Nearest Neighbor Search in Spatial
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1 Nearest Neighbor Search in Spatial and Spatiotemporal Databases 1
2 Content 1. Conventional NN Queries 2. Reverse NN Queries 3. Aggregate NN Queries 4. NN Queries with Validity Information 5. Continuous NN Monitoring 6. NN in Non-Euclidean Spaces 7. Private NN Queries 2
3 1. Conventional Nearest Neighbor (NN) Query Given a query location q, find the nearest object. a q Depth First and Best-First Search using R-(TPR-) trees Our goal: avoid visiting nodes that cannot contain results 3
4 Basic Pruning Metric: MINDIST Minimum distance between qand a minimum bounding rectangle MBR. E 1 mindist(e 1,q) q p mindist(e 1,q) is a lower bound of d(o,q) for every object o in E 1. If we have found a candidate NN p, we can prune every MBR whose mindist> d(p, q). 4
5 E 1 Depth-First (DF) NN Algorithm y axis e b c f g E 4 E 5 E 1 d E 3 h a E 2 E 6 i query m j k E 7 x axis Root E 1 E E 3 E 4 E 5 E 6 E a b c d e f g i j h k l m E E 3 4 E 5 E E 6 7 l Note: distances not actually stored inside nodes. Only for illustration E2
6 DF Search Visit E 1 y axis m E 2 g h E 6 e f d E b 3 a c E 4 E 5 E 1 i query E 7 l k j l m x axis Root E 1 E E E E E E E E2 a b c d e f g i j 5 h 2 k E 3 E 4 E 5 E 6 E 7
7 DF Search Find Candidate NN a y axis m E 2 g h E 6 e f d E b 3 a c E 4 E 5 E 1 i query E 7 l k j l m x axis Root E 1 E 2 First Candidate NN: a with distance E E E E E E E2 a b c d e f g i j h 5 2 k E 3 E 4 E 5 E 6 E 7
8 DF Search Backtrack to Root and Visit E y axis m E 2 g h E 6 e f d E b 3 a c E 4 E 5 E 1 i query E 7 l k j x axis Root E 1 E First Candidate NN: a with distance 5 E E E E E E a b c d e f g i j 5 h 2 E2 k E 3 E 4 E 5 E 6 E 7
9 E 1 DF Search Find Actual NN i y axis e b c f g d E 3 h E 4 E 5 E 1 a Root E 1 E E 2 E 6 i m query j k E 7 x axis E 3 E 4 E 5 E 6 E l First Candidate NN: a with distance 5 Actual NN: i with distance 2 a b c d e f g i j h k l m 5 2 E E 3 4 E 5 E E 6 7 E2
10 y axis e b c f g d E 3 Optimality h E 4 E 5 E 1 a E 2 E 6 i m query j Question: Which is the minimal set of nodes that must be visited by any NN algorithm? Answer: The set of nodes whose MINDIST is smaller than or equal to the distance between qand its NN (e.g., E 1, E 2, E 6 ). k E 7 x axis l 10
11 Best-First (BF) NN Algorithm (Optimal) Keep a heap Hof index entries and objects, ordered by MINDIST. Initially, H contains the root. While H φ Extract the element with minimum MINDIST If it is an index entry, insert its children into H. If it is an object, return it as NN. End while 11
12 BF Search Visit root y axis m E 2 g h E 6 e f d E b 3 a c E 4 E 5 E 1 i query E 7 l k j x axis Root E 1 E Action Visit Root Heap E 1 1 E 2 2 E E E E E E E a b c d e f g i j h k l m E E 3 4 E 5 E E 6 7
13 y axis e b c f g d E 3 h E 4 E 5 E 1 a E 2 E 6 i m query j BF Search Visit E 1 k E 7 x axis l Root Action Visit Root Heap E 1 1 follow E 1 E 2 2 E 2 2 E 3 5 E 5 5 E 9 4 E 1 E E E E E E E E a b c d e f g i j h k l m E E 3 4 E 5 E E 6 7
14 y axis e b c f g d E 3 h E 4 E 5 E 1 a E 2 E 6 i m query j BF Search Visit E 2 k E 7 x axis l Root E 1 E Action Visit Root Heap E 1 1 follow E 1 E 2 2 follow E 2 E 6 2 E 2 2 E 3 5 E 5 3 E 5 5 E 5 5 E 9 4 E 9 4 E 13 7 E E E E E E E a b c d e f g i j h k l m E E 3 4 E 5 E E 6 7
15 BF Search Visit E 6 y axis e b f g d E 3 h E 4 E 5 E 1 a E 2 E 6 i m query j k E 7 l Action Visit Root Heap E 1 1 follow E 1 E 2 2 follow E 2 E 6 2 follow E 6 E 2 2 E 3 5 E 5 3 i 2 E 5 3 E 5 5 E 5 5 E 5 5 E 9 4 E 9 4 E 9 4 E 7 13 j 10 E 7 13 k 13 2 c x axis Root E 1 E E E E E E E E a b c d e f g i j h k l m E E 3 4 E 5 E E 6 7
16 y axis e b c f g d E 3 h E 4 E 5 E 1 a BF Search Find Actual NN i E 2 E 6 i m query j k E 7 x axis l Root E 1 E Action Visit Root Heap E 1 1 follow E 1 E 2 2 follow E 2 E 6 2 follow E 6 E 2 2 E 3 5 E 5 3 i 2 E 5 3 E 5 5 E 5 5 E 5 5 Report i and terminate E 9 4 E 9 4 E 9 4 E 7 13 j 10 E 7 13 k 13 E E E E E E E a b c d e f g i j h k l m E E 3 4 E 5 E E 6 7
17 Discussion about Conventional NN Queries (1) Both DF and BF can be easily adapted to (i) extended (instead of point) objects and (ii) retrieval of k(>1) NN. BF is incremental; i.e., it can report the NN in increasing order of distance without a given value of k. Example: find the 10 closest cities to HK with population more than 1 million. We may have to retrieve many (>>10) cities around Hong Kong in order to answer the query. 17
18 Discussion about Conventional NN Queries (2) Both DF and BF can be used to process NN queries on moving objects indexed by TPR-trees: find my NN during the next minute (based on the current motion). q a b 18
19 2. Reverse NN Queries Monochromatic: given a multi-dimensional dataset Pand a point q, find all the points p Pthat have qas their nearest neighbor Bichromatic: given a set Qof queries and a query point q, find the objects p Pthat are closer to qthan any other point ofq p 1 q p 3 p 4 RNN(q)=p 1, p 2 NN(q)= p 3 p 2 19
20 KM Algorithm (for static datasets) Find the NN of every data point p-let the vicinitycircle(p, dist(p,nn(p))) centeredatpwithradius equalto the Euclideandistance betweenpand itsnn. Index the MBRsof all circleswithan R-tree, calledthe RNN-tree The reverse nearestneighborsof qare retrievedby a point location queryon the RNN-tree, whichreturnsall circles that contain q. 20
21 SAA Algorithm (supports updates) Divide the space around the query qinto six equal regions S 1 to S 6. Find the NN p i of qin each region S i Find the NN of each p i if distance(p i, NN(p i )) < distance(p i, q) there is no RNN of qin S i otherwise, the only RNN of qin S i is p i. p 2 p 1 60 o p 3 S 1 S 2 S 3 p 4 60 o q p 5 p 6 p7 S 4 S 6 S 5 21
22 TPL Algorithm (supports updates, >2 dimensions, k-rnn) Filter-refinement approach Find the set S cnd of candidate points Find neighbors of the query point q incrementally Every new neighbor prunes the search space Continue until the entire space is pruned Keep all the pruned points and nodes in a set S rfn Refinement step: eliminate false positives from S cnd First NN N 1 p' p1 N 2 p1 (p, q) perpendicular bisector ( p 1, q ) q ( p 2, q) p2 q Second NN 22
23 Example of TPL 23
24 Filter Step Visit Root Action Heap S cnd S rfn visit root {N 10,N 11,N 12 } 24
25 Filter Step Visit N 10 Action Heap S cnd S rfn visit N 10 {N 3,N 11,N 2,N 1,N 12 } 25
26 Filter Step Visit N 3 Action Heap S cnd S rfn visit N 3 {N 11,N 2,N 1,N 12 } {p 1 } {p 3 } 26
27 Filter Step Visit N 11 Action Heap S cnd S rfn visit N 11 {N 5,N 2,N 1,N 12 } {p 1 } {p 3,N 4,N 6 } 27
28 Filter Step Visit N 5 Action Heap S cnd S rfn visit N 5 {N 2,N 1,N 12 } {p 1,p 2 } {p 3,N 4,N 6,p 6 } 28
29 Filter Step Visit N 1 Action Heap S cnd S rfn visit N 1 {N 12 } {p 1,p 2,p 5 } {p 3,N 4,N 6,p 6,N 2,p 7 } 29
30 Example (end of filter step) Action Heap S cnd S rfn {p 1,p 2,p 5 } {p 3,N 4,N 6,p 6,N 2,p 7,N 12 } 30
31 Refinement Step Nodes outside the cicrles centered at the candidates can be discarded p 3 invalidates candidate p 1 In order to verify candidate p 2 we need to visit N 4 p 5 is an actual RNN Other Work: Bichromatic RNN Road Networks Metric Spaces Continuous RNN 31
32 3. Aggregate NN Queries Given a set Pof data points and a set Qof query points, an aggregate NN query returns the data point pwith the minimum aggregate distance. The aggregate distancebetween a data point pand Q={q 1,.., q n }is defined as adist(p,q) = f( pq 1,, pq n ), where pq i is the Euclidean distance of pand q i. We use f=sumin the examples. 32
33 Multiple Query Method (MQM) Idea based on the thresholdalgorithm for top-kqueries: Perform incrementalnnqueries for each point in Qand combine their results. <p 10, 7>, <p 11, 6>, Threshold T=5 (2+3) Every non-yet visited point must have distance T=5 Since the best candidate found p 11 has distance > T we must continue search 33
34 Multiple Query Method (MQM) We find p 11 again, this time as the second NN of q 1 The value of the threshold is updated to T=6 We already have a candidate with distance equal to T=6 Search terminates 34
35 Minimum Bounding Method (MBM) MQM applies multiple NN queries and may visit the same node and data point multiple times MBM Applies the MBR of Qto prune the search space with a single (DF or BF) traversal, using heuristics to prune the search space Heuristic 1:Let Mbe the MBR of Q, and best_distbe the distance of the best points found so far. A node Ncannot contain qualifying points, if: best_dist mindist( N, M ) n Heuristic 2: A node N cannot contain qualifying points, if: q Q i mindist( N, q ) i best_dist
36 4. NN Queries with Validity Information The result of conventional NN queries may be invalidated as soon as the query or some data object(s) move Goal: To return some additional information that determines the validityof the current result The validity regionis such that the NN set remains the same as long as q remains in the region. The validity region of a NN query qis the VoronoiCell of the NN o. 36
37 Settings: Static Objects, Moving Query. Client/server architecture A central server indexes the Voronoi diagram of the objects Given a query, the server returns the current NN and the validity timeof the result. The current NN is the object whose Voronoi cell covers the query point Restrictions:(i) assumes a maximum speed (ii) applicable only to single NN (iii) and static objects. Validity Time 37
38 Redundant Results Settings: moving client/query, static objects. The client needs to keep updated about the NN set as it moves. Goal: Minimize the number of queries by returning m>k NNs. Example: A client a location q wants to know its NN Server returns m=4 NNs {o,a,c,b} If 2 dist(q,q') dist(q,b)-dist(q,o), THEN the NN at q' is among the 4 NN of the first query. Problem: how to determine m. 38
39 Time parameterized NN Assuming that queries and/or objects move linearly, a TPNN returns: The current NN set R The validity period Tof R The change Cof the result at the end of T y axis e f g h i m k j l Result: R={i}, T=2, C={j} 4 2 b c d query (speed 1) a its position at time 2 x axis
40 Processing TP NN with R-(TPR-) trees Influence time of a MBR: the earliest possible time that any object in the MBR will become the new NN. Algorithm: first compute the conventional NN Then traverse the R-tree (TPRtree) again, visiting nodes in ascending order of their influence time (instead of the mindist). Cost of TPNN queries about the same as that of conventional queries because we have to visit the influencing nodes anyway (to find the NN) y axis P NN q (speed 1) x axis E position at time 2 at this time q has equal distance to P NN and E 40
41 Location Based NN queries (LBNN) A location-based knn query q returns The current knns The validity regionsuch that the result remains the same as long as qremains in the region. The validity region of qis the Voronoi Cell(VC) of the NN o. Based on TPNN queries.
42 Continuous Nearest Neighbors (CNN) Goal: Given a line segment q=[s,e], find the NN of everypoint on q. Result representation: {s(.nn=a), s 1 (.NN=c), s 2 (.NN=f), s 3 (.NN=h), e}. The points (s, s 1, s 2, s 3,e) are the split points, meaning that ais the NN for the interval [s, s 1 ), cis the NN for the interval [s 1, s 2 ) etc. b d s s 1 s 2 s 3 f e a c g h 42
43 Main idea Maintain the set of split points incrementally. After processing a After processing c 43
44 Processing with an R-(TPR-) tree Avoid examination of all points. Given an MBR Eand query segment q, Emust be searched if and only if there exists a split point s i SL such that dist(s i,s i.nn) > mindist(s i, E). 44
45 5. CNN monitoring Continuous NN monitoring: Assumeasetofk-NNqueriesoverasetofmovingobjects. Numerous queries and objects move in an unpredictable manner. A central server continuously monitors the k NNs of each query and reports to the clients. Goal: Minimize CPU overhead at the server and/or network overhead of location updates. Usually main-memory processing using simple grids(r-trees incur expensive updates) 45
46 YPK-CNN Initial Result Computation Visit the cells in a square R around the cell c q covering q until you find the first object (at distance d). Search the cells intersecting the square SR (search region) centered at c q with side length 2 d+δ, and determine the actual NN set. Result Update Compute the maximum distance d max of the current locations of the previous NNs (i.e., d max is the distance of the previous neighbor that moved furthest). The new SR is a square centered at c q with side length 2 d max +δ. 46
47 SEA-CNN SEA-CNN assumes that the initial result has been computed. Let best_dist be the distance of the NN Result Update Similar to YPK-CNN but with circular SR if the query q moves to a new location q, then SEA-CNN sets r =best_dist + dist(q,q ), and computes the new NN set by processing all the objects that lie in the circle centered at q with radius r. p 7 p 3 p 2 Answer region best_dist p 1 q p 6 p 4 p 5 best_dist+dist(q,q') q' SR p 8 p 10 p 9 47
48 CPM Conceptual Partitioning Update handling when Motivation: when computing outgoing objects are more the initial result we only visit than incoming ones the minimum number of cells (without computing the distance of all cells) Update handling when incoming objects are at least as many as outgoing ones p' 2 p 3 p' 3 p 2 best_dist q p 1 Influence region 48
49 Threshold-based NN 1. YPK-CNN, SEA-CNN, CPM assume that objects issue updates whenever they move (even if they do not influence a query). Their goal is to minimize the computation overhead at the server. 2. TB minimizes the network overhead for location updates(orthogonal to previous algorithms). 3. Main idea: avoid object updates when their movement does not affect any query. 4. Thresholds to objects: No violations no result change, no need for updates. 5. Applicable to non-euclidean spaces and arbitrary movement. 49
50 Related Work on Moniroring Range query monitoring. Approximate NN monitoring. Minimization of network overhead. Continuous Monitoring of Reverse NN. NN processing using air indexes in broadcasting environments. 50
51 6: NN queries in Non-Euclidean Spaces Road Networks Find my nearest hotel in terms of driving distance. Answer: Hotel b(the Euclidean NN is d) 51
52 Euclidean Restriction Algorithm 1st Euclidean NN 2nd Euclidean NN 52
53 Network Expansion Algorithm n 4 1 p 5 2 n n p 3 n 5 p 2 q 3 p 1 2 n 1 n 6 p n 8 n 7 n 9 53
54 Related Work on Road NN knns based on pre-computed network Voronoi cells. Path knn queries (k-nns for every point on a trajectory). SurfacekNN Query Processing. Continuous Monitoring. 54
55 NN in the presence of obstacles The NN of qin terms of obstructed distance is b, although the Euclidean NN is a. 55
56 Visibility graphs Have been used widely in Computational Geometry for shortest path problems (e.g., find the shortest path from p start to p end that does not cross any obstacle). p start p end Problem: We cannot maintain the entire visibility graph in memory for real spatial datasets. Solution: We only need the obstacles and objects that affect the result of the query. 56
57 Obstacle nearest neighbor algorithm Idea: Similar to the Euclidean Restriction algorithm for road networks. BUT how do we perform the obstructed distance computations? 57
58 Obstructed distance computation Goal: compute the obstructed distance between p and q. First retrieve obstacles o 1, o 2 in the Euclidean range. Compute a provisional distance d 1 (p,q) using only o 1, o 2. d 1 (p,q) is not enough because the shortest path is obstructed by o 3. Perform a second Euclidean range query on the obstacle R-tree using d 1 (p,q) and retrieve o 3, o 4. Compute a new obstructed distance d 2 (p,q) taking o 3, o 4 into account. Repeat the process until the obstructed distance remains the same for two consecutive iterations. 58
59 Related work Visible Nearest Neighbor Queries: We only care about objects visible from the query point. Visible Reverse Nearest Neighbors. Continuous Obstructed Nearest Neighbor Queries. 59
60 7. Private NN Queries Anonymizer model 1. A user/client Usends its NN query and anonymity requirementkto the anonymizer, which maintains the current locations of numerous clients. 2. The anonymizer removes the user ID and selects an anonymizing set(as) that contains Uand at least K-1 other clients in its vicinity. The K-anonymizing spatial region(k-asr or ASR) is an area that spatially encloses AS. 3. The anonymizer forwards the ASR to the LBS that stores the spatial data. 4. (iv) The LBS processes the query and returns to the anonymizer a set of candidate results. 5. The anonymizer removes the false hits and forwards the actual result to U. user U actual position, query, K secure connection actual results ASR, query n o n s e c u r e c o n n e c t i o n candidate results Anonymizer LBS location data objects
61 Casper The anonymizer maintains the locations of the clients using a pyramid data structure, similar to a Quad-tree, where the minimum cell size corresponds to the anonymity resolution. Once the anonymizer receives a query from U, it uses a hash table on the user ID pointing to the lowest-level cell cwhere U lies. If ccontains enough users (i.e., c K) for the anonymity requirements, it forms the ASR. Otherwise ( c <K), the horizontal c h and vertical c v neighbors of care retrieved. If c c h Kor c c v K, the corresponding union of cells becomes the ASR. On the other hand, if c c h < Kand c c v < K, the anonymizer retrieves the parent of c and repeats this process recursively. 61
62 Assume a query qwith K=2. Casper Example If qis issued by U 1 oru 2, the ASR is cell (0,2), (1,3). If qis issued by U 3 oru 4, the ASR is the union of cells (1,2), (2,3) (1,3), (2,4). If qis issued by U 5, the ASR is the entire data space. (1,4) U 4 (2,4) (4,4) Problem of Casper: (0,3) (0,2) U 1 (1,3) U 2 U3 (1,2) (2,1) (2,3) (2,2) U 5 (3,2) (4,2) Although U 1 to U 4 are in the 2-ASR of U 5, U 5 is not in the 2-ASR of any of those users. Consequently, an attacker that detects an ASR covering the entire space can infer with high probability that it originates from U 5. (0,0) (2,0) 62
63 Reciprocity Property Consider a user Uissuing a query with anonymity degree K, anonymizing set AS, and anonymizing spatial region ASR. AS satisfies reciprocityif (i) it contains Uand at least K-1 additional users, and (ii) every user in AS also generates the same anonymizing set AS for the given K. Consider an algorithm Center Cloak that, given a query from U, finds its K-1 closest users and sets the ASR as the MBR that encloses them. Given a query from U 3, U 4, or U 5, Center Cloak would generate the top ASR. If the query originates from U 6, it would generate the bottom ASR. The second ASR violates anonymity because an attacker can be sure that the query is issued by U 6, as the same ASR could not be generated for any other client. Specifically, any AS involving 3 users and created for U 4 or U 5, would contain U 3, but not U 6. 4 y y U 2 U 1 U 2 U 1 U U U 3 U 4 3-ASR for U 3,U 4, or U 5 U 5 U 3 U ASR for U 6 U x x
64 Hilbert Cloak Uses the Hilbert space filling curve to map the 2-D space into 1-D values. These values are then indexed by an annotated B+-tree, which supports efficient search by value or by rank (i.e., position in the 1-D sorted list). The algorithm partitions the 1-D sorted list into groups of Kusers (the last group may have up to 2K-1 users). For a querying user Uthe algorithm finds the group where Ubelongs, and returns the MBR of the group as the ASR. The same ASR is returned for any user in a given group; therefore the algorithm is reciprocal. U 10 U 8 Buckets for K=3 U 9 U 6 U 7 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 Buckets for K=4 U 2 U 5 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 U 3 U 1 Hilbert Curve U 4 64
65 Additional Topics Location privacy without anonymizer Private information retrieval Anonymization of trajectories Query processing techniques for anonymized queries 65
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