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 Anil Vullikanti (Virginia Tech) 1 / 12

Link Scheduling Link Scheduling Problem : choose largest subset I ∈ I v 2 u 1 Max-weight Link Scheduling u 3 v 3 u 2 e 2 Problem : choose I ∈ I s.t. e 1 e 3 wt ( I ) = � e ∈ I wt ( e ) is maximized v 1 v 4 — subroutine for maximizing u 4 e 4 throughput capacity ab Scheduling Complexity c of a set I : a conflict-free link set - all E ′ of links ( sc ( E ′ )): smallest k links in I can be scheduled such that E ′ = I 1 ∪ . . . ∪ I k , where simultaneously each I j is a conflict-free link set I : set of all possible conflict-free link sets 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 Physical model: based on SINR Disk based interference constraints v 2 u 1 v 2 u 3 u 1 u 3 v 3 u 2 e 2 v 3 u 2 e 1 e 2 e 1 e 3 e 3 v 1 v 4 v 1 u 4 v 4 e 4 u 4 e 4 Links e i can transmit simultaneously using power level Transmission radius for u i : r ( u i ) = c · ( J ( e i )) 1 /α J ( e i ) if Edges e 1 and e 2 interfere if J ( e i ) they are within interference d ( u i , v i ) α ∀ i , ≥ β J ( e j ) range in the resulting graph. N + � j � = i d ( u j , v i ) α Anil Vullikanti (Virginia Tech) 3 / 12

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 Anil Vullikanti (Virginia Tech) 4 / 12

Advantages and Disadvantages of Disk based interference models v 1 u 1 u 2 v 2 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 Anil Vullikanti (Virginia Tech) 5 / 12

Complexity of Link Scheduling Physical interference model Disk based models NP-complete NP-complete O (log ∆ log n ) approximation Uniform power levels: greedy to length of schedule in gives O (1) approximation general, where ∆ = max e ℓ ( e ) Non-uniform power levels: min e ′ ℓ ( e ′ ) O (log ∆) approximation for Inductive Scheduling for O (1) fixed uniform/linear power approximation levels Polynomial time approximation Scheduling complexity of schemes connectivity for any set of Distributed algorithms in radio nodes: O (log 2 n ) broadcast model with O (log n ) time Anil Vullikanti (Virginia Tech) 6 / 12

Greedy heuristics for scheduling in Physical interference model Generic Greedy Heuristic : While edges in current set E ′ are not conflict-free Remove edges e k satisfying CON from E ′ � � d ( u i , v i ) α Let Z = d ( u i , v j ) α SRA 1 : max { � j Z kj , � j Z jk } is minimized SMIRA 2 : max { � j � = k J ( e j ) Z kj , � j � = k J ( e k ) Z jk } WCRP 3 ; LISRA 4 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 Same power levels in both models Uniform power level for each Instances Γ with link: There is an instance Γ for sc disk (Γ) = Ω( n ) for any choice which sc Phy (Γ) sc disk (Γ) = O (1 / n ) of power levels Linear power level for each link Instances Γ where ( J ( e ) = c · ℓ ( e ) α ): There is an uniform/linear power levels instance Γ for which ⇒ sc Phy (Γ) = Θ( n ) sc Phy (Γ) sc disk (Γ) = Ω( n ) For any instance Γ, sc Phy (Γ) = O (log 2 n ) in Physical model, using non-linear power levels Anil Vullikanti (Virginia Tech) 8 / 12

Graph measures to characterize performance in Physical model Any set Γ of links can be scheduled in O ( χ ρ log n ) time, where χ ρ is the ρ -disturbance 5 ρ -disturbance of a link e i = ( u i , v i ), χ ρ ( e i ): # senders “close” to u i ρ -disturbance of Γ: max e i χ ρ ( e i ) 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 ≥ max e { 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 ( I in (Γ) log n ) in Physical model 8 Topology control algorithms to construct set of links with low I in Directed links: there exists connected set Γ with I in (Γ) = O (log n ) for any set V of nodes Symmetric links: instances where I in = 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 Anil Vullikanti (Virginia Tech) 11 / 12

Open problems Improving bounds for Physical model: graph based models with non-uniform power levels Distributed algorithms for scheduling Anil Vullikanti (Virginia Tech) 12 / 12