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Searching Trajectories by Locations – An Efficiency Study Zaiben Chen 1, Heng Tao Shen 1, Xiaofang Zhou 1, Yu Zheng 2, Xing Xie 2 1 The University of Queensland 2 Microsoft Research, Asia

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Outline Research problem & application scenarios Basic ideas K Best-Connected Trajectory (k-BCT) query The Incremental k-NN Algorithm (IKNN) Performance study Best-first Depth-first Optimization & extension Experiments Conclusion

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Research Problem: Searching Trajectory Databases GPS trajectories collected by GeoLife Project, MSRA How to retrieve the trajectories we want?

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Searching Trajectory Databases Search by a location Search by a sample trajectory Frentzos et al. Geoinfomatica07; Dfoser et al. VLDB00. (R-tree variants) Chen et al, SIGMOD05; Vlachos et al, ICDE02; Yi et al, ICDE98, etc. (Similarity)

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Searching Trajectory Databases The problem we study: Searching by multiple locations To find trajectories that are close to all the locations Technically, it is an extension of the single-location based query. But more complicated. Practically, it produces a more general way to search trajectories. Two extreme cases (one location, many locations)

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Application motivations The Microsoft GeoLife Project GeoLife is a location-based service built on Microsoft Virtual Earth. Our work benefits the following two functions (1) Travel recommendation E.g. To help a visitor planning a trip to multiple attractions by considering others traveling trajectories. (2) Sharing life experiences & friend recommendation E.g. To find out which users share the similar daily route through Queens Plaza, Central Stat., Mains St.

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Application motivations Geo-Coding: From Pictures to Coordinates The recommended route

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Application motivations Geo-Coding: From Pictures to Coordinates The recommended route The first step: to define the closeness (i.e. distance) between a trajectory and locations

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Similarity Function The similarity function reflects how close a trajectory is to the given locations, and we call the most similar trajectory the best-connected trajectory. Step 1. find out the closest trajectory point on R to each location q i Step 2. sum up the contribution of each matched pair. (unordered query) Dist q (q i, R) is the shortest distance from q i to R Q={q 1, q 2, … q m }, R={p 1, p 2, … p n }

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Problem Definition k-Best Connected Trajectory (k-BCT) query Given a set of trajectories T = {R 1, R 2, …, R n }, a set of query locations Q = {q 1, q 2, …, q m }, and the similarity function Sim(Q, R), the k-BCT query is to find the k trajectories among T that have the highest similarity. Assumption: The number of query locations is small. (m is a small constant) Intuition: The k-BCT result is the JOIN of m single-location based queries.

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Basic ideas Incremental k-NN Algorithm (IKNN) Step 1. Index all the trajectory points by one single R-tree Get the shortest distance from a query location to the trajectories Step 2. Search for the λ-nearest neighbor (λ-NN) of each query location (q 1 to q m ), by using any traditional k-nearest neighbor algorithm over R-tree. For any trajectory that scanned by a λ-NN, its shortest distance to the query point is known. Candidate set C = {all scanned trajectories}

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IKNN algorithm Step 3. Construct lower bounds of similarity. For a trajectory R1 in C, assume it got 3 points p1, p2 and p3 scanned by the λ-NN search of q1, q2. R1 p1p2 Sim(Q, R1) = e -|q1, p1| + e -|q2, p2| + e -|q3, p5| p3 q1 q2q3 p5 e -|q1, p1| + e -|q2, p2|

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The Incremental k-NN algorithm Step 4. Construct upper bound of similarity. For any trajectory that is not covered by the λ-NN search, e.g. R5 its distance to q i must be larger than the radius of q i R1 Sim(Q, R5) = e -|q1, R5| + e -|q2, R5| + e -|q3, R5| e -radius1 + e -radius2 + e -radius3 q1 q2q3 R5 radius1radius2radius3

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The Incremental k-NN algorithm Step 5. Check the STOP condition (pruning condition) For a k-BCT query, if we can get k candidate trajectories whose lower bounds are not less than the upper bound of similarity for all un-scanned trajectories, then the k best-connected trajectories must be included in the candidate set. if the condition is satisfied go to the refinement step else increase λ by some Δ repeat the search process With the search region of the λ-NN search enlarges, eventually k best-connected trajectories will be found.

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Problem The problem: we may need to increase λ and compute the lower/upper bounds for many rounds before we eventually find the k-BCT results. The λ-NN search will run for many rounds for every query location. (let λ be a constant k initially, and Δ be k as well) round 1: 1 – k nearest neighbors round 2: 1 – 2k nearest neighbors … round i: 1 – i*k nearest neighbors Trajectory points are visited multiple times. Normally, λ >> k, so the complexity is λ^2.

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Problem The problem: we may need to increase λ and compute the lower/upper bounds for many rounds before we eventually find the k-BCT results. The λ-NN search will run for many rounds for every query location. (let λ be a constant k initially, and Δ be k as well) round 1: 1 – k nearest neighbors round 2: 1 – 2k nearest neighbors … round i: 1 – i*k nearest neighbors Normally, λ >> k, so the complexity is lambda square. Can we reduce the overlapped search regions?

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Efficiency study of the IKNN Adaption of the λ-NN algorithm The best-first nearest neighbor search [Hjaltason et al., TODS99] A priority queue is maintained to store all the R-tree entries that have yet to be visited, using the MINDIST as a key. So it visits MBRs/Objects in the order of the MINDIST. The depth-first nearest neighbor search [Roussopoulos et al., SIGMOD95] It recursively traverses the R-tree level by level in a depth-first manner, while maintaining a global list of k nearest candidates found so far. Estimate the performance of the IKNN adopting different λ-NN algorithms

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Adaption of the λ-NN algorithm The best-first NN search Retrieve the λ, λ+, λ+2, … NN for each query location incrementally until the k best-connected trajectories are included in the candidate set. Benefit The λ-NN is returned in an incremental way I/O optimal, no overlap occurs, V sum = λ Shortcoming Memory consumption is NOT guaranteed. A priority queue is maintained to store all the R-tree entries that have yet to be visited. The queue may be as large as the whole dataset in an extreme case.

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The best-first strategy Performance (R-tree leaf access) Estimate the circle region (with radius r) that contains λ points [Belussi et al. VLDB95] Estimate the leaf access of a range query with radius r [Korn et al. TKDE2001] m independent λ-NN queries q λ objects radius

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Adaption of the lambda-NN algorithm The depth-first NN search Every time we search for the λ+ NN, we have to re-visit the search region of the λ-NN query. Benefit: Guaranteed memory usage, O(c Log c N) Drawback: Too many overlaps A simple improvement: Double λ at each round, to reduce the number of rounds and amortize cost. Pruning: All MBRs whose MAXDIST is even smaller than the current search range of λ-NN can be skipped in the search of λ+ NN.

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The depth-first strategy Performance (R-tree leaf access) The search region is not necessary a circle! So we can not use the previous method directly. Estimate the size of the first visited MBR (at any level) that contains not less than λ points Estimate the radius (MAXDIST) of the region that contains the MBR MBR 1 qiqi MAXDIST R-tree nodes outside the circle with radius MAXDIST wont be visited.

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The depth-first strategy (cont.) Performance Estimate the leaf access of a range query with radius MAXDIST [Korn et al. TKDE2001] Finally,

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Summary IKNN algorithmMemory usageObject visitsLeaf access The best-first strategy no guaranteem × O(λ) The depth-first strategy O(logN * c)m × O(λ) The best-first strategy, although has no guarantee in memory usage, it normally runs faster and the priority queue can still be accommodated in the main memory of a modern computer easily. The modified depth-first strategy reaches nearly the same performance as that of the best-first strategy, while it still preserves a low memory consumption

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Optimization & Extension Considering the importance of the query locations and assigning different weights in exploring objects. Extension to query locations with an order specified

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Experiments 12, 653 trajectories (1,147,116 points) collected by the Geolife project Number of query locations: 2 to 10 Tests are conducted on PC with 2.1GHz CPU and 1GB memory

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Experiments – Node Access

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Experiments – Query Time

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Experiments – Memory Usage

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