Power Conserving
Computation of Order-Statistics
over Sensor Networks
Michael B. Greenwald & Sanjeev Khanna
Dept. of Computer & Information Science
University of Pennsylvania
PODS 04, June 2004
MBG 1
Sensor Networks
• Cheap => plentiful, large
scale networks
• Low power => long life
• Easily deployed,
wireless =>
hazardous/inaccessible
locations
• Base Station
w/connection to outside
world => info flows to
station & we can extract data,
• Self-configuring => just
deploy, and network sets
itself up
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Management & Use of Sensor Nets
Power Consumption and
Network Lifetime
• Battery replacement is
difficult. Network lifetime
may be a function of worstcase battery life.
• Dominant cost is per-byte
cost of transmission
• Minimize worst-case power
consumption at any node,
maximize network lifetime.
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Management & Use of Sensor Nets
Primary operation is data
extraction
• Declarative DB-like interface
• Issue query, answers stream
towards basestation
• Not necessarily consistent
with power concerns
• COUNT, MIN, MAX easy to
optimize: aggregate in
network.
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Example of simple aggregate query:
MAX
Primary operation is data
extraction
• Aggregate in network
• Transmit only max
observation of all your
children
• O(1) per node.
But what about complicated, but natural, queries,
e.g. MEDIAN?
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Richer queries: Order-statistics
• Exact-order-statistic query:
-quantile(S,) returns the element with rank
r =  |S|
• Approximate order statistic query:
approx--quantile(S,,) returns an element with
rank r,
(1 -  ) |S| <= r <= (1 +  ) |S|
• Quantile summaries query:
quantile-summary(S,) returns a summary Q,
such that
-quantile(Q,) returns an element with rank r,
(1 -  ) |S| <= r <= (1 +  ) |S|
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Goals
• Respond accurately to quantile queries
• Balance power consumption over network
(load on any two nodes differs by at most a poly-log factor)
• Topology independent
• Structure solution in a manner amenable to
standard optimizations:
e.g. non-holistic/decomposable
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Naïve approaches
1. Send all |S| to basestation at root
– If root has few childre, then (|S|) cost
2. Sample w/probability 1/(|S|2)
–
–
–
Uniform sampling
Lost samples
Only probabilistic “guarantee”
Some way to handle summaries in such a way that they can
be aggregated in the network?
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Preliminaries: Representation
• Record min possible rank (rmin) and max rank
(rmax) for each stored value:
10 21 29 34 40 46… 189
1,1
2,2
3,3
4,4
5,5
6,6
…
10,10
10 24 34 40 44 …341
1,1
4,20 23,23 37,37 51,54 … 100,100
• Proposition: If for all 1 <= j < |Q|, rmax(j+1)- rmin(j)
<= 2|S|, then Q is an -approximate summary.
... 40 44 …
 = .1, |S| = 100
.. 37,37 51,54 …
Median = Q(S,.5)
rmin(4)
rmax(5) return entry w/rank .5 100 = 50
(.5-)*100 = 40, (.5+)*100 = 60
Choose largest tuple s.t.
rmin <= r - 
40 <= 51 <= rank(44) <= 54 <= 60
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Preliminaries:
COMBINE and PRUNE Operations
• COMBINE: merge n quantile summaries into one
• PRUNE: reduce number of entries in Q to B+1
• Q = COMBINE({Qi})
– Sort entries, compute new rmin and rmax for each entry.
– Let predecessor set for a given entry (v,rmin,rmax) consist of max
entry v’ from each input summary, such that v’ <= v (successor
set has min v’ such that v’ >= v)
– rmin =  rmin’ for v’ in predecessor set
– rmax = 1 +  (rmax’-1) for v’ in successor set
• Q’ = PRUNE(Q,B)
– Query for each of the [0, 1/B, 2/B, …, 1] quantiles from Q, and
return that set as Q’.
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Combining summaries
… 50
…
… 40
…
8,10
70 …
11,13
12,16
60 …
11,12
•
…
|S|=400,  = .01
… 50
…
…
… 43 51 …
… 121,125 130,140
|S|=300,  = .01
|S|=1000,  = .02
140,163
51 …
149,166
…
|S|=1700,  = .02
…
COMBINE({Qi}):
merge n quantile summaries into one
– Sort entries, and compute “merged” rmin and rmax.
• Theorem: Let Q = COMBINE({Qi}), then
|S| =  |Si|, |Q| =  |Qi|, and  <= max(i).
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Pruning Summary
… 60
…
[(1/B)-
160,183
81 …
334,353
•
…
(1/B)+]
[(2/B)-

|S|=1700,  = .02, 2|S| = 35
1/B
PRUNE(Q,B):
(2/B)+]

reduce number of entries in Q to B+1
• Theorem: Let Q’ = PRUNE(Q,B), then
|S| =  |Si|, |Q’| = B+1, and ’ <=  + 1/(2B)
– 2’|S| = (2 + (1/B))|S|
– ’ =  + 1/(2B)
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Naïve approaches
1. Send all |S| to basestation at root
– If root has few childre, then (|S|) cost
2. Sample w/probability 1/(|S|2)
–
–
–
Uniform sampling
Lost samples
Only probabilistic “guarantee”
3. COMBINE all children, and PRUNE before sending
to parent.
–
–
–
How do you choose B?
If h known and small (say, log |S|), then B=h/ and this has cost
O(h/)
But if topology is deep (linear?), this can be O(|S|).
Need an algorithm that is impervious to pathological
topologies
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Topology Independence
• Intuition: PRUNE based on the number of
observations, not on the topology. Only call PRUNE
when |S| doubles. then the # of PRUNE operations
is bounded by log(|S|)
• If the number of PRUNE operations is known, then
we can choose a value of B that is guaranteed to
never violate the precision guarantee. If there are
log(|S|) PRUNEs, then B=log(|S|)/(2).
• If there are restrictions on when we can perform
PRUNE, then the number/size of the summaries we
transmit to our parents will be larger than B.
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Algorithm
• The class of a quantile summary Q’ is floor(log (|S’|))
• Algorithm:
– Receive all summaries from children.
– COMBINE and PRUNE all summaries of each class with each other.
– Transmit the resulting summaries to your parent.
• Analysis
– At most log(|S|) classes
– B = log(|S|)/(2).
– Total communication cost to parent = log(|S|) log(|S|)/(2).
Theorem: There is a distributed algorithm to compute an
-approximate summary with a maximum transmission
load of O(log2(|S|)/(2) at each node.
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An Improved Algorithm
• New Algorithm:
If we know h in advance, then keep only the
log(8h/) largest classes.
Look only at min and max of each deleted
summary, and merge into the largest class,
introducing an error of at most /2h.
Maximum transmission load is
O((log(|S|)log(h/))/)
Details in paper….
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Computing Exact Order-Statistics
• Initially (can choose any ):
lbound = -, hbound = +, offset = 0,
rank = |S|, p = 1;
• Each round:
– query for
1. Q = an -approximate summary of [lbound,hbound]
2. offset = exact rank of lbound.
– r = rank-offset;
lbound = Q(r-p|S|/4), hbound = Q(r+p|S|/4); p = p+1;
• After at most p = log1/(|S|) passes, Q is exact,
and we can return the element with rank rank.
Theorem: There is a distributed algorithm to compute
exact order statistics with max transmission load of
O(log3(|S|)) values at each node.
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Practical Concerns
Median
• Amenable to general
optimization
techniques
• Exact: more precise e
increases msg size, but
reduces # of passes
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•Simulator of Madden &
Stanek
•Worst-case assumptions
for our algorithm
Summary
MBG 18
Related Work
• Power mgmt: e.g. [Madden et al, SIGMOD ‘03],
QUERY LIFETIME, sample rate adjusted to meet
lifetime requirement.
• Multi-resolution histograms: [Hellerstein et al,
IPSN ‘03], wavelet histograms, no guarantees,
multiple passes for finer resolution.
• Streaming data: [GK, SIGMOD ‘01], [MRL:
Manku et al, SIGMOD 98] avg-case cost, [CM:
Cormode, Muthukrishnan, LATIN ‘04] avgcase cost, probabilistic, space blowup.
• Multi-pass algorithms for exact orderstatistics, based on [Munro & Paterson,
Theoretical Computer Science, 80]
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Conclusions
• Substantial improvement in worst-case pernode transmission cost over existing
algorithms.
• Topology independent
• In simulation, nodes consume less power
than our worst-case analysis
• Some observations on streaming data vs.
sensor networks:….
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Sensor networks vs. Streaming
data
• If sensor values change slowly => multiple passes.
• Uniform sampling of sensor networks is hard;
unknown density, loss can eliminate entire subtree.
• Sensor nets are inherently parallel; no single processor
sees all values.
To compare 2 disjoint subtrees either (a) cost in communication
and memory, or (b) aggregate values in one subtree before
seeing other --- discarding info.
• Sensor net model is strictly harder than streaming:
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Input at any node is summary
of prior data plus new datum MBG 21
Questions?
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MBG 22
Power Consumption of Nodes
in Sensor Net
• 4000-4500 nJ to send a single bit.
• No noticeable startup penalty.
• A 5mW processor running at 4MHz uses 4-5nJ
per instruction.
• Transmitting a single-bit costs as much as
800-1000 instructions.
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Optimizations
• Conservative Pruning: Discard superfluous
i/B quantiles, if rmax(i+1)-rmin(i-1) < 2|S|
• Early Combining: COMBINE and PRUNE
different classes, as long as sum of |Si|
increases class of largest input summary
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