Collision-free scheduling: Complexity of Interference Models Anil - - PowerPoint PPT Presentation

Collision-free scheduling: Complexity of Interference Models

Anil Vullikanti

Department of Computer Science, and Virginia Bioinformatics Institute, Virginia Tech

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Link Scheduling

u1 v1 u2 v2 u3 v3 u4 v4 e1 e2 e3 e4 I: a conflict-free link set - all links in I can be scheduled simultaneously I: set of all possible conflict-free link sets Link Scheduling Problem: choose largest subset I ∈ I Max-weight Link Scheduling Problem: choose I ∈ I s.t. wt(I) =

e∈I wt(e) is maximized

— subroutine for maximizing throughput capacityab Scheduling Complexityc of a set E ′ of links (sc(E ′)): smallest k such that E ′ = I1 ∪ . . . ∪ Ik, where each Ij is a conflict-free link set

a[Tassiulas and Ephremides, 1992] b[Georgiadis, Neely and Tassiulas, 2006] c[Mosciborda, Wattenhofer and Zollinger, MobiHoc 2006] Anil Vullikanti (Virginia Tech) 2 / 12

Interference Models

Disk based interference

u1 v1 u2 v2 u3 v3 u4 v4 e1 e2 e3 e4

Physical model: based on SINR constraints u1 v1 u2 v2 u3 v3 u4 v4 e1 e2 e3 e4 Transmission radius for ui: r(ui) = c · (J(ei))1/α Edges e1 and e2 interfere if they are within interference range in the resulting graph. Links ei can transmit simultaneously using power level J(ei) if ∀i,

J(ei ) d(ui ,vi)α

N +

j=i J(ej) d(uj ,vi)α

≥ β

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Collision-free scheduling: models matter

All interference models are NP-complete to solve optimally in general

• need to explore polynomial time approximations

Disk based models:

Greedy works well: O(1) approximation Efficient distributed algorithms with low overhead

Physical model:

Natural Greedy schemes do not work well Constant factor approximations not known (yet) in general

Performance estimates depend crucially on interference model, and whether or not power levels are fixed to be the same in both models Performance in Physical model can be related to static graph measures in some cases

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Advantages and Disadvantages of Disk based interference models

u1 v1 u2 v2

Underestimate: close-by links cannot simultaneously transmit Overestimate: far-away links cannot influence a specific link transmission Local model: Simple distributed scheduling algorithms based on local degree

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Complexity of Link Scheduling

Disk based models NP-complete Uniform power levels: greedy gives O(1) approximation Non-uniform power levels: Inductive Scheduling for O(1) approximation Polynomial time approximation schemes Distributed algorithms in radio broadcast model with O(log n) time Physical interference model NP-complete O(log ∆ log n) approximation to length of schedule in general, where ∆ = maxe ℓ(e)

mine′ ℓ(e′)

O(log ∆) approximation for fixed uniform/linear power levels Scheduling complexity of connectivity for any set of nodes: O(log2 n)

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Greedy heuristics for scheduling in Physical interference model

Generic Greedy Heuristic: While edges in current set E ′ are not conflict-free Remove edges ek satisfying CON from E ′ Let Z =

• d(ui ,vi)α

d(ui ,vj)α

• SRA1: max{

j Zkj, j Zjk} is minimized

SMIRA2: max{

j=k J(ej)Zkj, j=k J(ek)Zjk}

WCRP3; LISRA4 Instances where all these heuristics have performance Ω(n) relative to OPT

1[Zander, 1992] 2[Lee et al., 1995] 3[Wang et al., 2005] 4[Zander, 1992] Anil Vullikanti (Virginia Tech) 7 / 12

Performance Limits: Physical vs Disk Based models

Power levels allowed to differ Instances Γ with scdisk(Γ) = Ω(n) for any choice

• f power levels

Instances Γ where uniform/linear power levels ⇒ scPhy(Γ) = Θ(n) For any instance Γ, scPhy(Γ) = O(log2 n) in Physical model, using non-linear power levels Same power levels in both models Uniform power level for each link: There is an instance Γ for which scPhy (Γ)

scdisk(Γ) = O(1/n)

Linear power level for each link (J(e) = c · ℓ(e)α): There is an instance Γ for which

scPhy(Γ) scdisk(Γ) = Ω(n)

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Graph measures to characterize performance in Physical model

Any set Γ of links can be scheduled in O(χρ log n) time, where χρ is the ρ-disturbance5

ρ-disturbance of a link ei = (ui, vi), χρ(ei): # senders “close” to ui ρ-disturbance of Γ: maxei χρ(ei) Can be much larger than OPT

Different congestion measure based on Inductive Scheduling 6 7

C(e) = {e′ = (u′, v ′) ∈ Γ : ℓ(e′) ≥ ℓ(e), ℓ(e′) ≥ c · d(u, u′)} OPT ≥ maxe{C(e)}/ log n Set Γ can be scheduled in O(OPT log ∆ log n) time Scheduled in a distributed manner in polylogarithmic rounds

5[Moscibroda, Oswald and Wattenhofer, 2007] 6[Chafekar, Anil Kumar, Marathe, Parthasarathy, Srinivasan, MobiHoc 2007] 7[Chafekar, Anil Kumar, Marathe, Parthasarathy, Srinivasan, INFOCOM 2008] Anil Vullikanti (Virginia Tech) 9 / 12

Graph measures to characterize performance in Physical model

Scheduling complexity of any set Γ of links is O(Iin(Γ) log n) in Physical model8

Topology control algorithms to construct set of links with low Iin Directed links: there exists connected set Γ with Iin(Γ) = O(log n) for any set V of nodes Symmetric links: instances where Iin = Ω(√n).

8[Moscibroda, Wattenhofer and Zollinger, MobiHoc 2006] Anil Vullikanti (Virginia Tech) 10 / 12

Collision-free scheduling: summary

Approximate solutions necessary Computationally, Disk based models much simpler than Physical Performance estimates in disk model can be significantly different from Physical model; relative performance inconsistent Performance in Physical model can be related to static graph measures in some cases

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Open problems

Improving bounds for Physical model: graph based models with non-uniform power levels Distributed algorithms for scheduling

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