Computing and Communicating Functions over Sensor Networks

							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
     are adjacent in G(n).
• 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
• Data Downloading Problem:
   – Consider the identity function, corresponding to the sink
     downloading all the raw measurements of all the sensors.
   – |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.