{{p}^{2}}$ | p^{2} | $\frac{3p-1}{2}$ |

Searchlight [15] | $\frac{2}{t}$ | $\frac{{t}^{2}}{2}$ | t |

Combinatorial + Multiples of k | $\frac{\left(k+1\right)+\alpha}{{k}^{2}+k+1}$ | ${k}^{2}+k+1$ | $\left(k+1\right)+\alpha $ |

Protocol | DC | |||
---|---|---|---|---|

10% | 5% | 2% | 1% | |

Disco | p_{1} = 13,p _{2} = 31 | p_{1} = 29,p _{2} = 61 | p_{1} = 97,p _{2} = 101 | p_{1} = 191,p _{2} = 211 |

U-Connect | p = 11 | p = 23 | p = 53 | p = 101 |

Searchlight | t = 20 | t = 40 | t = 100 | t = 200 |

Combinatorial + Multiples of k | k = 9 | k = 19 | k = 49 | k = 97 |

Protocol | DC | |||
---|---|---|---|---|

10% | 5% | 2% | 1% | |

Disco | 403 | 1769 | 9797 | 40,301 |

U-Connect | 121 | 529 | 2809 | 10,201 |

Searchlight | 200 | 800 | 5000 | 20,000 |

Combinatorial + Multiples of k | 91 | 381 | 2451 | 9507 |

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Choi, S.; Yi, G.
Neighbor Discovery Optimization for Big Data Analysis in Low-Power, Low-Cost Communication Networks. *Symmetry* **2019**, *11*, 836.
https://doi.org/10.3390/sym11070836

**AMA Style**

Choi S, Yi G.
Neighbor Discovery Optimization for Big Data Analysis in Low-Power, Low-Cost Communication Networks. *Symmetry*. 2019; 11(7):836.
https://doi.org/10.3390/sym11070836

**Chicago/Turabian Style**

Choi, Sangil, and Gangman Yi.
2019. "Neighbor Discovery Optimization for Big Data Analysis in Low-Power, Low-Cost Communication Networks" *Symmetry* 11, no. 7: 836.
https://doi.org/10.3390/sym11070836