Maximizing a submodular utility for deadline constrained data collection in sensor networks
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Maximizing a Submodular Utility for Deadline Constrained Data Collection in Sensor Networks. Zizhan Zheng and Ness B. Shroff Presenter: Wenzhuo Ouyang Department of Electrical and Computer Engineering The Ohio State University. TexPoint fonts used in EMF.

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Maximizing a Submodular Utility for Deadline Constrained Data Collection in Sensor Networks

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Maximizing a submodular utility for deadline constrained data collection in sensor networks

Maximizing a Submodular Utility for Deadline Constrained Data Collection in Sensor Networks

ZizhanZheng and Ness B. Shroff

Presenter: WenzhuoOuyang

Department of Electrical and Computer Engineering

The Ohio State University

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.: AAAAAAAAA


Outline

Outline

  • Motivation

  • System Model and Problem Formulation

  • Approximation Algorithms

  • Simulations

  • Conclusion and Future Work


Motivation

Motivation

  • Data collection in a sensor network

    • Each node holds some sensing data for an event

    • A sink collects data through a routing tree

  • Utilitymaximization

    • Collecting data from all the nodes is often infeasible

      • delay: large network, real-time data request

      • energy, …

    • Trading off data quality and communication cost

      • Redundancy in the data: spatial, temporal

  • Data Collection under a Deadline Constraint


System model

System Model

  • A routing tree T = (V, E) rooted at the sink s0

  • Each node has at most one data packet ready to deliver

  • Time slotted system, 1-hop interference model

    • One packet can be forwarded in each time slot

    • Links are reliable

  • Two data collection schemes

    • Raw data forwarding: complicated post-processing of data needed

    • In-network data aggregation: MAX, MIN, SUM, etc.

      • aggregation is perfect

  • A utility function defined on subsets of nodes

    • f: 2V! R+ , where f(S) gives the utility of nodes in a subset S µ V

    • e.g., f(S) = i2Swifor some wi2 R+ (additive utility)


Problem formulation

Problem Formulation

  • Deadline constrained utility maximization

    max f(S) s.t.SµV, LF(S) ·D (resp. LA(S) ·D) (1)

    • D– deadline constraint (# time slots allowed)

    • LF(S) (resp. LA(S))– minimum number of time slots needed for forwarding (resp.aggregating)all the packets in SµV to the sink

    • Examples

LF(S) = 5

LA(S) = 3

LF(S) >> LA(S) in a larger setting

Data forwarding

Data aggregation


Beyond additivity monotone submodular utility

Beyond Additivity: Monotone SubmodularUtility

  • For additive utility and for data aggregation only, an efficient polynomial time solution to Problem (1) is known using dynamic programming (Hariharan and Shroff’09).

  • An additive utility largely ignores the spatial correlation of sensor nodes

    • Our contribution: Efficient algorithms for maximizing a more general form of utility that captures a large class of spatial correlation for both data forwarding and data aggregation.

    • Assumptions about utility function f: 2V! R+

  • Normalized: f(;) = 0

  • Monotone: f(S) ·f(T) 8SµTµV

  • Submodular: f(S[{a}) – f(S) ¸f(T[ {a}) –f(T) 8S µ T µ V and

    a2V \ T

    • A discrete counterpart of concavity (‘diminishing return’)

    • Includes additive utility as a special case


  • S ubmodular utility for sensor selection

    Submodular Utility for Sensor Selection

    • Area and point coverage in a disk sensing model

    • Mutual information (Krause et al.’07)

      • Given random variables defined on nodes, X1,…,X|V|

      • f(S) = I(XS; XV nS) = H(XVnS) – H(XV nS | XS)

    • Variance reduction for modeling sensing uncertainty (under a mild condition) (Das and Kempe’08, Krause et al.’08)

    • Maximum a posteriori (MAP) estimate and a variant of maximum likelihood (ML) estimate for parameter estimation (Shamaiah et al.’10)

    f(S) = # points covered by nodes in S

    Ex: S = {b}, T = {b, c},

    f(S[ {a}) – f(S) = 2,

    f(T[ {a}) – f(T) = 1.

    c

    b

    a


    Challenges

    Challenges

    • Most of previous works on sensor selection focus on maximizing a submodular function subject to a cardinality constraint:

      max f(S) s.t.SµV, |S| ·D

    • Problem (1) reduces to this special case for a tree of height 1.

      • There is no (1-1/e+²)-approximation for any ² > 0 unless P = NP for a general monotone submodular function (Feige’98).

    • Problem (1) in its general form is more challenging due to the multi-hop data forwarding nature and 1-hop interference.

  • For additive utility, an efficient solution to Problem (1) is known for data aggregation (Hariharan and Shroff’09), but remains open for data forwarding.


  • Main results

    Main Results

    • For data forwarding, a simple greedy algorithm achieves a factor 1/2-approximation when the sink has a single child, and 1/3-approximation in general. For additive utility, the greedy algorithm is optimal in the first case, and has a factor 1/2 in general.

  • For data aggregation, a bi-criteria approximation can be achieved, which finds a solution with a utility at least a fraction of the optimal utility and a delay at most ½TD.

    • D – deadline constraint

    • hT – height of the tree

    • ½T – a parameter determined by tree structure and bounded by the maximum node degree. We expect it is typically small (< 2).


  • Deadline constrained data forwarding

    Deadline Constrained Data Forwarding

    • Problem: max f(S) s.t.SµV, LF(S) ·D

    • A simple greedy algorithm (Algorithm 1)

    1: SÃ;.

    2: while true do

    3: AÃ {a: a2V \ S and LF(S[ {a}) ·D }.

    4: ifA = ;then break.

    5: aà argmaxa2Af(S[ {a}) –f(S).

    6: SÃ S[ {a}.

    7: returnS.

    • Need to know LF(S) for SµV

      • For a tree network subject to 1-hop interference, the minimum delay schedule can be determined (Florens et al.’04)

      • But, a key construction in the above approach needs to be fixed, which is critical both for ensuring the correctness of the algorithm and for its analysis

    Focus on membership oracle

    Assume a value oracle is given


    Analysis of the greedy algorithm

    Analysis of the Greedy Algorithm

    • Submodular maximization over p-systems

    • Our problem: max f(S) s.t.S2IF , whereIF꞉= {SµV : LF(S) ·D}

    • Proposition: (V, IF) is a 1-system when the sink has only one child, and a 2-system in general.

      • Approximation factors of the greedy algorithm follow directly from the proposition and the lemma

    • p-system –A pair (A, I) , Iµ 2A, such that

    (i) ;2I,

    (ii) 8S µA, if S2I and S’µS, then S’ 2I (sets in I are called independent sets),

    (iii) 8 SµA, the size of the maximum independent set in S is at most p times the size of any maximal independent set in S.

    – Ex: For the edge set E and the matchingsMin a graph, (E, M) is a 2-system.

    • Lemma: For a p-system (A, I) and f: 2A! R+, the problem of max f(S) s.t.S2Ican be approximated by the greedy algorithm within a factor of 1/(p+1) if fis monotone submodular and f(;) = 0, and a factor 1/p if f is additive. (Fisher et al.’78)


    Simulations

    Simulations

    • 1000£1000 2d area (5£5 grid), 1000 target points, 200 sensor nodes (sensing range 100, communication range 200).

    • f(S)– number of target points covered by nodes in S.

    • A randomly selected node as the sink, a routing tree built by breadth-first search.

    • Compared with a random node selection algorithm

      • Greedy algorithm performs 70% better

      • Bi-criteria algorithm performs up to 50% better, and 25% better in average.

    • Minimum delay / deadline < 2.5 and ¼ 1.5in average.


    Conclusion and future work

    Conclusion and Future Work

    • We have proposed efficient approximation algorithms for two data collection schemes over a tree network subject to 1-hop interference, for maximizing a submodular utility subject to a deadline constraint.

    • We plan to extend our work to more general settings, e.g.,

      • Unreliable wireless links

      • Imperfect aggregation

      • Other interference models

      • Other types of constraints, e.g., an energy constraint on each node

    • Joint optimization of tree construction and sensing set selection.


    Maximizing a submodular utility for deadline constrained data collection in sensor networks

    Thank you !


    Differences with network utility maximization

    Differences with Network Utility Maximization

    • Traditional NUM Model for a multi-hop wireless network

      • A set of users (flows), each with a source, a destination, a real data rate xs, and a utilityUs(xs),typically non-decreasing and strictly concave

      • Objective: max sUs(xs)s.t. the system is stable subject to some interference constraint

    • Major differences in our setting

      • No exogenous arrivals (packets ready at time 0)

      • A deadline constraint: bounding the delay for data collection

      • Binary decision variables: for each node, whether to deliver data or not

        • A separable utility)an additive utility

          • s Us(xs) = s wsxsfrom some ws2R+

          • largely ignores the spatial correlation of sensed data

        • A set function is more natural: f(S) gives the utility of nodes in set S.


    Challenges1

    Challenges

    • Most of previous works on sensor selection focus on maximizing a submodular function subject to a cardinality constraint:

      max f(S) s.t.SµV, |S| ·D

    • Problem (1) reduces to this special case for a tree of height 1.

    • There is no (1-1/e+²)-approximation for any ² > 0 unless P = NP for a general monotone submodular function (Feige’98).

  • For additive utility, an efficient solution to Problem (1) is known for data aggregation (Hariharan and Shroff’09), but remains open for data forwarding.

  • For a general monotone submodular utility and when the 1-hop interference model is replaced by the ‘clique’ model, Problem (1) for data aggregation is closely related to the group Steiner problem, and the latter is hard to approximate within a logarithmic factor (Halperin and Krauthgamer’03).


  • Minimum delay data forwarding

    Minimum Delay Data Forwarding

    • Main ideas (Florens et al.’04)

      • Consider data dissemination instead

      • Map a tree network to a multi-line network

    Our construction

    5 time slots

    3 time slots

    3 hops

    5 hops


    Deadline constrained data aggregation

    Deadline Constrained Data Aggregation

    • Problem: max f(S) s.t.SµV, LA(S) ·D

    • Observations

      • Without loss of optimality, a node should wait until it receives all the packets from its children, and then forward one aggregated packet.

      • LA(S) = LA(T(S)), where T(S) is the minimum subtree spanning S[ s0

    s0

    • The simple greedy algorithm is still applicable.

      • LA(S) for any SµV can be determined recursively

    • However, proving a performance bound for it eludes us.

      • Obscure structure of the feasible set under the 1-hop interference model


    Deadline constrained data aggregation cont

    Deadline Constrained Data Aggregation (cont.)

    • Main idea: Approximation by the ‘clique’ interference model.

      • At most one node can transmit at any time

      • The minimum delay for aggregating packets on SµVequals |T(S)| .

    • Consider the new Problem: max f(S) s.t.SµV, |T(S)| ·D’.

      • Identify a proper D’ to connect the two problems.

        Proposition:Let I1 ꞉= {SµV : |T (S)| ·D’}.Then (V, I1) is a hT–system, where hTis the height of the tree.

        Corollary: The greedy algorithm is a factor 1/(hT+1) approximation to the new problem.


    A bi criteria approximation algorithm 2

    A Bi-criteria Approximation (Algorithm 2)

    1: hÃmin (hT, D), and remove nodes in Tat level larger than h.

    2: D’Ã the maximum cardinality of any subtreeT1 µTwith root s0and minimum delay bounded by D(Hariharan’09).

    3: Find a maximum utility subtreeT2µTwith root s0and size bounded by D’ using the greedy algorithm.

    4: Expand T2greedily without further increasing the minimum delay.

    Proposition: The algorithm finds a subtree with a utility at least a fraction of the optimal utility and a minimum delay at most ½TD, with

    ·¢T(the maximum node degree).


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