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An Effective Coreset Compression Algorithm for Large Scale Sensor Networks Dan Feldman, Andrew Sugaya Daniela Rus MIT. Data. Data. Data. Data. =. Data. Data. Data. How much data?. 1 GPS Packet = 100 bytes. (latitude, longitude, time). 1 GPS Packet = 100 bytes.

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slide1
An Effective Coreset Compression Algorithm for Large Scale Sensor NetworksDan Feldman, Andrew SugayaDaniela RusMIT
slide2

Data

Data

Data

Data

=

Data

Data

Data

slide5

1 GPS Packet

=

100 bytes

(latitude, longitude, time)

slide6

1 GPS Packet

=

100 bytes

every 10 seconds

slide7

~40 Mb / hour

or

~1 Gb / day

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~1 Gb / day

per device

slide9

~300 million

smart phones

sold in 2010

http://mobithinking.com/mobile-marketing-tools/latest-mobile-stats

slide10

For

100 million devices

slide11

For

100 million devices

~ 100 petabytes

per day

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~ 100 thousand

terabytes

per day

gps points data
GPS-points Data
  • iPhones can collect high-frequency GPS traces
  • GPS-point = (latitude, longitude, time)
3 d visualization
3-D Visualization
  • axes: (latitude, longitude)
  • axis: time
challenges
Challenges
  • Storing data on iPhone is expensive
  • Transmission data is expensive
  • Hard to interpret raw data
  • Dynamic real-time streaming data
key insight identify critical points
Key Insight: Identify Critical Points
  • Approximate the n points by k << n semantically meaningful connected segments
our approach
Our Approach
  • Approximate the input GPS-points by connected segments using a k-spline
  • Output the text description of the endpoints (e.g., using Google Maps)
solution overview
Solution overview
  • Semantically compress data points
    • Use coresets
  • Fit lines to the semantic points
    • Use splines on coreset
  • Reverse geo-cite to get directions
definition spline
Definition:-Spline

A -spline is a sequence of connected segments in

optimal spline
Optimal -spline
  • over every k-spline
optimal spline1
Optimal -spline
  • over every k-spline
  • is an optimal -splineof if :
problem statement
Problem Statement
  • Input: set P of n data points in Rd and integer k
  • Output: optimal k-spline for P that provides semantic compression for large data set P
our main compression theorem
Our Main Compression Theorem
  • For every set of points in there is a subset C such that:
    • The maximum distance between a point in to its closest point in is at most
    • can be computed in time

Example application

  • The optimal -spline of is an -approximation of
  • An -approximation for can be computed in time using time algorithm
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Streaming Compression using merge & reduce

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Parallel computation

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summary
Summary

-spline

points

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Compress points: to

Input: points

Approximate points by segments

Extend to segments

Project points onto segments

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Optimal -spline of:

5000 points

300 points

running time
Running time
  • The “slow” optimization algorithm is applied only on small samples or
  • The bottleneck is removing half of in every iteration
  • Total time:
  • Distances can be computed using matrix multiplication  GPU and Parallel computations
space
Space
  • -spline for each one of the iterations  segments
  • Grids of size for each segment
slide89

The Experiment

Input: A set of points

Compute “optimal” -spline using heuristic.

Compute maximum distance from the points to the -spline

Compute compression for

Compute optimal of

Compute Dist(P,)

Compression ratio:

Error:

slide90

Optimal -spline of:

Compression ratio:

Error:

slide91

Experiments:

Subject in Singapore

Error Ratio

Compression Ratio

slide93

Website

Data Display

Coreset Display

Visualization of Result of Algorithm - A Coreset

contribution
Contribution
  • Semantic compression of data from sensors
  • Line simplification using
    • One pass over data
    • Logarithmic space (for massive data sets)
    • Linear time
    • Provable bounded error