Cluster Nested Loop kFarthest Neighbor Join Algorithm for Spatial Networks
Abstract
:1. Introduction
 This paper presents a cluster nested loop join algorithm for quickly evaluating spatial network kFN join queries. The CNLJ algorithm clusters query points before retrieving candidate data points for clustered query points all at once. As a result, it does not retrieve candidate data points for each query point multiple times.
 The CNLJ algorithm’s correctness is demonstrated through mathematical reasoning. In addition, a theoretical analysis is provided to clarify the benefits and drawbacks of the CNLJ algorithm concerning query point spatial compactness.
 An empirical study with various setups was conducted to demonstrate the superiority and scalability of the CNLJ algorithm. The CNLJ algorithm outperforms the conventional join algorithms by up to 50.8 times according to the results.
2. Background
2.1. Related Work
2.2. Notation and Formal Problem Description
3. Clustering Points and Computing Distances
3.1. Clustering Query and Data Points Using Spatial Network Connection
3.2. Computing Maximum and Minimum Distances from a Border Point to a Data Cluster
4. Cluster Nested Loop Join Algorithm for Spatial Networks
4.1. Cluster Nested Loop Join Algorithm
Algorithm 1 CNLJ($k,Q,P$). 
Input:k: number of FNs for q, Q: set of query points, and P: set of data points Output:$\mathrm{\Omega}\left(Q\right)$: Set of ordered pairs of each query point q in Q and a set of k FNs for q, i.e., $\mathrm{\Omega}\left(Q\right)=\left\{\u2329q,\mathrm{\Omega}\left(q\right)\u232a\rightq\in Q\}$.

Algorithm 2$kFN\_join(k,\overline{{Q}_{C}},\phantom{\rule{4pt}{0ex}}\overline{P})$. 
Input:k: number of FNs for q, $\overline{{Q}_{C}}$: query cluster, and $\overline{P}$: set of data clusters Output:$\mathrm{\Omega}\left(\overline{{Q}_{C}}\right)$: Set of ordered pairs of each query point q in $\overline{{Q}_{C}}$ and a set of k FNs for q, i.e., $\mathrm{\Omega}\left(\overline{{Q}_{C}}\right)=\left\{\u2329q,\mathrm{\Omega}\left(q\right)\u232a\rightq\in \overline{{Q}_{C}}\}$

Algorithm 3$find\_candidates(k,l,{b}_{q},\overline{P})$. 
Input:k: number of FNs for q, l: maximum distance between border points in $\overline{{Q}_{C}}$, ${b}_{q}$: border point of $\overline{{Q}_{C}}$, and $\overline{P}$: set of data clusters Output:$\mathrm{\Omega}\left({b}_{q}\right)$: Set of k FNs for ${b}_{q}$

Algorithm 4$retrieve\_kFN(k,q,\mathrm{\Omega}\left(B\left(\overline{{Q}_{C}}\right)\right))$. 
Input:k: number of FNs for q, q: query point in $\overline{{Q}_{C}}$, $\mathrm{\Omega}\left(B\left(\overline{{Q}_{C}}\right)\right)$: set of candidate data points for q Output: $\mathrm{\Omega}\left(q\right)$: set of k FNs for q

4.2. Evaluating kFN Queries at Border Points
4.3. Evaluating an Example kFN Join Query
5. Performance Evaluation
5.1. Experimental Settings
5.2. Experimental Results
6. Discussion and Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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References  Space Domain  Query Type  Data Type 

[8,9,14,19]  Euclidean space  RkFN search  Monochromatic 
[14,22]  Euclidean space  RkFN search  Bichromatic 
[6,9,17,18,20,25]  Euclidean space  kFN search  
[7]  Euclidean space  AkFN search  
[26]  Euclidean space  FDL search  
[13]  Spatial network  RkFN search  Monochromatic 
[10,13]  Spatial network  RkFN search  Bichromatic 
[35]  Spatial network  kFN search  
[11]  Spatial network  AkFN search  
This study  Spatial network  kFN join 
Symbol  Definition 

k  Number of requested FNs 
Q and q  A set Q of query points and query point q in Q, respectively 
P and p  A set P of data points and data point p in P, respectively 
$\overline{{v}_{l}{{v}_{l+1}\cdots v}_{m}}$  Vertex sequence where ${v}_{l}$ and ${v}_{m}$ are either an intersection vertex or a terminal vertex and the other vertices, ${v}_{l+1},\cdots \phantom{\rule{0.166667em}{0ex}}{,v}_{m1}$, are intermediate vertices 
$\overline{{q}_{i}{q}_{i+1}{\cdots q}_{j}}$  Query segment connecting query points ${q}_{i},{q}_{i+1},\cdots \phantom{\rule{0.166667em}{0ex}},{q}_{j}$ in a vertex sequence (in short, $\overline{{q}_{i}{q}_{j}}$) 
$\overline{{p}_{l}{p}_{l+1}{\cdots p}_{m}}$  Data segment connecting data points ${p}_{l},{p}_{l+1},\cdots \phantom{\rule{0.166667em}{0ex}},{p}_{m}$ in a vertex sequence (in short, $\overline{{p}_{l}{p}_{m}}$) 
$\overline{{Q}_{C}}$ and $\overline{{P}_{C}}$  Set of query segments and set of data segments, respectively 
$\overline{Q}$ and $\overline{P}$  Set of query clusters and set of data clusters, respectively 
$B\left(\overline{{Q}_{C}}\right)$ and $B\left(\overline{{P}_{C}}\right)$  Sets of border points of $\overline{{Q}_{C}}$ and $\overline{{P}_{C}}$, respectively 
${b}_{q}$ and ${b}_{p}$  Border points of $\overline{{Q}_{C}}$ and $\overline{{P}_{C}}$, respectively 
$\mathrm{\Omega}\left(q\right)$  Set of k data points farthest from a query point q 
$dist(q,p)$  Length of the shortest path connecting points q and p 
$len\left(\overline{qp}\right)$  Length of the segment $\overline{qp}$ 
${\mathit{b}}_{\mathit{q}}$  $\left\{\overline{{\mathit{p}}_{1}{\mathit{p}}_{2}{\mathit{p}}_{3}}\right\}$  $\{\overline{{\mathit{p}}_{4}{\mathit{p}}_{5}},\overline{{\mathit{p}}_{5}{\mathit{p}}_{6}}\}$ 

${q}_{1}$  $maxdist({q}_{1},\left\{\overline{{p}_{1}{p}_{2}{p}_{3}}\right\})=28$  $maxdist({q}_{1},\{\overline{{p}_{4}{p}_{5}},\overline{{p}_{5}{p}_{6}}\})=11$ 
$mindist({q}_{1},\left\{\overline{{p}_{1}{p}_{2}{p}_{3}}\right\})=24$  $mindist({q}_{1},\{\overline{{p}_{4}{p}_{5}},\overline{{p}_{5}{p}_{6}}\})=5$  
${v}_{1}$  $maxdist({v}_{1},\left\{\overline{{p}_{1}{p}_{2}{p}_{3}}\right\})=24$  $maxdist({v}_{1},\{\overline{{p}_{4}{p}_{5}},\overline{{p}_{5}{p}_{6}}\})=12$ 
$mindist({v}_{1},\left\{\overline{{p}_{1}{p}_{2}{p}_{3}}\right\})=19$  $mindist({v}_{1},\{\overline{{p}_{4}{p}_{5}},\overline{{p}_{5}{p}_{6}}\})=9$  
${v}_{2}$  $maxdist({v}_{2},\left\{\overline{{p}_{1}{p}_{2}{p}_{3}}\right\})=25.5$  $maxdist({v}_{2},\{\overline{{p}_{4}{p}_{5}},\overline{{p}_{5}{p}_{6}}\})=8$ 
$mindist({v}_{2},\left\{\overline{{p}_{1}{p}_{2}{p}_{3}}\right\})=23$  $mindist({v}_{2},\{\overline{{p}_{4}{p}_{5}},\overline{{p}_{5}{p}_{6}}\})=5$ 
CNLJ Algorithm  Nonclustering Join Algorithm  

Number of kFN queries to be evaluated  $M\xb7\overline{Q}$  $\leftQ\right$ 
Time complexity to evaluate the kFN search  $O(\leftE\right+\leftV\right\mathrm{log}\leftV\right\phantom{\rule{4pt}{0ex}}+\leftP\right\mathrm{log}\leftP\right\phantom{\rule{4pt}{0ex}})$  $O(\leftE\right+\leftV\right\mathrm{log}\leftV\right\phantom{\rule{4pt}{0ex}}+\leftP\right\mathrm{log}\leftP\right\phantom{\rule{4pt}{0ex}})$ 
Time complexity to evaluate the kFN join  $O\left(\right\overline{Q}\xb7(\leftE\right+\leftV\right\mathrm{log}\leftV\right\phantom{\rule{4pt}{0ex}}+\leftP\right\mathrm{log}\leftP\right\phantom{\rule{4pt}{0ex}}))$  $O\left(\rightQ\xb7(\leftE\right+\leftV\right\mathrm{log}\leftV\right\phantom{\rule{4pt}{0ex}}+\leftP\right\mathrm{log}\leftP\right\phantom{\rule{4pt}{0ex}}))$ 
${\mathit{b}}_{\mathit{q}}$  $\mathit{dist}({\mathit{b}}_{\mathit{q}},\mathit{p})$  $\mathit{sntl}\_\mathit{dist}\left({\mathit{b}}_{\mathit{q}}\right)$  $\mathbf{\Omega}\left({\mathit{b}}_{\mathit{q}}\right)$ 

${q}_{1}$  $dist({q}_{1},{p}_{1})=24$  $sntl\_dist\left({q}_{1}\right)=20$  $\mathrm{\Omega}\left({q}_{1}\right)=\{{p}_{1},{p}_{2},{p}_{3}\}$ 
$dist({q}_{1},{p}_{2})=25$  
$dist({q}_{1},{p}_{3})=27$  
${v}_{1}$  $dist({v}_{1},{p}_{1})=19$  $sntl\_dist\left({v}_{1}\right)=15$  $\mathrm{\Omega}\left({v}_{1}\right)=\{{p}_{1},{p}_{2},{p}_{3}\}$ 
$dist({v}_{1},{p}_{2})=20$  
$dist({v}_{1},{p}_{3})=24$  
${v}_{2}$  $dist({v}_{2},{p}_{1})=23$  $sntl\_dist\left({v}_{2}\right)=18$  $\mathrm{\Omega}\left({v}_{2}\right)=\{{p}_{1},{p}_{2},{p}_{3}\}$ 
$dist({v}_{2},{p}_{2})=24$  
$dist({v}_{2},{p}_{3})=23$ 
q  $\mathit{dist}(\mathit{q},\mathit{p})$  $\mathbf{\Omega}\left(\mathit{q}\right)$ 

${q}_{1}$  $dist({q}_{1},{p}_{1})=24$  $\mathrm{\Omega}\left({q}_{1}\right)=\{{p}_{2},{p}_{3}\}$ 
$dist({q}_{1},{p}_{2})=25$  
$dist({q}_{1},{p}_{3})=27$  
${q}_{2}$  $dist({q}_{2},{p}_{1})=25$  $\mathrm{\Omega}\left({q}_{2}\right)=\{{p}_{1},{p}_{2}\}$ or 
$dist({q}_{2},{p}_{2})=26$  $\mathrm{\Omega}\left({q}_{2}\right)=\{{p}_{2},{p}_{3}\}$  
$dist({q}_{2},{p}_{3})=25$  
${q}_{3}$  $dist({q}_{3},{p}_{1})=21$  $\mathrm{\Omega}\left({q}_{3}\right)=\{{p}_{2},{p}_{3}\}$ 
$dist({q}_{3},{p}_{2})=22$  
$dist({q}_{3},{p}_{3})=25$  
${q}_{4}$  $dist({q}_{4},{p}_{1})=21$  $\mathrm{\Omega}\left({q}_{4}\right)=\{{p}_{2},{p}_{3}\}$ 
$dist({q}_{4},{p}_{2})=22$  
$dist({q}_{4},{p}_{3})=26$ 
Name  Description  Vertices  Edges  Vertex Sequences 

NA  Highways in North America (NA)  175,813  179,179  12,416 
SJ  City streets in San Joaquin (SJ), California  18,263  23,874  20,040 
Parameter  Range 

Number of query points ($\leftQ\right$)  1, 2, 3, 4, 5, 7, 10 ($\times {10}^{3}$) 
Number of data points ($\leftP\right$)  1, 2, 3, 4, 5, 7, 10 ($\times {10}^{3}$) 
Number of FNs required (k)  1, 2, 4, 8, 16 
Distribution of query and data points  Centroid distribution 
Number of centroids for query points in Q (${C}_{Q}$)  1, 3, 5, 7, 10 
Number of centroids for data points in P (${C}_{P}$)  1, 3, 5, 7, 10 
The standard deviation for normal distribution ($\sigma $)  ${\mathbf{10}}^{\mathbf{2}}$ 
Roadmap  NA, SJ 
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Cho, H.J. Cluster Nested Loop kFarthest Neighbor Join Algorithm for Spatial Networks. ISPRS Int. J. GeoInf. 2022, 11, 123. https://doi.org/10.3390/ijgi11020123
Cho HJ. Cluster Nested Loop kFarthest Neighbor Join Algorithm for Spatial Networks. ISPRS International Journal of GeoInformation. 2022; 11(2):123. https://doi.org/10.3390/ijgi11020123
Chicago/Turabian StyleCho, HyungJu. 2022. "Cluster Nested Loop kFarthest Neighbor Join Algorithm for Spatial Networks" ISPRS International Journal of GeoInformation 11, no. 2: 123. https://doi.org/10.3390/ijgi11020123