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

1. 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

2. Outline • Motivation • System Model and Problem Formulation • Approximation Algorithms • Simulations • Conclusion and Future Work

3. 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

4. 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)

5. 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

6. 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

7. 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

8. 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.

9. 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).

10. 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: Aà {a: a2V \ S and LF(S[ {a}) ·D }. 4: ifA = ;then break. 5: aà argmaxa2Af(S[ {a}) –f(S). 6: Sà 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

11. 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)

12. 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.

13. 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.

14. Thank you !

15. 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.

16. 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).

17. 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

18. 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

19. 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.

20. 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).