1. iSEE: Efficient k-Nearest-Neighbor Monitoring over Moving Obejcts [SSDBM 2007] Wei Wu, Kian-Lee Tan National University of Singapore

2. Problem Settings Given a query point q, continuously report k nearest objects of q Objects and query points move in an unpredictable fashion Objects and query points can be indexed in main memory

3. Related Work Divide space into conceptual rectangles Initialize heap with first level rectangles and c q For top entry e in heap unless mindist(e,q) > best_dist; If e is rectangle, enheap all cells in e If e is cell; check objects in it Conceptual Partitioning: An efficient Method for Continuous nearest neighbor monitoring Kyriakos Mouratidis, Marios Hadjieleftheriou, Dimitris Papadias [SIGMOD 2005]

4. Motivations CPM visits un-necessary cells during update If some nearest neighbor moves away from the query, the new answer cannot be among any cell c for which maxdist(c,q)< best_dist where best_dist is the distance of kth nearest neighbor from q All the shaded cells are visited Ideally, the update should start from the cells that intersect the circle

5. Motivations Heap size is large because rectangles do not approximate circles very well All the cells in any rectangle Rec are inserted in heap when r becomes greater than mindist(Rec,q) All cells in the four rectangles are enheaped. Ideally, only the cells that intersect the circle should be enheaped How to alleviate these problems? CircularTrip VOB

6. Visit Order Build (VOB) Each group L i G j has either 4 or 8 cells The cells in any group L i G j have similar min-dist from q The min-dist of a group L i G j from q is smaller than min-dist of L i+1 G j The min-dist of a group L i G j from q is smaller than min-dist of L i G j+1 Min-dist of L i G j is the minimum of all min-distances between the cells in L i G j from q.

7. Initial Computation Initially en-heap c q and group L 1 G 1 in heap with its min-dist For each de-heaped entry If it is a cell c Look in c for potential NNs store c in visit_list If it is a group L i G j en-heap all cells in L i G i with their min-distance en-heap next level group L i+1 G j with its min-dist if i=j en-heap next group of same level L i G j+1

9. Data Structure Each cell in grid stores object list and influence list best_NN stores the Nearest neighbors among the visited cells search heap H contains the cells and groups that were en-heaped but not de-heaped (Enables quick updates) visit_list stores min and max-distances of all the cells that were de- heaped

10. Update Handling If an object x moves inside the circle include x in result and delete current kth-NN (Update influence List) go backward (descending order) in visit_list deleting q from all the cells c for which min-dist(c,q) > new best_dist If a result object x moves outside the circle delete x from the result set start from the beginning of visit_list and skip the cells for which max-dist(c,q)< best_dist

11. Experiments

13. Memory Comparison with CPM Data structure of iSEE is same as CPM but CPM has a larger search heap visit_list of both CPM and iSEE contains same number of cells but iSEE also stores max-dist for cells Let r be the distance of kth NN from q CPM memory search heap 2.(4r 2 – πr 2 ) = 1.71r 2 visit_list 2(πr 2 ) Total: 8r 2 iSEE memory: visit_list 3(πr 2 ) Total: 3(πr 2 ) + search heap

14. iSEE vs CircularTrip CircularTrip doesn’t need any visit_list or search heap When a query moves, iSEE computes from scratch whereas CircularTrip uses previous information VOB is almost as expensive as CircularTrip (if two consecutive circularTrips visit some (at most 27%) cells twice VOB computes max-dist for each cell) Let C be the number of cells that are needed to be visited during computation, CircualrTrip computes distances for at most 1.27C cells whereas VOB computes distances for 2C cells. Update of influence list by iSEE is faster than CircularTrip because they store visit_list (but lazy update approach can be used in both of these algorithms) CircularTrip uses less memory (50%-85% of iSEE) and the running time of both algorithm is estimated to be similar

15. CircularTrip is more flexible ArcTrip returns all the cells that intersect a circle and lie in a specified angle range ArcTrip can be used to continuously monitor constrained nearest neighbor queries optimally (visiting minimal set of cells) Extension of iSEE for constrained NN queries makes it in-efficient Recall that six constrained NN queries are needed to be continuously monitored for continuous monitoring of RNN queries

16. CircularTrip is more flexible ArcTrip can be used to monitor constrained NN queries over irregular regions efficiently CircularTrip can also be used to efficiently monitor farthest neighbor queries Farthest neighbors in constrained regions can also be monitored For all algorithms, CircularTrip preserves its property that it needs no book-keeping information and visits minimum number of cells