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Continuous Spatiotemporal Trajectory Join

Petko Bakalov, Vassilis J. Tsotras

University of California, Riverside

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Overview Motivation CSTJ definition Indexing Spatiotemporal streams along the

temporal axe Evaluation Framework and algorithms Experimental results

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Motivation Streaming

Spatiotemporal data is generated in many real life applications

There is need for more complex continuous spatial predicates

Result

QueryClients

GPSDevices

Antenna

Server

Spatiotemporal stream

Objectlocation

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Continuous SpatiotemporalTrajectory Join The problem of identifying all pairs of moving

objects which follow similar movement pattern for specified period of time.

Example: "Identify the pairs of (security officers, visitors) that have followed each other in the last 10 minutes.“

Difference from existing continuous predicates: The predicate involves relative time constraint “ ………….last 10 minutes.“

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Formal Definition Input: A stream of location/time-instant <li, ti>

pairs from two sets of objects or and os a threshold ε, and a relative time interval δt

Output: set V of pairs <oi, oj>, where oi Є or, oj Є os, such that for every time instant ti in the interval [tnow; tnow – δt]

D( location(oi, ti), location(oi, ti), ) < ε

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Overview Motivation CSTJ definition Key Ideas Indexing Spatiotemporal streams along the

temporal axe Evaluation Framework and algorithms Experimental results

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Key Ideas Initial formation of the result (MDM05, GIS05)

Use trajectory approximations to reduce the size of the problem

Define a lower bound distance function for this approximation

Using this distance function, prune as many trajectory pair similarity evaluations as possible.

Incremental reevaluation. Reuse as much as possible the result from the previous iteration

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Trajectory Approximation Use of a uniform spatial grid to discretize the

spatial domain. Each object location li in the stream can be

approximated with the grid cell in the boundaries of which it is.

Trajectory approximation

11 12 13

21 22 23

31 32 3322 22 22 23 13 13

<22,3><23,4><13,6>

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Trajectory Approximation Organize the trajectory approximations in a

way that enables the use of incremental approach

<22,3><23,4><13,6>

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<32,2><33,3><23,5><13,6>

<31,1><21,3><11,7>

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Time from

Time to

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Trajectory Approximation For any time period ( for example (3;6)) this

space can be divided on 4 regions

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Region 2

Region 3

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Lower bound distance function We proposed the use of the following distance

function, defined over trajectory approximations

d() is the minimal distance between grid cells Slightly different form the functions in MDM05,

GIS05

);(

2, )())(),((

ii

ittti

jisjritt srdoToTD

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Overview Motivation CSTJ definition Key Ideas Indexing Spatiotemporal streams along the

temporal axe Evaluation Framework and algorithms Experimental results

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Initial formation of the result Step 1. Define an order among the trajectories

in the system by using as a score the distance between these strings and M randomly picked trajectories called origins Oi: The origins can be artificially created

Step 2. By using sliding window algorithm find all pairs between the two sets having score less then the threshold.

Step 3. Verify the result with the actual trajectory data

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Step 1 – compute the scores

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Choose originO1 <33,6>

Create partitioning for time period (tstart-δt; tstart) e.g. (2;5)I1 – set of points in region 1I2 – set of points in region 2I3 – set of points in region 3I4 – set of points in region 4

Assume that a query with δt=3 is introduced in the system in time instance tstart=5

Time to

Time from

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Step 2 & 3 sliding window and verificationCompute squared sum of distances between trajectory approximations and origin using the formula

Compute the object scores

W1(o1) = 1,4 W1(o2) = 1 W1(o3) = 2,4

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Sliding window algorithm

Object 1 and Object 2 might have a join – report as a candidate pair

0 1 2 3

Object 1 – Set 1 Object 2 – Set 2Object 3 – Set 2

Object 1 and Object 2 are tested if they indeed satisfy the criteria

W1(o1) = 1,4 W1(o2) = 1 W1(o3) = 2,4

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Continuous reevaluation Once the initial result is formed the query

moves to a second phase where it is constantly reevaluated

To minimize the cost of the reevaluation we keep the results form the previous step and apply to them the changes which have occurred to the stream

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Continuous reevaluation

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Experimental Results Synthetic datasets on the

freeways of Illinois Up to 150 000 moving

objects Measure the average

number of disk accesses for the index and the total number of candidate trajectories that need to be retrieved

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Experimental Results

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Experimental Results

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