# Computing and Communicating Functions over Sensor Networks

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```							Computing and Communicating
Functions over Sensor Networks
A. Giridhar and P. R. Kumar
Presented by
Srikanth Hariharan
Overview
•   Introduction
•   System Model
•   Divisible Functions
•   Symmetric Functions
– Type-Sensitive
– Type-Threshold
• Conclusion
Introduction
• Wireless Sensor Networks: Sensors sense the
environment and communicate a relevant summary of
the data to a sink.
• Summary of data is, in general, a function of the raw
sensor measurements.
• Examples – Mean, Max, Min, Sum etc.
• Question: How should nodes cooperate to efficiently
compute the desired function at the sink?
• Interested in computing the maximum rate at which the
function can be communicated and its dependency on
the number of sensors.
Model
• n sensor nodes
• ij – Distance between nodes i and j.
• Protocol model () for wireless communication.
• Successful communication between two nodes takes
place at a rate W bits/second.
• Sensor measurements belong to a fixed finite set, .
• fn : n n – Function of interest.
• Block Coding is permitted – each sensor has an
associated block of N readings known a priori.
Notations
• R(f,n) – Range of fn.
• SN,n – Strategy to pass messages between sensors
which results in computation of fn at the sink.
• T(SN,n) – Worst case time taken by scheme SN,n.
• R(SN,n) = N/T(SN,n) – Rate of scheme SN,n.
• R(n)max – Supremum over all schemes and block-
lengths.
• G(n) – Graph associated with the wireless network.
• d(G(n)) – Maximum of the degrees of nodes.
Network Topologies
• Collocated Networks:
– Networks whose graph is complete.
– Every transmission is heard by all nodes.
• Random Planar Networks:
– Nodes and sink are uniformly and independently
distributed on a unit square.
– Common range, r(n), is chosen such that, by using
multihop communication, the graph is connected.
Review
Lemma: For random planar networks, if range
r(n) =  (2 log n / n), then G(n) is connected
w.h.p. and d(G(n)) ≤ c log n w.h.p.

Result follows from earlier paper on critical
power for asymptotic connectivity.
Divisible Functions
• Functions that can be computed in a divide-and-
conquer fashion.
• Trivial upper bound on rate:
• A function f: n  (n) is divisible if:
– |R(f,n)| is non-decreasing in n;
– Given any partition (S) = {S1,S2,…,Sj} of S  {1,2,…,n},
there exists a function g(S), such that for any x  n,
f(xS) = g(S)(f(xS1),f(xS2),…,f(xSj)).
• Example:
– max(1,2,3,4,5) = max(max(1,2),max(3,4),max(5)).
Divisible Functions
• Theorem: Let f be a divisible function. Suppose G(n) is
connected, and d(G(n)) ≤ k1 log |R(f,n)|, for some k1 >
0. Then,

• Implication: On the above class of graphs, the simple
upper bound is actually achievable for divisible
functions.
Proof Outline
• Tessellation of plane into square
cells of side r/2.
• Cell graph: Defined on the set of
non-empty cells as vertices.
– Two cells are adjacent if there are
two nodes within each cell which
• Construct a rooted spanning tree of the cell graph with
the cell containing the sink node as the root.
• Let max be the maximum depth of the tree and (c)
denote the depth of cell c.
Proof Outline
• In each cell, designate as the relay node, a node u, which
is adjacent to a node v in the parent of c, and designate v
as the relay parent of u.
• Each cell has one relay node (picked out of possibly
multiple choices) and possibly multiple relay parents.
• For each node u, define the descendant set Du as:
– If u is a relay node of a cell c, Du is the set of all nodes that
either belong to c or to descendants of c.
– Else if u is the relay parent of u1,…,ul, Du := {u} U Du1 U … U Dul.
– Else Du := u.
Proof Outline
• Between times jT1 and (j+1)T1, the following
transmissions take place for each cell c.
• Let m := j – 2(max - (c)).
• If m ≤ 0 or m > N, cell c does not transmit.
• Otherwise,
– Each non-relay node v in c transmits to the relay node of c.
– If m > 1 and c does not contain the sink, the relay node of c
transmits to its relay parent.
• It can be shown that if
the transmissions can be feasibly scheduled and the sink
will obtain the desired function.
Applications
– Consider the identity function, corresponding to the sink
– |R(f,n)| = ||n. Degree condition in Theorem is satisfied for any
connected graph.
– f can be communicated at a rate O(1/n).
• Frequency Histogram of Sensor Measurements:
– Define the type-vector,(x) := [1(x),2(x),…, ||(x)], where i(x) is
the number of occurrences of i in x.
–  can be computed at a maximum rate of (1/log n), if d(G(n)) =
O(log n).
– From earlier Lemma, this means that in a random planar
network which is connected w.h.p., the maximum rate at which
 can be computed is (1/log n) w.h.p.
Symmetric Functions
• Functions which are invariant with respect to
permutations of their arguments:
• Data generated by a sensor is of primary importance.
Sensor identity is not important.
• A symmetric function f(x) depends on x only through its
type-vector(x).
• Let f’() denote the value of f(x), for any x with(x) = .
• Because of this dependence on the type-vector, maximum
rate for any symmetric function is (1/log n) w.h.p., in
random planar networks.
Classes of Symmetric Functions
• Two disjoint classes:
– Type-Sensitive
– Type-Threshold
• Type-Sensitive Functions:
– A symmetric function is type-sensitive if there exists some  with
0<  <1, and an integer k, and any j ≤ n – [n], given any subset
{x1,x2,…,xj}, there are two subsets of values {yj+1,yj+2,…,yn} and
{zj+1,zj+2,…,zn}, such that

– Examples: Mode, Mean, Median, Standard deviation.
– A type-sensitive function cannot be determined if a large
enough fraction of the arguments are unknown.
Classes of Symmetric Functions
• Type-Threshold Functions:
– A symmetric function f is said to be type-threshold if there exists
a non-negative ||-vector, , called the threshold vector, such
that f(x) = f’((x)) = f’(min((x), )), for all x  n.
– The value of a type-threshold function can be determined by a
fixed number of known arguments.
– Examples:
•   Max, Min, Range – Threshold vector, [1,1,…,1].
•   kth largest value – Threshold vector, [1,1,…,1].
•   Mean of k largest values – Threshold vector, [k,k,…,k].
•   Indicator function I{xi = k, for some i} – Threshold vector, [0,0,…,1,0,0,…,0].
– There exist symmetric functions that are neither type-sensitive
nor type-threshold.
Collocated Networks - Results
• Theorem: The maximum rate for computing a
type-sensitive function in a collocated network,
using any CFS (Collision-Free Strategy) is (1/n).
• Theorem: The maximum rate for computing a
non-constant type-threshold function in a
collocated network, using any CFS is (1/log n).
• Thus, type-sensitive functions are maximally
difficult to compute, in a collocated network.
• Because of block coding, type-threshold functions
can be computed at an exponentially faster rate
than type-sensitive functions.
Random Planar Networks - Results
• Consider a random planar network, with common range
r(n), chosen to be large enough such that the network is
connected. Let f and g be type-sensitive and type-
threshold functions, respectively.
Conclusion

• Future work includes considering other types of
functions, obtaining lower bounds, introducing
correlations among sensor readings, introducing an
information theoretic approach and considering power
control.

```
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