Energy-Latency Tradeoff for In-Network Function Computation in Random Networks

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1 Energy-Latency Tradeoff for In-Network Function Computation in Random Networks P. Balister 1 B. Bollobás 1 A. Anandkumar 2 A.S. Willsky 3 1 Dept. of Math., Univ. of Memphis, Memphis, TN, USA 2 Dept. of EECS, University of California, Irvine, CA, USA. 3 Dept. of EECS, Massachusetts Institute of Technology, Cambridge, MA, USA. Presented by Dr. Ting He. IEEE INFOCOM 2011 Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 1 / 21 Dep

2 In-network Function Computation Internet PSTN Traditional Wire-line Networks Over-provisioned links Layered architecture Data forwarding: no processing at intermediate nodes Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 2 / 21 Dep

In-network Function Computation Internet PSTN Traditional Wire-line Networks Over-provisioned links Layered architecture Data forwarding: no processing at intermediate nodes Decision Node (sink)

3 In-network Function Computation Internet PSTN Traditional Wire-line Networks Over-provisioned links Layered architecture Data forwarding: no processing at intermediate nodes Decision Node (sink) Energy-Constrained Sensor Networks Multihop wireless communication Transmission energy costs Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 2 / 21 Dep

4 In-network Function Computation Internet PSTN Traditional Wire-line Networks Over-provisioned links Layered architecture Data forwarding: no processing at intermediate nodes Decision Node (sink) Energy-Constrained Sensor Networks Multihop wireless communication Transmission energy costs In-network computation for energy savings Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 2 / 21 Dep

5 Energy-Latency Tradeoff for In-network Computation Transmission Energy Costs for Wireless Communication Cost for direct transmission between i and j scales as R ν (i,j), where 2 ν 6 and ν is known as path-loss exponent. Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 3 / 21 Dep

6 Energy-Latency Tradeoff for In-network Computation Transmission Energy Costs for Wireless Communication Cost for direct transmission between i and j scales as R ν (i,j), where 2 ν 6 and ν is known as path-loss exponent. Achieving Energy Efficiency Multi-hop routing instead of direct transmission In-network computation to reduce amount of data transmitted Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 3 / 21 Dep

7 Energy-Latency Tradeoff for In-network Computation Transmission Energy Costs for Wireless Communication Cost for direct transmission between i and j scales as R ν (i,j), where 2 ν 6 and ν is known as path-loss exponent. Achieving Energy Efficiency Multi-hop routing instead of direct transmission In-network computation to reduce amount of data transmitted Latency of Data Reception Number of hops required for data transmission Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 3 / 21 Dep

8 Energy-Latency Tradeoff for In-network Computation Transmission Energy Costs for Wireless Communication Cost for direct transmission between i and j scales as R ν (i,j), where 2 ν 6 and ν is known as path-loss exponent. Achieving Energy Efficiency Multi-hop routing instead of direct transmission In-network computation to reduce amount of data transmitted Latency of Data Reception Number of hops required for data transmission Energy-Latency Tradeoff Direct transmission: Higher cost but lower latency Multihop routing: Lower cost but higher latency Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 3 / 21 Dep

9 Problem Formulation Goal Design policy π to communicate certain function of data at nodes to the fusion center Energy Consumption of a Policy π Total energy costs R ν (i,j) (i,j) G π n Latency of Function Computation Delay for function value to reach fusion center Optimal Energy-Latency Tradeoff Minimize energy consumption subject to latency constraint Can we design policies which achieve optimal energy-latency tradeoff? Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 4 / 21 Dep

10 Summary of Results Stochastic Node Configuration n nodes placed uniformly at random in R d over area n Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 5 / 21 Dep

11 Summary of Results Stochastic Node Configuration n nodes placed uniformly at random in R d over area n Sum Function Computation Deliver sum of data at nodes to fusion center Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 5 / 21 Dep

12 Summary of Results Stochastic Node Configuration n nodes placed uniformly at random in R d over area n Sum Function Computation Deliver sum of data at nodes to fusion center Energy-Latency Tradeoff for Sum Function Computation Propose novel policies which meet latency constraint Prove order-optimal energy-latency tradeoff Characterize scaling behavior with respect to path-loss exponent ν Order-optimal Energy-Latency Tradeoff Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 5 / 21 Dep

13 Summary of Results Contd., Stochastic Node Configuration n nodes placed uniformly at random in R d over [0,n 1/d ] d Clique-Based Function Computation Function which decomposes over cliques of a graph Relevant for statistical inference of graphical models (correlated sensor data) Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 6 / 21 Dep

14 Summary of Results Contd., Stochastic Node Configuration n nodes placed uniformly at random in R d over [0,n 1/d ] d Clique-Based Function Computation Function which decomposes over cliques of a graph Relevant for statistical inference of graphical models (correlated sensor data) Energy-Latency Tradeoff for Clique Function Computation Extend previous policy for this class of functions Prove order optimality under following conditions: 1 Latency constraints belong to a certain range 2 The graph governing the function is a proximity graph, e.g. k-nearest neighbor graph, random geometric graph Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 6 / 21 Dep

15 Related Work Capacity of In-network Function Computation Rate of computation (Giridhar & Kumar 06) Single-shot computation considered here Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 7 / 21 Dep

16 Related Work Capacity of In-network Function Computation Rate of computation (Giridhar & Kumar 06) Single-shot computation considered here Minimum Broadcast Problem Minimize time of broadcast to all nodes from a single source (Ravi 94) Equivalent to latency of sum function computation Energy-latency tradeoff not considered before Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 7 / 21 Dep

17 Related Work Capacity of In-network Function Computation Rate of computation (Giridhar & Kumar 06) Single-shot computation considered here Minimum Broadcast Problem Minimize time of broadcast to all nodes from a single source (Ravi 94) Equivalent to latency of sum function computation Energy-latency tradeoff not considered before Energy Optimization for Clique Function Computation Steiner-tree reduction (Anandkumar et. al. 08, 09) Order-optimality for random networks (Anandkumar et. al. 09) Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 7 / 21 Dep

18 Related Work Capacity of In-network Function Computation Rate of computation (Giridhar & Kumar 06) Single-shot computation considered here Minimum Broadcast Problem Minimize time of broadcast to all nodes from a single source (Ravi 94) Equivalent to latency of sum function computation Energy-latency tradeoff not considered before Energy Optimization for Clique Function Computation Steiner-tree reduction (Anandkumar et. al. 08, 09) Order-optimality for random networks (Anandkumar et. al. 09) Novelty: Energy-Latency Tradeoff for Function Computation Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 7 / 21 Dep

19 Outline 1 Introduction 2 Detailed Model and Formulation 3 Sum Function Computation 4 Conclusion Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 8 / 21 Dep

20 Detailed System Model Communication Model Half-duplex nodes: no simultaneous transmission and reception Dedicated reception: Cannot receive data from multiple nodes No other interference constraints: orthogonal channels/directional antenna Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 9 / 21 Dep

21 Detailed System Model Communication Model Half-duplex nodes: no simultaneous transmission and reception Dedicated reception: Cannot receive data from multiple nodes No other interference constraints: orthogonal channels/directional antenna Propagation Model Unit transmission delay at all links Stochastic Node Configuration V n n nodes placed uniformly at random in R d over [0,n 1/d ] d Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University ofieee California, INFOCOM Irvine, 11CA, USA., 9 / 21 Dep

22 Energy-Latency Tradeoff Energy Consumption of a Policy π E π (V n ) := R ν (i,j) (i,j) G π n Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 10 / 21 Dep

23 Energy-Latency Tradeoff Energy Consumption of a Policy π E π (V n ) := R ν (i,j) (i,j) G π n Latency of Function Computation L π (V n ) Delay for function value to reach fusion center Minimum Latency L (V n ) := min π L π (V n ) Optimal Energy-Latency Tradeoff E (V n ;δ) := min π E π (V n ), s.t. L π L +δ. Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 10 / 21 Dep

24 Outline 1 Introduction 2 Detailed Model and Formulation 3 Sum Function Computation 4 Conclusion Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 11 / 21 Dep

25 Preliminaries for Sum Function Computation Computation Along a Tree T Links directed towards fusion center (root) Each node waits to receive data from children It then computes sum of values (along with own data) and forwards along outgoing link Process stops when data reaches fusion center Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 12 / 21 Dep

26 Preliminaries for Sum Function Computation Computation Along a Tree T Links directed towards fusion center (root) Each node waits to receive data from children It then computes sum of values (along with own data) and forwards along outgoing link Process stops when data reaches fusion center Latency Along a Tree... Root r 1 2 k T 1 T 2 T k Latency L T along tree T is L T = max i=1,...,k {i+l T i } T i : subtree rooted at node i 1,...,k : are of root such that L T1 L T2... L Tk Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 12 / 21 Dep

27 Minimum Latency Result Minimum Latency Tree Minimum latency for sum function computation over n nodes is L (n) = log 2 n. max. # of nodes in tree with latency L is 2 L. Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 13 / 21 Dep

28 Minimum Latency Result Minimum Latency Tree Minimum latency for sum function computation over n nodes is L (n) = log 2 n. max. # of nodes in tree with latency L is 2 L. Construction Minimum Latency Tree T Recursively add child to each node already in tree Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 13 / 21 Dep

29 Minimum Latency Result Minimum Latency Tree Minimum latency for sum function computation over n nodes is L (n) = log 2 n. max. # of nodes in tree with latency L is 2 L. Construction Minimum Latency Tree T Recursively add child to each node already in tree Level l(e;t) of link e in tree T l(e;t) = L T t e. t e : time of transmission at link e Process starts at time 0. Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 13 / 21 Dep

30 Example of Minimum Latency Tree T Shown with edge-level labels Root r 1 Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 14 / 21 Dep

31 Example of Minimum Latency Tree T Shown with edge-level labels Root r Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 14 / 21 Dep

32 Example of Minimum Latency Tree T Shown with edge-level labels Root r Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 14 / 21 Dep

33 Example of Minimum Latency Tree T Shown with edge-level labels Root r Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 14 / 21 Dep

34 General Policy for Energy-Latency Tradeoff Observations Minimum Latency L independent of node locations V n Energy consumption depends on node locations V n Construct aggregation tree T depending on V n Overview of Algorithm π AGG Iteratively bisect region under consideration Choose child in the other half Connect to the child along least energy route with at most w k intermediate nodes Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 15 / 21 Dep

35 Example for π AGG policy Root r 1 Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 16 / 21 Dep

36 Example for π AGG policy Root r Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 16 / 21 Dep

37 Example for π AGG policy Root r Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 16 / 21 Dep

38 Example for π AGG policy Root r Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 16 / 21 Dep

39 Analysis of π AGG policy Latency under π AGG policy log 2 n 1 L π = L (n)+ k=0 w k Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 17 / 21 Dep

40 Analysis of π AGG policy Latency under π AGG policy log 2 n 1 L π = L (n)+ k=0 Optimal Energy-Latency Tradeoff Problem Minimize energy subject to latency constraint w k E (V n ;δ) := min π E π (V n ), s.t. L π L +δ. Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 17 / 21 Dep

41 Analysis of π AGG policy Latency under π AGG policy log 2 n 1 L π = L (n)+ k=0 Optimal Energy-Latency Tradeoff Problem Minimize energy subject to latency constraint w k E (V n ;δ) := min π E π (V n ), s.t. L π L +δ. Choice of weights for π AGG for optimal tradeoff For k = 0,..., log 2 n 1 w k = { ζδ2 k(1/ν 1/d) if ν > d, 0 o.w. Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 17 / 21 Dep

42 Main Result: Optimal Energy-Latency Tradeoff Optimal Energy-Latency Tradeoff Minimize energy subject to latency constraint E (V n ;δ) := min π E π (V n ), s.t. L π L +δ. Theorem For given δ, path-loss ν, dimension d, as number of nodes n, Θ(n) ν < d, E(E (V n ;δ))= O ( max{n,n(logn)(1+ δ logn )1 ν } ) ν = d, Θ ( max{n,n ν/d (1+δ) 1 ν } ) ν > d, Expectation is over node locations V n of n Achieved by the policy π AGG Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 18 / 21 Dep

43 Outline 1 Introduction 2 Detailed Model and Formulation 3 Sum Function Computation 4 Conclusion Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 19 / 21 Dep

44 Conclusion Summary of Results Considered energy-latency tradeoff for function computation Considered sum function and function over cliques Proposed novel aggregation policies Proved order-optimal energy-latency tradeoff Outlook Extensions beyond single-shot computation Multiple fusion centers with multiple functions for computation Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 20 / 21 Dep

45 Thank You! Balister et. al. (Dept. of Math., Univ. of Memphis, Memphis, Energy-Latency TN, USA, Tradeoff Dept. of EECS, University of IEEE California, INFOCOM Irvine, 11CA, USA., 21 / 21 Dep

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