Collision-free scheduling: Complexity of Interference Models

Transcription

1 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

2 Link Scheduling u 1 v 1 v 2 u 2 e u 3 v 3 e 2 1 e 3 u 4 v 4 e 4 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: choosei I s.t. wt(i )= e I wt(e) is maximized subroutine for maximizing throughput capacity ab Scheduling Complexity c of a set E of links (sc(e )): smallest k such that E = I 1... I k,where each I j 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

3 Interference Models Disk based interference u 1 v 2 e u 2 u3 v 3 1 e 2 e3 v 1 u 4 v 4 e 4 Transmission radius for u i : r(u i )=c (J(e i )) 1/α Edges e 1 and e 2 interfere if they are within interference range in the resulting graph. Physical model: based on SINR constraints u 1 v 1 v 2 u 2 e u 3 v 3 e 2 1 e 3 u 4 v 4 e 4 Links e i can transmit simultaneously using power level J(e i )if i, J(e i ) d(u i,v i ) α N + j i J(e j ) d(u j,v i ) α β Anil Vullikanti (Virginia Tech) 3/12

4 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

5 Advantages and Disadvantages of Disk based interference models u 1 v 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

6 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) min e l(e ) O(log ) approximation for fixed uniform/linear power levels Scheduling complexity of connectivity for any set of nodes: O(log 2 n) Anil Vullikanti (Virginia Tech) 6/12

7 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 Let Z = d(ui,v i ) α 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

8 Performance Limits: Physical vs Disk Based models Power levels allowed to differ Instances Γ with sc disk (Γ) = Ω(n) foranychoice of power levels Instances Γ where uniform/linear power levels sc Phy (Γ) = Θ(n) For any instance Γ, sc Phy (Γ) = O(log 2 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 sc Phy (Γ) sc disk (Γ) = O(1/n) Linear power level for each link (J(e) =c l(e) α ): There is an instance Γ for which sc Phy (Γ) sc disk (Γ) =Ω(n) Anil Vullikanti (Virginia Tech) 8/12

9 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 ei 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

10 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

11 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

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