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1 DEIM Forum 2017 C8-2 k s @u.tsukuba.ac.jp, tou@dblab.is.tsukuba.ac.jp, chx@cc.tsukuba.ac.jp, furuse@cs.tsukuba.ac.jp aggregate nearest neighbor, ANN Q P Q.ANN ANN k AkNN L 1 AkNN AkNN 1. ANN, AkNN,, (Aggregate Nearest Neighbor ANN) P Q Q () ANN ANN ANN ANN 1 ANN k AkNN L 1 AkNN AkNN ANN AkNN ANN ANN k 1 ANN AkNN ANN k. k Q AkNN 2. D p 1 = (x 1,, x D) p 2 = (y 1,, y D) p 1p 2.. p 1 p 2 p 1p 2 1. p 1p 2 = D x i y i 2 (1) i=1, sum max, min f, p Q = q 1, q 2,, q n}

2 adist (p, Q) 2. adist (p, Q) = f ( pq 1, pq 2,, pq n ) = f ( pq i )(2) 1< = i< = n 1 ( k : AkNN ) P = p 1, p 2,, p N } Q = q 1, q 2,, q n} k, P Q k k (AkNN). 3. (aggregate nearest neighbor, ANN), [1] sum, [2] max min. f = sum, ANN Q. f = max,. f = min,. [1] [2], MQM(multiple query method),spm(single point method), MBM(minimal bounding method) ANN., P R-tree. mindist(n j, M) n(= Q ). f = max f = min, amindist(n j, M) = mindist(n j, M) ( 3). amindist(n j, M) > = best dist n mindist(nj, M) > = best dist (f = sum) mindist(n j, M) > = best dist (f = max or min) (3) 2: p j M amindist(p j, M) best dist, p j,. ANN f = sum, amindist(p j, M) p j M mindist(p j, M) n(= Q ). f = max f = min, amindist(p j, M) = mindist(p j, M) ( 4). amindist(p j, M) > = best dist n mindist(nj, M) > = best dist (f = sum) mindist(p j, M) > = best dist (f = max or min) (4) 3: N j, M amindist(n j, M) best dist, Q amindist(n j, Q) best dist, N j. amindist(n j, Q) q i Q N j mindist(n j, q i)., amindist(n j, Q) = f (mindist(n j, q i)). ( 5). 1< = i< = n 1 2 R-tree amindist(n j, Q) > = best dist f (mindist(n j, q i)) > = best dist (5) 1< = i< = n R-tree B-tree, 1 2 R-tree. R-tree (MBR). R-tree,,, O(N). MBM MBM MBM(minimal bounding method) SPM. SPM,, ANN Q ANN. MBM q m, Q Q M. MBM 3. 1: N j Q M amindist(n j, M) best dist, N j best NN,. f = sum, amindist(n j, M) N j M 2 MBM ( f = sum) [1] [2] MQM,SPM,MBM, MBM MBM MBM Q 3 SPM. Q R-tree R-tree ANN [5], [6], [9], [7], [8], [10], [11]. [5] vp-ann Projection-Based Pruning. [6]

3 ANN [7], [9] Q P [8], [10] ANN 3. 3 AkNN AkNN [12], [13] [12] f = max L 1 AkNN [13] P AkNN AkNN 4. MQM,SPM,MBM ANN AkNN AkNN ANN k ANN best dist best NN MBM best dist AkNN AkNN MBM MBM MBM Q R-tree AkNN k Q AkNN 5. ANN AkNN k best dist best maxdist 4: N best p k Q N j amindist(q, N j) best maxdist N j best maxdist amindist(n j, Q) > = best maxdist n mindist(nj, Q) > = best maxdist (f = sum) (6) mindist(n j, Q) > = best maxdist (f = max or min) 6. MBM Q Q MBR M 1 2 Q 6.1 Q G p j G p j Q 6.1 (G Q ) adist(p j, G) = n adist(p j, G) = n Gq i + n q ip j = n q ip j q ip j < n = q ip j = adist(p j, Q) [9] Q G 1 : G N j amindist(g, N j) best dist N j best NN amindist(n j, G) > = best dist n mindist(nj, G) > = best dist (f = sum) mindist(n j, G) > = best dist (f = max or min) (7) 2 : G p j amindist(g, p j) best dist p j best NN amindist(p j, G) > = best dist n mindist(pj, G) > = best dist (f = sum) mindist(p j, G) > = best dist (f = max or min) (8) Q G N j Q G N j Q G 6.2 dist(q i, N j) < = dist(q i, G) + dist(g, N j) amindist(q, N j) < = amindist(q, G) + adist(g, N j)

4 6.2 best maxdist = amaxdist(g, N j)+adist(q, G) 4 best maxdist Q : N best p k G N j amindist(g, N j) best maxdist N j N best AkNN amindist(n j, G) > = best maxdist n mindist(nj, G) > = best maxdist (f = sum) (9) mindist(n j, G) > = best maxdist (f = max or min) Q amindist (f = max) 7.1 () Q Q p q f 7.1 Q c q i p q i q x q i p q x pq i < = pq x q x q c1, q c2 pq x < = pq c1, pq x < = pq c2 pq i < = pq x < = pq c ANN max Q c Q c = Q x Q x Q MBR MBR Q x 2 2 Q x 2 MBR Q x Q x Q x 7. 2 (f = sum) 1,2 Q i Q div G i n i 1 : N j Q G i div amindist(gi, Nj) best dist N j best NN amindist(n j, G i) > = best dist n i mindist(n j, G i) > = best dist (f = sum) (10) 3 2 : p j Q G amindist(g, p j) best dist p j best NN amindist(p j, G i) > = best dist n i mindist(p j, G i) > = best dist (f = sum) (11) 7.2 ANN max Q Qc ANN(Q) = ANN(Q c) Q Q c p q f ANN max adist(q, p) = adist(q c, p),ann(q) = ANN(Q c) ANN f = sum Algorithm1 f = sum f = max O(div n)o(n) k O(n N)

5 (f = max n nx ) f = sum,f = max O(div n + n N)O(n + nx N) MBM O(n N) f = sum div << N f = max n k MBM Algorithm 1 (f=sum) (root: R-tree root, Q: query set) 1: /* Q i Q */ 2: /* G i Q i */ 3: /* H amindist(n j, Q) */ 4: Node root; best maxdist ; best dist[k] ; best NN[k] null 5: while Node = null && amindist(node, Q) < best dist[k 1] do 6: if Node.isIndex then 7: /* Node R-tree */ 8: for N j Node do 9: if amindist(n j, G i ) < best dist[k 1] then 10: if amaxdist(n j, G i ) < best maxdist then 11: if amindist(n j, Qc) < best dist[k 1] then 12: H.push(N j ) 13: best maxdist amaxdist(n j, G i] )+adist(g i, Q) /* best maxdist */ 14: end if 15: end if 16: end if 17: end for 18: else if Node.isLeaf then 19: /* Node R-tree */ 20: for p j Node do 21: if amindist(p j, G i ) < best dist[k 1] then 22: if adist(p j, Qc) < best dist[k 1] then 23: best NN[k 1] p j ; best dist[k 1] adist(p j, Q) /* ANN */ 24: end if 25: end if 26: end for 27: end if 28: Node H.pop() 29: end while 30: return (best NN) , MBM AkNN MBM..,R-tree spatialindexsrc OS: OS X Yosemite CPU: 2.6 GHz Intel Core i5 : 8 GiB 1600 MHz DDR3 : C , 2, (0,1). 13,(0,1). x max x min X. y max y min Y. x = (x x min)/(x max x min) (12) y = (y y min)/(y max y min) (13) Q n A(Q) Dkf = sum div, n = Q 16, 32, 64, 128, 256, 512 A(Q) 0.01, 0.09, 0.16, 0.25, 0.49, 1 D 2, 4, 8, 16 k 1, 5, 10, 15, 20 div 1, 5, 10, 15, 20, 25, 100,. 1http://download.osgeo.org/libspatialindex/spatialindex-src tar.gz 2populated places,

6 [a]f = sum (div=20) 4 vary k (n = 128, A(Q) = 1.0, dim = 2) [a]f = sum (div=2) 6 vary n (A(Q) = 1.0, dim = (2or8), k = 20, div = 20) [a]f = sum (div=2) 7 vary area (n = 128, dim = (2or8), k = 20, div = 20) k MBM f = max MBM k 5 vary div (n = 128, A(Q) = 1.0, dim = 2, k = 20) 5 div div = 20 div = 25 div = 20 6 dim = 2 dim = 8 Q n MBM sum n MBM sum max dim = 2 n MBM dim = 8 7 dim = 2 dim = 8 A(Q) sum dim = 2 A(Q) MBM Q dim = 8 max dim = 2 A(Q) dim = 8

7 [a]f = sum (div=2) 8 vary dim (n = 128, A(Q) = (0.25or1.0), k = 20) [a]vary n (A(Q)=1.0, dim=2, k=20, div=20) 9 [b]vary area (n=128, dim=2, k=20, div=20) 8 A(Q) = 0.25 A(Q) = 1.0 dim sum max MBM max A(Q) n n sum A(Q) summax 9. ANN AkNN AkNN R-tree sum max [1] D. Papadias, Y. Tao, K. Mouratidis and C. Kit Hui. Group Nearest Neighbor Queries. In: ICDE, pp (2004) [2] D. Papadias, Y. Tao, K. Mouratidis and C. Kit Hui. Aggregate Nearest Neighbor Queries in Spatial Databases. ACM Trans. Database Syst. 30(2), (2005) [3] H. Chen, R. Shi, K. Furuse and N. Ohbo. Finding RkNN Straightforwardly with Large secondary Storage. In: INGS, pp.77-82(2008) [4] Y. Luo, J. Liu, C. Lian and H. Chen. Finding RkNN by Compressed Straightforward Index. In: ISKE, pp (2008) [5] Y. Luo, H. Chen, K. Furuse and N. Ohbo. Efficient Methods in Finding Aggregate Nearest Neighbor by Projection-Based Filtering. In: ICCSA, pp (2007) [6]. On efficient approximate aggregate nearest neighbor search..,(2012) [7] H. Li, H. Lu, B. Huang and Z. Huang. Two Ellipse-based Pruning Methods for Group Nearest Neighbor Queries. In: GIS, pp (2005) [8] M.Yiu, N. Mamoulis and D. Papadias. Aggregate Nearest Neighbor Queries in Road Networks. IEEEE Trans. Knowl. Data Eng. pp (2005) [9] S. Namnandorj, H. Chen, K. Furuse and N. Ohbo. Efficient Bounds in Finding Aggregate Nearest Neighbors. In: DEXA, pp (2008) [10] L. Zhu, Y. Jing, W. Sun and P. Liu. Voronoi-Based Aggregate Nearest Neighbor Query Processing in Road Networks In: GIS, pp (2010) [11] F. Li, B. Yao and P. Kumar. Group Enclosing Queries. In: TKDE, pp, (2011) [12] Haitao Wang. Aggregate-MAX Top-k Nearest Neighbor Searching in the L1 Plane. Int. J. Comput. Geometry Appl. 25(1): 57- (2015) [13] Maytham Safar. Group K -Nearest Neighbors queries in spatial network databases. Journal of Geographical Systems 10(4): (2008) [14] Camila Ferreira Costa, Javam C. Machado, Mario A. Nascimento, Jos Antnio Fernandes de Macdo. Aggregate k- nearest neighbors queries in time-dependent road networks. MobiGIS 2015: 3-12

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