1. Computing and Communicating Functions over Sensor Networks A.Giridhar and P. R. Kumar Presented by Srikanth Hariharan

2. Overview Introduction System Model Divisible Functions Symmetric Functions – Type-Sensitive – Type-Threshold Conclusion

3. 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.

4. 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, . f n :  n  n – Function of interest. Block Coding is permitted – each sensor has an associated block of N readings known a priori.

5. Notations R(f,n) – Range of f n. S N,n – Strategy to pass messages between sensors which results in computation of f n at the sink. T(S N,n ) – Worst case time taken by scheme S N,n. R(S N,n ) = N/T(S N,n ) – Rate of scheme S N,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.

6. 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.

7. 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.

8. 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) = {S 1,S 2,…,S j } of S  {1,2,…,n}, there exists a function g  (S), such that for any x   n, f(x S ) = g  (S) (f(x S 1 ),f(x S 2 ),…,f(x S j )). Example: – max(1,2,3,4,5) = max(max(1,2),max(3,4),max(5)).

9. Divisible Functions Theorem: Let f be a divisible function. Suppose G (n) is connected, and d(G (n) ) ≤ k 1 log |R(f,n)|, for some k 1 > 0. Then, Implication: On the above class of graphs, the simple upper bound is actually achievable for divisible functions.

10. 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.

11. 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 D u as: – If u is a relay node of a cell c, D u is the set of all nodes that either belong to c or to descendants of c. – Else if u is the relay parent of u 1,…,u l, D u := {u} U D u 1 U … U D u l. – Else D u := u.

12. Proof Outline Between times jT 1 and (j+1)T 1, 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.

13. 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.

14. 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.

15. 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 {x 1,x 2,…,x j }, there are two subsets of values {y j+1,y j+2,…,y n } and {z j+1,z j+2,…,z n }, 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.

16. 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]. k th largest value – Threshold vector, [1,1,…,1]. Mean of k largest values – Threshold vector, [k,k,…,k]. Indicator function I{x i = k, for some i} – Threshold vector, [0,0,…,1,0,0,…,0]. – There exist symmetric functions that are neither type-sensitive nor type-threshold.

17. 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.

18. 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.

19. 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.