Medians and Beyond New Aggregation Techniques for Sensor Networks

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Medians and Beyond New Aggregation Techniques
for Sensor Networks
N. Shrivastava, C. Buragohain, D. Agrawal, and S.
Suri(Sensys 2004)
Presented by Dohyung Kim in CNLAB
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Contents

• Introduction
• Quantile Digest (q-digest)
• Building, merging, and representation
• Space complexity and error bound
• Queries on q-digest
• Confident factor
• Experimental evaluation
• Conclusion

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Introduction

• Design constraints of the communication and
  information infrastructure
• limited capability of individual sensor
• computation capability
• communication bandwidth
• battery power
• Periodical delivery of sensing data leads to
  excessive communication, which is wasteful.

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Introduction (contd)

• Alternatives
• Routing in a tree shape
• Route combined messages from multiple nodes to BS
  (base station)
• Problem large message size
• Energy efficient query processing architecture
  (UC Berkeley, Cornell)
• In network aggregation (aggregated measure)
• Minimizing both the number and the size of
  messages
• Problems in the case of requiring the
  distribution of sensor values at BS (median)

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Quantile Digest (q-digest)

• Approximation scheme for non single valued
  queries.
• Error of O(log(s)/m)
• Message size m
• Returned value of sensors 1, s
• Carries with itself the estimate of error
• Strictly bounded error
• Provides Error-Memory Trade-off
• Confidence Factor
• Multiple Queries possible

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Quantile Digest (contd)

• Conceptual complete tree
• Every bucket has a counter , count(v)
• Node v in q-digest iff. following q-digest
  property
• Vp parent, Vs sibling K compress pmtr,
• n sum of frequency of all sensor values
• Exception case leaf node root node

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Building Q-digest
Data loss !!
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Merging Q-digests
Small subset of q-digest with n 400 (n1 n2
200), k 10, s64
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Representation of Q-digest

• A set of tuples like (nodeid(v), count(v))
• (1,1), (6,2), (7,2), (10,4), (11, 6)
• Nodeid(v) level order
• O(log(2s)log(n)) bits for each tuple.

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Queries on q-digest

• Quantile query
• Given a fraction q ? (0, 1), find the value whose
  rank in sorted sequence of the n values is qn.
• Examples (median query, that is 0.5n)
• (1,1), (6,2), (7,2), (10,4), (11,6)
• (10.4), (11,6), (6,2), (7,2), (1,1) result of
  post order traverse
• The count of (11,6) is more than 0.5n(8),
  therefore estimated median is 4 (nodeid 11)
• Inverse quantile query
• Given a value x, determine its rank in the sorted
  sequence of the input values
• Range query
• Fine the number of values in the given range
  low, high
• Consensus query
• Given a fraction s ? (0, 1), find all the values
  which are reported by more than sn sensors. That
  is called finding Frequent items.

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Space Complexity Error Bound

• (lemma 1) A q-digest constructed with compression
  parameter k has a size at most 3k
• (lemma 2) In a q-digest created using the
  compression factor k, the maximum error in count
  of any node is ancestorsk/n, that is log(s)k/n
• (lemma 3) Given p q-digest Q1, Q2,.. Qp built on
  n1, n2, .., np values, each with maximum relative
  error of log(s)/k, the algorithm MERGE combines
  them into a q-digest for ? ni values, with the
  same relative error
• (theorem 1) Given memory m to build a q-digest,
  it is possible to answer quantile query with
  error e such that e lt 3 log(s)/m, when e is
  defined as r-qn/n. (r true rank, qn quantile
  query)

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Confident Factor

• Theoretical worst case error leads to useless
  transmission of large messages because of rear
  occurrence.
• Use m for which it is expected to deliver the
  required error guarantee
• To verify the error guarantee is met, confident
  factor is used for calculation.
• Confident factor ? maximum weight of any path
  from root to leaf in Q)/n
• Weight of the path means the sum of counts in the
  path
• Maximum error is present in the path with maximum
  weight

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Experimental Evaluation

• Simulation of algorithm written by C
• Assumption
• Sensors with Fixed radio range
• Randomly deployed in square area
• Density constant network
• 16-bit Random and correlated sensor value with
  geographical location
• Averaged over 5 different topologies
• Compared with list (unaggregated data
  summarization)

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Range Queries and Histogram
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Accuracy

• Error percentage Ratio of rank error and number
  of values.

• Confident factor

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Message Size

• Message size in list vs. q-digest
• (400 bytes with 2 error in former graph)

• Actual transmission of the node in list
• 5 of nodes bigger than 400 bytes but heavy
  load.
• 1 of nodes size up to 30K

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Total Data Transmission
scalability

• The message size of q-digest is set to 160 bytes

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Residual Power

• Common metrics
• Total power consumption
• super-linearly with total data transmitted
• Lifetime
• The time at which network partition occur
• Considering transmission power plus computation
  power at node close to BS
• P residual power/initial power

• More than one node (0.02 in 8000) have residual
  power fraction less
• than ½ due to the just one query !!

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Conclusion

• Q-digest is distributed data summarization
  technique for approximate queries using limited
  memory
• Q-digest can preserves information about high
  frequency values while compressing information
  about low frequency ones
• Q-digest makes it possible to save much bandwidth
  and power, that is, scalable