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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 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.

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posted: | 2/21/2013 |

language: | English |

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