# A Continuous Region-Based Skyline Computation for a Group of Mobile Users

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- We formally introduce the problem of finding the optimal meeting points, also known as skylines, of a group of users who are on the move and justify the significance of addressing the problem. In general, the optimal meeting points, i.e., skyline objects, that best meet the spatial and non-spatial preferences of the users are identified at a specified time interval until, finally, a skyline object is selected by the group of users as a meeting point. Several definitions are formally put forward, which include properties of a user, properties of an object, dominance, non-spatial dominance, spatial dominance, dominance in a space, skylines of a space, region of interest, search region, and object of interest.
- We propose an efficient solution, named Region-based Skyline for a Group of Mobile Users (RSGMU) method, which is designed mainly for deriving skyline objects for a group of users while they are on the move. RSGMU assumes a centroid-based movement where users are assumed to be moving towards a centroid. Unlike previous works [3,4,7,8,9,10] that require users to frequently report their latest locations, RSGMU utilises the dynamic motion formula to predict the locations of the users at a specified time interval, which results in the skyline objects being continuously updated. In addition, the skyline results are derived in advance before the users have actually arrived at the location. Meanwhile, to avoid re-computation of skylines at each time interval, the objects of interest are organized in a Kd-tree data structure.
- We conduct extensive experiments with various parameter settings to prove RSGMU′s capabilities in deriving the skyline objects for a group of users while they are on the move. These parameter settings are time interval, number of objects, data dimensionality, density, space size, number of users in a group, number of groups of users, and number of skyline objects.

## 2. Related Works

#### 2.1. Skyline Queries for a Mobile User

#### 2.2. Skyline Queries for a Group of Static Users

^{2}S

^{2}) and Voronoi-based Spatial Skyline (VS

^{2}) for static query points and Voronoi-based Continuous SSQ (VCS

^{2}) for streaming query points. This work was then extended in [5] for spatial network databases. Two algorithms were proposed, namely, SNS

^{2}and VSNS

^{2}, which compute the spatial skyline with respect to the network distance in a spatial network database. Later, [6] proposed a new index structure, termed VoR-Tree, which incorporates Voronoi diagram and Delaunay graph of a set of objects into an R-tree that indexes their geometries. The data structure is experimented on various Nearest Neighbour (NN) queries that include kNN, Reverse kNN, k Aggregate NN, and spatial skyline queries on point data.

#### 2.3. Skyline Queries for a Group of Mobile Users

^{2}) was proposed for streaming query points, $Q$, whose points change location over time (i.e. mobile). The spatial skyline was updated by exploiting the pattern of change in $Q$ to avoid unnecessary recomputation of the skyline.

- (a)
- (b)
- The works by [2,5,6,27,28,29,30] that focus on non-continuous skyline query for a group of users used the initial locations of the users in determining the skyline objects. Hence, the type of movement either arbitrary or centroid-based was not significant in those studies. However, users in the group can always update their locations by submitting a new query.
- (c)
- Continuous skyline query for a group of users as studied by [4] assumed an arbitrary movement where users are required to frequently update their latest locations.
- (d)
- Most of the works used the spatial attributes of the users as well as the spatial and non-spatial attributes of the objects in obtaining the skyline objects [2,3,7,8,9,10,27,28,29,30], except for the works by [4,5,6], in which the non-spatial attributes of the objects were not taken into consideration. Hence, [4,5,6] defined skyline objects as those objects that best meet the spatial preferences of the group of users.
- (e)
- Our proposed method, RSGMU, differs from the works reported in this paper regarding the following: (i) RSGMU assumes a centroid-based movement to ensure that the travelling distance of each user is almost the same, so that they are able to meet on time. (ii) Unlike [2,4,6,27,28,29,30], which utilised a specific data structure to organise the objects in the whole space, RSGMU uses Kd-tree to organise the identified objects of interest that fall within a certain subspace. Hence, the changes in the users′ locations involve a smaller number of objects to be traversed. (iii) RSGMU utilises the dynamic motion formula to predict the locations of the users at a specified time interval, which results in the skyline objects to be continuously updated. On the other hand, users are required to frequently update their latest locations in [3,4,7,8,9,10] in order to continuously update the skyline objects. (iv) Although most works [2,3,7,8,9,10,27,28,29,30] used the spatial attributes of the users as well as the spatial and non-spatial attributes of the objects in obtaining the skyline objects, they are limited as their solutions are unable to cater the movement of the users.

Author | Skyline Algorithm | Evaluation Criteria | Type of Query | Method of Tracking User′s Location | Data Structure Used | Single/ Group | Type of Movement | ||
---|---|---|---|---|---|---|---|---|---|

User | Object | ||||||||

Spatial | Spatial | Non- Spatial | |||||||

[8] | Continuous Skyline Query | √ | √ | √ | Continuous skyline query | Update | Kinetic | Single | Arbitrary |

[9] | I-SKY, N-SKY, Incremental I-SKY | √ | √ | √ | Range-based skyline query (RSQ), continuous RSQ, Probabilistic RSQ query | Update | × | Single | Arbitrary |

[10] | Location-based arbitrary-subspace skyline queries (LASQs) | √ | √ | √ | Continuous range-based skyline query | Update | × | Single | Arbitrary |

[3] | Landmark-based (LBA), index-based (IBA) | √ | √ | √ | Continuous range-based skyline query | Update | × | Single | Arbitrary |

[7] | Intersection node aggregation algorithm (INAA), link remolding algorithm (LMA), link fitting algorithm (LFA) | √ | √ | √ | Continuous range-based skyline | Update | × | Single | Arbitrary |

[4] | Branch-and-Bound Spatial Skyline (B^{2}S^{2}), Voronoi-based Spatial Skyline (VS^{2}), Voronoi-based Continuous SSQ (VCS^{2}) | √ | √ | × | Spatial skyline query (SSQ), Continuous SSQ | Update | R-tree, Voronoi diagram | Group | Arbitrary |

[5] | SNS^{2}, VSNS^{2} | √ | √ | × | Spatial skyline query | × | × | Group | × |

[6] | Voronoi-based and R-tree Spatial Skyline (VoR-Tree) | √ | √ | × | Spatial skyline query | × | Voronoi diagram, Delaunay graph | Group | × |

[27] | VR (Voronoi and R-tree) | √ | √ | √ | Spatial skyline query | × | R-tree, Voronoi diagram | Group | × |

[28] | VR (Voronoi and R-tree) | √ | √ | √ | Spatial skyline query | × | R-tree and Voronoi diagram | Group | × |

[29] | Group Skyline Algorithm (GSA) | √ | √ | √ | Imperfect spatial skyline query | × | R-tree and Voronoi diagram | Group | × |

[2,30] | Region-based Skyline for a Group of Users (RSGU) | √ | √ | √ | Spatial skyline query | × | R-tree, fragment | Group | × |

Our Method | Region-based Skyline for a Group of Mobile Users (RSGMU) | √ | √ | √ | Continuous SSQ | Predict/ Update | Kd-tree | Group | Centroid-based |

## 3. Preliminaries

#### 3.1. Definitions and Notations

**Definition**

**1.**

**Definition**

**2.**

ID | Location |
---|---|

u_{1} | (2, 6) |

u_{2} | (1.6, 3.2) |

u_{3} | (6, 3.2) |

ID | Location | Rating | Fee | ID | Location | Rating | Fee |
---|---|---|---|---|---|---|---|

${o}_{1}$ | (8, 6) | 2 | 80 | ${o}_{9}$ | (14, 9) | 2 | 90 |

${o}_{2}$ | (1, 4) | 2 | 80 | ${o}_{10}$ | (8, 1) | 3 | 95 |

${o}_{3}$ | (1, 2) | 3 | 80 | ${o}_{11}$ | (5.9, 5.8) | 2 | 100 |

${o}_{4}$ | (8, 8) | 1 | 60 | ${o}_{12}$ | (4, 8) | 3 | 95 |

${o}_{5}$ | (7, 3) | 2 | 90 | ${o}_{13}$ | (2, 7) | 3 | 92 |

${o}_{6}$ | (6, 6) | 2 | 80 | ${o}_{14}$ | (−2, 9) | 3 | 100 |

${o}_{7}$ | (3.1, 4) | 3 | 65 | ${o}_{15}$ | (5, 7) | 2 | 93 |

${o}_{8}$ | (10, 3.5) | 3 | 65 |

**Definition**

**3**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

#### 3.2. Sample Data

## 4. The Proposed Method

#### 4.1. Identify the Centroid

Algorithm 1: Identify the Centroid |

Input: A group of users, ${G}_{p}=\left\{{u}_{1},{u}_{2},\dots ,{u}_{p}\right\}$ with each user given as ${u}_{i}({x}_{i},{y}_{i})$ |

Output: Centroid, $C\left({x}_{C},{y}_{C}\right)$ |

1. Begin |

2. $C\left({x}_{C}=\frac{{{\displaystyle \sum}}_{i=1}^{n}xi}{n},{y}_{C}=\frac{{{\displaystyle \sum}}_{i=1}^{n}yi}{n}\right)$ /* Equation (1) |

3. End |

#### 4.2. Predict the Location of a User

Algorithm 2: Predict the Location of a User |

Input: A group of users, ${G}_{p}=\left\{{u}_{1},{u}_{2},\dots ,{u}_{p}\right\}$ with each user given as ${u}_{i}({x}_{i},{y}_{i})$; Centroid $C\left({x}_{C},{y}_{C}\right)$ |

Output: A group of users, ${G}_{p}=\left\{{u}_{1},{u}_{2},\dots ,{u}_{p}\right\}$ with each user’s location being updated as ${u}_{i}(x{\prime}_{i},y{\prime}_{i}$) |

1. Begin |

2. For each ${u}_{i}\in {G}_{p}$ do |

3. Calculate the displacement ${\mathcal{R}}_{{u}_{i}C}=\sqrt{{\left({x}_{C}-{x}_{i}\right)}^{2}+{\left({y}_{C}-{y}_{i}\right)}^{2}}$ /* Equation (7) |

4. Calculate the cosine $\theta $ and sine $\theta $ as follows: $\mathrm{cos}\theta =({x}_{C}-{x}_{i})/{\mathcal{R}}_{{u}_{i}C}$ /* Equation (8) $\mathrm{sin}\theta =({y}_{C}-{y}_{i})/{\mathcal{R}}_{{u}_{i}C}$ /* Equation (9) |

5. Calculate the displacement $\Delta R$ at time ${t}_{n}$ with speed ${v}_{i}$, $\Delta R={v}_{i}\ast \Delta t$ where $\Delta t={t}_{n}-{t}_{n-1}$ /* Equation (11) |

6. Calculate $x{\prime}_{i}$ and $y{\prime}_{i}$ at time ${t}_{n}$: $x{\prime}_{i}={x}_{i}+\Delta R\ast \mathrm{cos}\theta $ /* Equation (14) $y{\prime}_{i}=$ ${y}_{i}+\Delta R\ast \mathrm{sin}\theta $ /* Equation (16) |

7. End |

8. End |

#### 4.3. Construct a Search Region

#### 4.3.1. Identify the Search Region for Each User, ${S}_{{u}_{\mathrm{i}}}$

#### 4.3.2. Identify the Search Region Given a Group of Users, S_{G}

_{G}, which is derived as follows: S

_{G}= ${\mathrm{U}}_{i=1}^{n}{S}_{ui}$.

**Figure 6.**(

**a**) The search region for a group of users. (

**b**) The general region, $GR$, with radius ${R}_{GR}$.

#### 4.4. Construct a Kd-Tree

#### 4.4.1. Identify the General Region of a Group of Users

#### 4.4.2. Construct a Minimum Bounding Rectangle (MBR) Based on the General Region

- (i)
- In calculating the coordinates of a MBR, eight operations as described above need to be performed, i.e., $\mathrm{min}x$, $\mathrm{max}x$, $\mathrm{min}y$, $\mathrm{max}y$, $bl$, $tl$, $br$, and $tr$. The calculation of the coordinates of the MBR for MMBR requires $8\times n$ operations, while for SMBR only eight operations are needed.
- (ii)
- The MMBR is a subset of SMBR and thus the objects that fall within the area of MMBR are also the objects that fall within the area of SMBR but not vice versa. As such, the number of objects covered by MMBR is lesser than those covered by SMBR.
- (iii)
- Both MMBR and SMBR might contain uninterested objects; however, the number of uninterested objects in MMBR is lesser than or equal to the number of uninterested objects in SMBR.

#### 4.4.3. Build a Kd-Tree Based on the MBR

- (1)
- Traverse the Kd-tree in a depth first traversal manner.
- (2)
- For each visited object in the Kd-tree, ${o}_{i}$, if the object falls within the region of any search region of an individual user, ${S}_{{u}_{j}}$, then the object is considered as one of the objects of interest for the group of users. There are three possible cases based on the location of an object and its relevance position to a search region. These cases are as follows:
- (a)
- The object is outside the search region of an individual user, ${S}_{{u}_{i}}$. Here, if the Euclidean distance between the object ${o}_{i}$ and the user ${u}_{j}$ is greater than the radius of the search region, ${R}_{{u}_{j}}$, then ${o}_{i}$ is said to be outside the boundary of ${S}_{{u}_{j}}$. This condition is written as $Ed({o}_{i}$, ${u}_{j})>{R}_{{u}_{j}}$.
- (b)
- The object falls within the search region of an individual user, ${S}_{{u}_{j}}$. Here, if the Euclidean distance between the object ${o}_{i}$ and the user ${u}_{j}$ is less than the radius of the search region, ${R}_{{u}_{j}}$, then ${o}_{i}$ is said to be within the boundary of ${S}_{{u}_{j}}$. This condition is written as $Ed({o}_{i}$, ${u}_{j})<{R}_{{u}_{j}}$.
- (c)
- The object intersects with the boundary of a search region of an individual user, ${S}_{{u}_{j}}$. Here, if the Euclidean distance between the object ${o}_{i}$ and the user ${u}_{j}$ is equal to the radius of the search region, ${R}_{{u}_{j}}$, then ${o}_{i}$ is said to intersect with the boundary of ${S}_{{u}_{j}}$. This condition is written as $Ed({o}_{i}$, ${u}_{j})={R}_{{u}_{j}}$.

Algorithm 3: Construct a Kd-tree Algorithm |

Input: A group of users, ${G}_{p}=\left\{{u}_{1},{u}_{2},\dots ,{u}_{p}\right\}$ with each user given as ${u}_{i}({x}_{i},{y}_{i});$ A set of objects $O=\left\{{o}_{1},{o}_{2},\dots ,{o}_{m}\right\}$; The nearest object to $C$, ${o}_{n}$; A search region of each user ${u}_{i}$$,{S}_{{u}_{i}}$ with radius ${R}_{{u}_{i}}$ |

Output: A list of objects of interest, ${L}_{O}$ |

1. Begin |

2. ${L}_{O}=\left\{\right\}$ |

3. For each ${u}_{i}\in {G}_{p}$ do |

4. Obtain the Euclidean distance between ${u}_{i}$$\mathrm{and}{o}_{n}$$,Ed\left({o}_{n},{u}_{i}\right)$ |

5. End |

6. Obtain the farthest user, ${u}_{f}$, from ${o}_{n}$, where $\left\{{u}_{f}|{u}_{f}\in {G}_{p}\wedge \forall {u}_{j}\in {G}_{p}-\left\{{u}_{f}\right\}:Ed\left({o}_{n},{u}_{f}\right)\rangle Ed\left({o}_{n},{u}_{j}\right)\right\}$ |

7. Identify the radius of the general region ${R}_{GR}={R}_{{u}_{f}{o}_{n}}\times 2$ |

8. Construct the general region $GR=$ area bound by a circle with radius ${R}_{GR}$ and ${o}_{n}$ as the center point |

9. Construct an $MBR$ with the following vertices: $\mathrm{min}x={x}_{{o}_{n}}-{R}_{GR}$; $\mathrm{max}x={x}_{{o}_{n}}+{R}_{GR}$; $\mathrm{min}y={y}_{{o}_{n}}-{R}_{GR}$; $\mathrm{max}y={y}_{{o}_{n}}+{R}_{GR}$; $bl=(\mathrm{min}x,\mathrm{min}y)$; $tl=(\mathrm{min}x,\mathrm{max}y)$; $br=(\mathrm{max}x,\mathrm{min}y)$, and $tr=(\mathrm{max}x,\mathrm{max}y)$ |

10. For each ${o}_{i}\in O$ do |

11. If ${o}_{i}$ is within the $MBR$, then insert ${o}_{i}$ into the Kd-tree |

12. End |

13. Obtain the object of interest, ${L}_{O}$, by traversing the Kd-tree: If $\exists {u}_{j}\in {G}_{p},Ed({o}_{i}$, ${u}_{j})<{R}_{{u}_{j}}$ or $\exists {u}_{j}\in {G}_{p},Ed({o}_{i}$, ${u}_{j})={R}_{{u}_{j}}$, then ${L}_{O}={L}_{O}{{\displaystyle \cup}}^{}{o}_{i}$ |

14. End |

#### 4.5. Derive the Skylines

#### 4.5.1. Derive the Spatial Skylines

#### 4.5.2. Derive the Non-Spatial Skylines

#### 4.5.3. Derive the Final Skylines

Algorithm 4: Derive the Skylines Algorithm |

Input: A list of objects of interest, ${L}_{O}=\left\{{o}_{1},{o}_{2},\dots ,{o}_{o}\right\}$; A group of users, ${G}_{p}=\left\{{u}_{1},{u}_{2},\dots ,{u}_{p}\right\}$ |

Output: Final skylines, $Sk{y}_{{G}_{p}}$ |

1. Begin |

2. Let $T{L}_{O}={L}_{O}$ |

3. For each ${o}_{i}\in T{L}_{O}$ do |

4. For each ${o}_{j}\in T{L}_{O}$ do |

5. If ${o}_{i}{\prec}_{ns}{o}_{j}$ then $T{L}_{O}=T{L}_{O}-{o}_{j}$ /* Definition 4 |

6. Else If ${o}_{j}{\prec}_{ns}{o}_{i}$ then $T{L}_{O}=T{L}_{O}-{o}_{i}$ /* Definition 4 |

7. End |

8. End |

9 $Sk{y}_{n{s}_{{G}_{p}}}=T{L}_{O}$ |

10. Let $T{L}_{O}={L}_{O}$ |

11. For each ${o}_{i}\in T{L}_{O}$ do |

12. For each $\mathrm{For}\mathrm{each}{u}_{j}\in {G}_{p}$ do |

13. Get the distance between ${o}_{i}$ and ${u}_{j}$$,{o}_{i}-{u}_{j}$ |

14. End |

15. $SumDistance-{o}_{i}={\displaystyle \sum}_{j=1}^{p}{o}_{i}-{u}_{j}$ |

16. End |

17. Sort the objects of $T{L}_{O}$ based on the $SumDistance-{o}_{i}$ in ascending order |

18. For each ${o}_{i}\in T{L}_{O}$ do |

19. For each ${o}_{j}\in T{L}_{O}$ do |

20. If ${o}_{i}{\prec}_{s}{o}_{j}$ then $T{L}_{O}=T{L}_{O}-{o}_{j}$ /* Definition 5 |

21. Else If ${o}_{j}{\prec}_{s}{o}_{i}$ then $T{L}_{O}=T{L}_{O}-{o}_{i}$ /* Definition 5 |

22. End |

23. End |

24. $Sk{y}_{{s}_{{G}_{p}}}=T{L}_{O}$ |

25. $Sk{y}_{{G}_{p}}=Sk{y}_{n{s}_{{G}_{p}}}{{\displaystyle \cup}}^{}Sk{y}_{{s}_{{G}_{p}}}$ /* Definition 7 |

26. End |

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of (

**a**) a search region based on the initial locations and (

**b**) search region based on the new locations of a group of users.

**Figure 3.**(

**a**) The regions of interest, ${\mathcal{R}}_{{u}_{1}}$ and ${\mathcal{R}}_{u{\prime}_{1}}$, when ${u}_{1}$ changed their location, (

**b**) The regions of interest, ${\mathcal{R}}_{{u}_{3}}$ and ${\mathcal{R}}_{u{\prime}_{3}}$, when user ${u}_{3}$ moves towards $C$.

**Figure 4.**The displacement $\overrightarrow{{R}_{{u}_{i}C}}$ when user ${u}_{i}$ is moving towards the centroid $C$.

**Figure 5.**The search region of user u

_{i}: (

**a**) the nearest object o

_{n}is nearer to the user u

_{i}compared to C, (

**b**) the nearest object o

_{n}is further from the user u

_{i}compared to C, and (

**c**) the nearest object o

_{n}is at the same point as C.

**Figure 7.**(

**a**) Single minimum bounding Rectangle (SMBR). (

**b**) Multiple minimum bounding rectangle (MMBR). (

**c**) Minimum bounding rectangle (MBR).

**Figure 12.**The results of number of skyline objects and number of objects of interest with varying time intervals. (

**a**) Synthetic. (

**b**) TIGER. (

**c**) Synthetic. (

**d**) TIGER.

**Figure 17.**The results of processing time with varying number of users in a group. (

**a**) Synthetic. (

**b**) TIGER.

**Figure 18.**The results of processing time with varying number of groups of users. (

**a**) Synthetic. (

**b**) TIGER.

**Figure 19.**The results of number of skyline objects under different number of users in a group. (

**a**) Synthetic. (

**b**) TIGER.

ID | Initial Location | Speed v _{i} | Centroid C | Step 1 R _{uiC} | Step 2 | Step 3 ∆R | x′ | y′ | |
---|---|---|---|---|---|---|---|---|---|

cos θ | sin θ | ||||||||

u_{1} | (2, 6) | 72 | (3.2, 4.1) | 2.24 | 0.53 | −0.84 | 0.2 | 2.1 | 5.83 |

u_{2} | (1.6, 3.2) | 108 | 1.83 | 0.87 | 0.49 | 0.3 | 1.86 | 3.34 | |

u_{3} | (6, 3.2) | 90 | 2.94 | −0.95 | 0.3 | 0.25 | 5.76 | 3.27 |

ID | Initial Location | Speed v _{i} | Centroid C | Step 1 R _{uiC} | Step 2 | Step 3 ∆R | x′ | y′ | |
---|---|---|---|---|---|---|---|---|---|

cos θ | sin θ | ||||||||

u_{1} | (2.1, 5.83) | 72 | (3.2, 4.1) | 2.04 | 0.53 | −0.84 | 0.2 | 2.2 | 5.66 |

u_{2} | (1.86, 3.34) | 108 | 1.53 | 0.87 | 0.49 | 0.3 | 2.12 | 3.48 | |

u_{3} | (5.76, 3.27) | 90 | 2.69 | −0.95 | 0.3 | 0.25 | 5.52 | 3.34 |

Nearest Object | ${\mathit{x}}_{{\mathit{o}}_{\mathit{n}}}$ | ${\mathit{y}}_{{\mathit{o}}_{\mathit{n}}}$ | ${\mathit{R}}_{\mathit{G}\mathit{R}}$ | MBR | |||
---|---|---|---|---|---|---|---|

$\mathit{b}\mathit{l}\left({\mathit{x}}_{\mathit{b}\mathit{l}},{\mathit{y}}_{\mathit{b}\mathit{l}}\right)$ | $\mathit{b}\mathit{r}\left({\mathit{x}}_{\mathit{b}\mathit{r}},{\mathit{y}}_{\mathit{b}\mathit{r}}\right)$ | $\mathit{t}\mathit{l}\left({\mathit{x}}_{\mathit{t}\mathit{l}},{\mathit{y}}_{\mathit{t}\mathit{l}}\right)$ | $\mathit{t}\mathit{r}\left({\mathit{x}}_{\mathit{t}\mathit{r}},{\mathit{y}}_{\mathit{t}\mathit{r}}\right)$ | ||||

${o}_{7}$ | 3.2 | 4 | 6 | (−2.8, −2) | (9.2, −2) | (−2.8, 10) | (9.2, 10) |

User | $\mathit{x}$ | $\mathit{y}$ | ${\mathit{R}}_{{\mathit{u}}_{\mathit{i}}{\mathit{o}}_{\mathit{n}}}$ | MBR | |||
---|---|---|---|---|---|---|---|

$\mathit{b}\mathit{l}\left({\mathit{x}}_{\mathit{b}\mathit{l}},{\mathit{y}}_{\mathit{b}\mathit{l}}\right)$ | $\mathit{b}\mathit{r}\left({\mathit{x}}_{\mathit{b}\mathit{r}},{\mathit{y}}_{\mathit{b}\mathit{r}}\right)$ | $\mathit{t}\mathit{l}\left({\mathit{x}}_{\mathit{t}\mathit{l}},{\mathit{y}}_{\mathit{t}\mathit{l}}\right)$ | $\mathit{t}\mathit{r}\left({\mathit{x}}_{\mathit{t}\mathit{r}},{\mathit{y}}_{\mathit{t}\mathit{r}}\right)$ | ||||

${u}_{1}$ | 2 | 6 | 2.28 | (−0.28, 3.72) | (−0.28, 8.28) | (4.28, 3.72) | (4.28, 8.28) |

${u}_{2}$ | 1.6 | 3.2 | 1.7 | (−0.1, 1.5) | (−0.1, 4.9) | (3.3, 1.5) | (3.3, 4.9) |

${u}_{3}$ | 6 | 3.2 | 3 | (3, 0.2) | (3, 6.2) | (9, 0.2) | (9, 6.2) |

ID | ${\mathit{o}}_{\mathit{i}}-{\mathit{u}}_{1}$ | ${\mathit{o}}_{\mathit{i}}-{\mathit{u}}_{2}$ | ${\mathit{o}}_{\mathit{i}}-{\mathit{u}}_{3}$ | $\mathit{S}\mathit{u}\mathit{m}\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}-{\mathit{o}}_{\mathit{i}}$ |
---|---|---|---|---|

${o}_{2}$ | 2.13 | 1.07 | 4.81 | 8.01 |

${o}_{3}$ | 3.98 | 1.58 | 4.92 | 10.48 |

${o}_{5}$ | 5.66 | 5.14 | 1.26 | 12.06 |

${o}_{6}$ | 3.90 | 4.91 | 2.73 | 11.54 |

${o}_{7}$ | 2.08 | 1.40 | 2.75 | 6.23 |

${o}_{11}$ | 3.80 | 4.72 | 2.53 | 11.05 |

${o}_{13}$ | 1.17 | 3.66 | 5.29 | 10.12 |

Parameter Settings | Datasets | |
---|---|---|

Synthetic | Long Beach Tiger | |

Number of dimensions | 2, 4, 6, 8, 10 | 2, 4, 6, 8, 10 |

Number of users in a group | 4, 8, 15, 20, 25 | 4, 8, 15, 20, 25 |

Number of objects | 20,000, 50,000, 80,000 | 50,747 |

Space | [0, 250]*[0, 250], [0, 500]*[0, 500], [0, 750]*[0, 750], [0, 1000]*[0, 1000] | [0, 250]*[0, 250], [0, 500]*[0, 500], [0, 750]*[0, 750], [0, 1000]*[0, 1000] |

Density | - | 0.56%, 1.60%, 7%, 15%, 34% |

Time interval $\u2206t$ | 10 s | 10 s |

Velocity | 40–120 km/h | 40–120 km/h |

Types of Objects | % of the Type of Object in the Whole Population | No. of Objects |
---|---|---|

Hospital | 0.56 | 284 |

Restaurant | 1.60 | 812 |

Church | 7 | 3552 |

School | 15 | 7612 |

Institution | 34 | 17,254 |

Building | 10.84 | 5502 |

Hotel | 13 | 6597 |

Populated place | 18 | 9134 |

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## Share and Cite

**MDPI and ACS Style**

Dehaki, G.B.; Ibrahim, H.; Alwan, A.A.; Sidi, F.; Udzir, N.I.; Lawal, M.M.
A Continuous Region-Based Skyline Computation for a Group of Mobile Users. *Symmetry* **2022**, *14*, 2003.
https://doi.org/10.3390/sym14102003

**AMA Style**

Dehaki GB, Ibrahim H, Alwan AA, Sidi F, Udzir NI, Lawal MM.
A Continuous Region-Based Skyline Computation for a Group of Mobile Users. *Symmetry*. 2022; 14(10):2003.
https://doi.org/10.3390/sym14102003

**Chicago/Turabian Style**

Dehaki, Ghoncheh Babanejad, Hamidah Ibrahim, Ali A. Alwan, Fatimah Sidi, Nur Izura Udzir, and Ma′aruf Mohammed Lawal.
2022. "A Continuous Region-Based Skyline Computation for a Group of Mobile Users" *Symmetry* 14, no. 10: 2003.
https://doi.org/10.3390/sym14102003