Approximate Aggregation Techniques for Sensor Databases

1
Approximate Aggregation Techniques for Sensor
Databases

• John Byers
• Department of Computer Science
  Boston University
• Joint work with Jeffrey Considine, George
  Kollios, Feifei Li

2
Sensor Network Model

• Large set of sensors distributed in a sensor
  field.
• Communication via a wireless ad-hoc network.
• Node and links are failure-prone.
• Sensors are resource-constrained
• Limited memory, battery-powered, messaging is
  costly.

3
Sensor Databases

• Treat sensor field as a distributed database
• But data is gathered, not stored nor saved.
• Perform standard queries over sensor field
• COUNT, SUM, GROUP-BY
• Exemplified by work such as TAG and Cougar
• For this talk
• One-shot queries
• Continuous queries are a natural extension.

4
Tiny Aggregation (TAG) ApproachMadden,
Franklin, Hellerstein, Hong

• Aggregation component of TinyDB
• Follows database approach
• Uses simple SQL-like language for queries
• Power-aware, in-network query processing
• Optimizations are transparent to end-user.
• TAG supports COUNT, SUM, AVG, MIN, MAX and others

5
TAG (continued)

• Queries proceed in two phases
• Phase 1
• Sink broadcasts desire to compute an aggregate.
• Nodes create a routing tree with the sink as the
  root.
• Phase 2
• Nodes start sending back partial results.
• Each node receives the partial results of its
  children and computes a new partial result.
• Then it forwards the new partial result to its
  parent.
• Can compute any decomposable function
• f (v1, v2, , vn) g( f (v1, .., vk), f (vk1,
  , vn))

6
Example for SUM
sink
20

• Sink initiates the query
• Nodes form a spanning tree
• Each node sends its partial
• result to its parent
• Sink computes the total sum

7
Classification of Aggregates

• TAG classifies aggregates according to
• Size of partial state
• Monotonicity
• Exemplary vs. summary
• Duplicate-sensitivity
• MIN/MAX (cheap and easy)
• Small state, monotone, exemplary,
  duplicate-insensitive
• COUNT/SUM (considerably harder)
• Small state and monotone, BUT duplicate-sensitive
• Cheap if aggregating over tree without losses
• Expensive with multiple paths and losses

8
Basic approaches to computing SUM

• Separate, reliable delivery of every value to
  sink
• Extremely costly in energy and energy consumption
• Aggregate values back to sink along a tree
• A single fault eliminates values of an entire
  subtree
• Split values and route fractions separately
• Send (value / k) to each of k parents
• Better variance, but same expectation as approach
  (2)
• Send values along multiple paths
• Duplicates need to be handled.
• ltID, valuegt pairs have limited in-network
  aggregation.

9
Design Objectives for Robust SUM

• Admit in-network aggregation of partial values
• Let aggregates be both order-insensitive and
  duplicate-insensitive
• Be agnostic to routing protocol
• Trust routing protocol to be best-effort.
• Routing and aggregation logically decoupled NG
  03.
• Some routing algorithms better than others.

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Design Objectives (cont)

• Final aggregate is exact if at least one
  representative from each leaf survives to reach
  the sink.
• This wont happen in practice.
• It is reasonable to hope for approximate results.
• We argue that it is reasonable to use aggregation
  methods that are themselves approximate.

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Outline

• Motivation for sensor databases and aggregation.
• COUNT aggregation via Flajolet-Martin
• SUM aggregation
• Experimental evaluation

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Flajolet / Martin sketches JCSS 85

• Goal Estimate N from a small-space
  representation of a set.
• Sketch of a union of items is the OR of their
  sketches
• Insertion order and duplicates dont matter!

Prerequisite Let h be a random, binary hash
function. Sketch of an item For each unique
item with ID x, For each integer 1 i k in
turn, Compute h (x, i). Stop when h (x, i)
1, and set bit i.
X 0 0 1 0 0

Z 1 0 0 0 0

X Z 1 0 1 0 0
n
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Flajolet / Martin sketches (cont)
Estimating COUNT Take the sketch of a set of N
items. Let j be the position of the leftmost zero
in the sketch. j is an estimator of log2 (0.77 N)
S
1
1
1
0
1
j 3
Best guess COUNT 11

• Fixable drawbacks
• Estimate has faint bias
• Variance in the estimate is large.

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Flajolet / Martin sketches (cont)

• Standard variance reduction methods apply.
• Compute m independent sketches in parallel.
• Compute m independent estimates of N.
• Take the mean of the estimates.
• Provable tradeoffs between m and variance of the
  estimator

15
Application to COUNT

• Each sensor computes k independent sketches of
  itself (using unique ID x)
• Coming next sensor computes a sketch of its
  value.
• Use a robust routing algorithm to route sketches
  up to the sink.
• Aggregate the k sketches via union en-route.
• The sink then estimates the count.

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Multipath Routing

• Braided Paths

Two paths from the source to the sink that differ
in at least two nodes
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Routing Methodologies

• Considerable work on reliable delivery via
  multipath routing
• Directed diffusion IGE 00
• Braided diffusion GGSE 01
• GRAdient Broadcast YZLZ 02
• Broadcast intermediate results along gradient
  back to source
• Can dynamically control width of broadcast
• Trade off fault tolerance and transmission costs
• Our approach similar to GRAB
• Broadcast. Grab if upstream, ignore if downstream
• Common goal try to get at least one copy to sink

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Simple Upstream Routing

• By expanding ring search, nodes can compute their
  hop distance from the sink.
• Refer to nodes at distance i as level i.
• At level i, gather aggregates from level i1.
• Then broadcast aggregates to level i - 1
  neighbors.
• Ignore downstream and sidestream aggregates.

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Extending Flajolet / Martin Sketches

• Also interested in approximating SUM
• FM sketches can handle this (albeit clumsily)
• To insert a value of 500, perform 500 distinct
  item insertions
• Our observation We can simulate a large number
  of insertions into an FM sketch more efficiently.
• Sensor-net restrictions
• No floating point operations
• Must keep memory usage and CPU time to a minimum

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Simulating a set of insertions

• Set all the low-order bits in the safe region.
• First S log c 2 log log c bits are set to 1
  w.h.p.
• Statistically estimate number of trials going
  beyond safe region
• Probability of a trial doing so is simply 2-S
• Number of trials B(c,2-S). Mean O(log2 c)
• For trials and bits outside safe region, set
  those bits manually.
• Running time is O(1) for each outlying trial.
• Expected running time
  O(log c) time to draw from B(c,2-S)
  O(log2 c)

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Fast sampling from discrete pdfs

• We need to generate samples from B(n, p).
• General problem sampling from a discrete pdf.
• Assume can draw uniformly at random from 0,1.
• With an event space of size N
• O(log N) lookups are immediate.
• Represent the cdf in an array of size N.
• Draw from 0, 1 and do binary search.
• Cleverer methods for O(log log N), O(log N) time

Amazingly, this can be done in constant time!
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Constant Time Sampling

• Theorem Walker 77 For any discrete pdf D
  over a sample space of size N, a table of size
  O(N) can be constructed in O(N) time that enables
  random variables to be drawn from D using at most
  two table lookups.

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Sampling in O(1) time Walker 77

• Start with a discrete pdf. 0.40, 0.30, 0.15,
  0.10, 0.05
• Construct a table of 2N entries.

Algorithm Pick a column at random. Pick x
uniformly from 0, 1. If x lt pi ? output i.
Else output Qi
A B C D E
pi
0.25
1
1
0.75
0.5
__
__
Qi
A
B
A
In table above PrB 1 0.2 0.5 0.2
0.3 PrC 0.75 0.2
0.15
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Methods of Walker 77 (cont.)

• Ok, but how do you construct the table?

Table construction Take below-average i.
Choose pi to satisfy xi pi /n. Set j with
largest xj as Qi Reduce xj accordingly. Repeat.
A B C D E
0.05
0.15
0.10
0.40
0.30
0.20
0
0
0.25
0
0.20
A B C D E
pi
0.25
1
1
0.75
0.5
__
__
Qi
A
B
A
Linear time construction.
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Back to extending FM sketches

• We need to sample from B(c, 2-S) for various
  values of S.
• Using Walkers method, we can sample from B(c,
  2-S) in O(1) time and O(c) space, assuming tables
  are pre-computed offline.

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Back to extending FM sketches (cont)

• With more cleverness, we can trade off space for
  time. Recall that,
• Running time time to sample from B O(log2 c)
• Sampling in O(log2 c) time leads to O(c / log2 c)
  space.
• With max sensor value of 216, saving a log2 c
  term is a 256-fold space savings.
• Tables for S 1, 2,, 16 together take 4600
  bytes
• (without this optimization, tables would be gt1MB)

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Intermission

• FM sketches require more work initially.
• Need k bits to represent a single bit!
• But
• Sketched values can easily be aggregated.
• Aggregation operation (OR) is both
  order-insensitive and duplicate-insensitive.
• Result is a natural fit with sensor aggregation.

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Outline

• Sensor database motivation
• COUNT aggregation via Flajolet-Martin
• SUM aggregation
• Experimental evaluation

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Experimental Results

• We employ the publicly available TAG simulator.
• Basic topologies grid (2-D lattice) and random
• Can modulate
• Grid size default 30 by 30
• Node, packet, or link loss rate default 5 link
  loss rate
• Number of bitmaps default twenty 16-bit
  sketches.
• Transmission radius default 8 neighbors on the
  grid

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Experimental Results

• We consider four main methods.
• TAG transmit aggregates up a single tree
• DAG-based TAG Send a 1/k fraction of the
  aggregated values to each of k parents.
• SKETCH broadcast an aggregated sketch to all
  neighbors at level i 1
• LIST explicitly enumerate all ltkey, valuegt
  pairs and broadcast to all neighbors at level i
  1.
• LIST vs. SKETCH measures the penalty associated
  with approximate values.

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Message Comparison

• TAG transmit aggregates up a single tree
• 1 message transmitted per node.
• 1 message received per node (on average).
• Message size 16 bits.
• SKETCH broadcast a sketch up the tree
• 1 message transmitted per node.
• Fanout of k receivers per transmission (constant
  k).
• Message size 20 16-bit sketches 320 bits.

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COUNT vs Link Loss (Grid)
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COUNT vs Link Loss (Grid)
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COUNT vs Network Diameter (Grid)
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COUNT vs Link Loss (Random)
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SUM vs Link Loss
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Compressability

• The FM sketches are amenable to compression.
• We employ a very basic method
• Run length encode initial prefix of ones.
• Run length encode suffix of zeroes.
• Represent the middle explicitly.
• Method can be applied to a group of sketches.
• This alone buys about a factor of 3.
• Better methods exist.

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Compression
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Space Usage
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Future Directions

• Spatio-temporal queries
• Restrict queries to specific regions of space,
  time, or space-time.
• Other aggregates
• What else needs to be computed or approximated?
• Better aggregation methods
• FM sketches have rather high variance.
• Many other sketching methods can potentially be
  used.