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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

ZizhanZheng and Ness B. Shroff

Presenter: WenzhuoOuyang

Department of Electrical and Computer Engineering

The Ohio State University

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- Motivation
- System Model and Problem Formulation
- Approximation Algorithms
- Simulations
- Conclusion and Future Work

- 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

- Collecting data from all the nodes is often infeasible
- Data Collection under a Deadline Constraint

- 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)

- 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

- 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).

- 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+

a2V \ T

- A discrete counterpart of concavity (‘diminishing return’)
- Includes additive utility as a special case

- 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

- 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 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.

- 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).

- 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: aÃ 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

- 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)

- 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.

- 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.

Thank you !

- 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.

- A separable utility)an additive utility

- 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).

- 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

- 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

- 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.

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

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).