1. Tracking Moving Objects in Anonymized Trajectories Nikolay Vyahhi 1, Spiridon Bakiras 2, Panos Kalnis 3, and Gabriel Ghinita 3 1 St. Petersburg State University 2 John Jay College, City Univ. of New York 3 National University of Singapore

2. 2 Motivation  Collection of Trajectory Data Example: Traffic monitoring system  GPS or Sensors deployed across a city  Queries: Predict traffic conditions Data expected to be anonymous  Remove ID  Reconstruction of original trajectories E.g., Police tracking a suspect

3. 3 Problem Statement  Given a large database with anonymized spatio-temporal measurements, reconstruct the original object trajectories  Requirements Efficiency (large databases) Accuracy (useful results)

4. 4 Problem Statement  Input: A series of M snapshots S i, each containing exactly N measurements from timestamp t i  Output: A set of N trajectories  Each measurement can be associated with a single trajectory M = N = 3

5. 5 Related work: Multiple Target Tracking  This problem is closely related to multiple target tracking (MTT) algorithms Studied in the field of radar technology  Three major categories Nearest neighbor (NN) Joint probabilistic data association (JPDA) Multiple hypothesis tracking (MHT)

6. 6 Related work: NN and JPDA  They work in a single scan of the dataset  Greedy approach: in each timestamp, every sample is associated with a single track  Objective: minimize the error across all associations in the current timestamp  Performance: Efficient – can work in polynomial time Greedy approach results in many false associations

7. 7 Related work: MHT  Multiple hypotheses are maintained Joint probabilities are calculated recursively when new measurements are received  Each association is based on both previous and subsequent data (multiple scans)  Unfeasible hypotheses are eventually eliminated  Performance: Very accurate Computational and space complexity is exponential to the number of measurements

8. 8 Comparison  Very accurate  Very slow  Large errors  Fast  Very accurate  Much faster than MHT

9. 9 Our Approach MCMF: Min-cost Max-flow  Transform the tracking problem into a min-cost max-flow problem  Min-cost max-flow (graph algorithm) Input: a weighted graph G with two special nodes (source s and destination t) Objective: find the maximum flow that can be sent from s to t that results in the minimum cost Well-known algorithms exist that work in polynomial time

10. 10 Transformation  All edges have capacity 1  Node id (t i, p i, p j ): the object moves from location p i in timestamp t i to location p j in timestamp t i+1

11. 11 Calculating the Cost Values  Assume two successive measurements (p i and p j ) belong to the same track  Use these values to predict the next location  Calculate the error (i.e., cost) for every possible location p k

12. 12 Limitation of this Approach  Problem: A single measurement can be associated with multiple tracks!

13. 13 Solution: Create a Block for each Measurement  Corresponds to all partial tracks p m-1,i  p m,k  p m+1,j  A block containing a flow is marked as active  The only possible route inside an active block, is through the reverse path of the existing flow Block for k th measurement of m th timestamp (p m,k )

14. 14 Block Functionality Block for p 3,1 Block for p 2,1 Original track: p 2,1  p 3,1  p 4,1 New track: p 2,2  p 3,1  p 4,1 Original track: p 1,1  p 2,1  p 3,1 New track: p 1,1  p 2,1  p 3,2

15. 15 Improving the Running Time  Flow network is too large Inefficient, since solution requires multiple shortest path calculations  Assume any object can travel at most R max distance between two consecutive timestamps. R max depends on The maximum speed of the objects The time interval between two timestamps  This reduces significantly the number of vertices and edges inside each block

16. 16 The Tracking Algorithm  Successive Shortest Path Algorithm At each iteration, send a single flow unit across the shortest path from s to t Total of N iterations in our case  Most efficient implementation: Dijkstra with Fibonacci heap for priority queue Graph contains negative weights, but can utilize vertex potentials to avoid this (provided that there are no negative weight cycles)  Bellman-Ford also works very well

17. 17 Dealing with Negative Weight Cycles  Negative weight cycles do appear in MCMF calculations  In this case, follow a greedy approach: Output all the tracks that are discovered so far  they might not be optimal Remove all vertices and edges associated with these tracks from the flow network Start a new min-cost max-flow calculation on the reduced graph

18. 18 Complexity  Computational: N iterations of a shortest path algorithm O(MN 2 K(log(MNK) + K)) for Dijkstra with Fibonacci heap K is the average number of feasible associations (due to R max ) per measurement  Space: O(MNK 2 ) for storing the graph

19. 19 Experimental Evaluation  Data generator: Road map of San Francisco city For each object, randomly select a starting point and a destination point The object then follows the shortest path between the two points At each timestamp, every object i covers a distance d i  [0,R max ] Number of measurements: 50,000 to 500,000

20. 20 Experimental Evaluation  Competitor: Global Nearest Neighbor (GNN) Employs clustering within each snapshot Considered the best single scan algorithm – runs in O(MNC 2 ) time (C is the average cluster size)  Performance metrics: CPU time Success rate – percentage of partial tracks (triplets) that agree with original data

21. 21 Variable N CPU time [sec] Success rate [%]

22. 22 Variable R max (speed) CPU time [sec] Success rate [%]

23. 23 Points to Remember  Multiple-Target Tracking Large Anonymized Trajectory Databases  Existing methods are either inefficient or inaccurate  We proposed a polynomial time solution based on a novel transformation of the MTT problem into a min-cost max-flow problem  Very accurate Need to improve the running time

24. 24 Bibliography on LBS Privacy http://anonym.comp.nus.edu.sg