Joint Routing, Channel Assignment, and Scheduling for Throughput Maximization

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Joint Routing, Channel Assignment, and Scheduling
for Throughput Maximization
Slides from Mahmoud Al-ayyoub, based on our
recent work.
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Outline

• Introduction Related Work
• JRCAS with Pairwise Interference
• JRCAS with Physical Interference
• TDMA Link Scheduling with Physical Interference.
• Single-Path Flows

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Throughput Maximization

• Maximizing Networks Throughput
• Given a set of source-destination pairs, whats
  the max supported data transfer rate?
• Challenges in wireless networks
• Wireless Interference.
• Multiple Channels.
• Limited Number of Radio Interfaces.

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Throughput Maximization

• Entails solving the joint problem of
• Routing (single-path or multi-path)
• Channel Assignment (to links)
• Scheduling (of links)
• Hence, the name Joint Routing, Channel
  Assignment and Scheduling (JRCAS) problem.
• We give several approximation algorithms for
  various settings of the JRCAS problem.

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JRCAS Problem Description

• Input
• Network Graph, Channels, Source-destination
  pairs, number of radio interfaces/nodes,
  interference model.
• Output Schedule of links (with assigned
  channels) into time slots s.t.
• Each time slot is interference-free
• Interface constraint is satisfied
• Resulting flow satisfies conservation laws
• Total flow is maximized

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Related Work

• Major limitations of prior work on JRCAS problem
• Static channel assignment
• Pairwise interference only
• Only multi-path routing

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Related Work Pairwise Model

• Jain et al. 03 gave exponential time/space
  algorithm.
• Kumar et al. 05 design a constant-approximation
  algorithm for single-channel
• Alicherry et al. 05 gave an O(1)-apx for static
  channel assignment (too involved)

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Approach Overview

• Formulate and solve LP that determines optimal
  link "data rates" such that interface and the
  interference constraints are satisfied.
• Scale down the LP-computed data rates by a
  certain factor.
• Use greedy placement to schedule the scaled-down
  data rates.

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LP Interference Constraint

• Let a(e,k) denote the link utilization (fraction
  of time link e is active on channel k)
• Interference Constraint
• C(e) is the set of links that interfere with e
• c is the maximum independent set (number of
  mutually non-interfering links) in any C(e).
• Why is the above a constraint?

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LP Interface Constraint

• Interface Constraint
• I(u) is the number of radios at node u
• Why is above a constraint?

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LP Formulation

• LPs constraints
• Flow conservation constraints
• Interface constraints
• Wireless-interference constraints
• Objective is to

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LP solution ? JRCAS solution

• Scale down LP-computed link utilizations by a (c
  2) factor.
• Let the resulting link utilizations be
• Pick integer W s.t. is also
  integer for each (e,k).
• Consider period S of W time slots.
• For each (e,k), place it in the first
  time slots of S, s.t. no interface or
  interference constraint is violated.

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Feasibility and Approximation

• For (e,k), max time slots wherein (e,k) can not
  be placed
• due to interference is
• due to interface constraint violationis

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Correctness and Performance Proof

• Since (from LP equations)
• Thus, the greedy placement is feasible, and is
  (c2) approximate (because LP solution is best
  possible).

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Generalizations

• Above techniques easily generalize for
• Directional antenna.
• Varying power transmissions.
• Incorporating other constraints.
• Other objective functions
• Must be linear combination of link flows or
  utilizations.

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Outline

• Introduction Related Work.
• JRCAS with Pairwise Interference.
• JRCAS with Physical Interference.
• TDMA Link Scheduling with Physical Interference.
• Single-Path Flows.
• Simulations.
• Conclusion Future Work.

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Related Work Physical Model

• TDMA Link Scheduling
• Moscibroda Wattenhofer 06 gave O(log4n)-apx
• Requires power control.
• Brar et al. 06 gave O(n? ln1-?n)-apx.
• Recently, Goussevskaia et al. 07 gave O(1)-apx
• Constant factor depends on max/min link lengths
  ratio.
• For single-channel networks with uniform link
  weights.
• Chafekar et al. 07 gave a polylog-apx for the
  joint (single-path) routing and scheduling
  problem
• Their objective is to minimize end-to-end delay.

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Physical Interference

• A packet from u is successfully received at v if
• Based on this equation we define notion of
  weights for any pair of links as
• Then, transmission on e is successful if
  .
• Let
• C can be bounded under reasonable assumptions
  (shown later).

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Weight-based Algorithm

• Same approach as Pairwise, except
• Redefine interference constraintconsider 2
  cases
• If a link is active, S weights of other active
  links must be 1.
• o.w. S weights of other active links is C.
• Thus,
• Scaling factor is (C 3).
• Above algorithm is a (C 3)-apx

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Bounding C

• Assuming
• Constant max (min) link length is dmax (dmin )
• Density is bounded
• Log-distance path model for signal propagation
  with path loss exponent ? gt 2.
• Total signal strength at u due to all other nodes
  is at most
• Ignoring the ambient noise

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Length-class-based Algorithm

• Based on techniques from Goussevskaia et al.
  07.
• Classify links into length classes
• Links of length in 2j,2j1) belong to length
  class Lj.
• For each Lj, plane is divided into square cells
  of side µ2j
• µ constant depending on ? and ß.
• For a cell Aj in Lj, let ?(Aj) be the set of link
  in Lj whose receivers lie in Aj.

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Length-class-based Algorithm

• General approach is the same
• LP Formulation
• Only difference is the interference constraint
• Scaling-down LP solution
• Scaling factor is (q 2).
• Use modified greedy placement (based on
  Goussevskaia et al. 07) for scheduling.

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LP solution ? JRCAS solution

• Scale down link utilizations by a (q 2) factor.
• Pick integer W s.t. is also integer
  for each (e,k).
• For each length class Lj, plane is divided into
  square cells of side µ2j.
• Cells are 4-colored s.t. adjacent cells have
  different colors.

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LP solution ? JRCAS solution

• For each (e,k) ? Ljh (length class Lj and color
  h), placing it in the first time
  slots of Sjh (of length W) s.t. no interface or
  interference constraint is violated, and no 2
  links (e,k) and (e',k) are placed in the same
  time slot ife,e' ? ?(Aj).
• Theorem 5.1 of Goussevskaia et al. 07
  guarantees feasibility.
• Above algorithm is a 4(q 2)g(L)-apx
• g(L) is the number of non-empty length classes.

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Outline

• Introduction Related Work.
• JRCAS with Pairwise Interference.
• JRCAS with Physical Interference.
• TDMA Link Scheduling with Physical Interference.
• Single-Path Flows.
• Simulations.
• Conclusion Future Work.

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TDMA Link Scheduling

• Given a set of weighted links, find the minimum
  length interference-free schedule of this set.
• We specialize our previous 2 techniques to get 2
  different O(1)-apx algorithms.
• Same approach for both weight-based
  length-class-based
• LP formulation.
• Scaling-down LP solution.
• Use greedy placement for scheduling.

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Weight-based Algorithm

• LP formulation
• Weight constraints
• Interfaces constraints
• Interference constraints
• Objective is
• Scaling-down factor is (C 3).
• Above algorithm is a (C 3)-apx for the TDMA
  Link Scheduling problem.

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Length-class-based Algorithm

• Similarly
• LP Formulation
• Only difference is interference constraint
• Scale down LP solution by (q 2) factor.
• Use modified greedy placement for scheduling.
• Above algorithm is a 4(q 2)g(L)-apx
• Generalization of Goussevskaia et al. 07s
  result for single-channel networks with
  uniform-weight links.

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Outline

• Introduction Related Work.
• JRCAS with Pairwise Interference.
• JRCAS with Physical Interference.
• TDMA Link Scheduling with Physical Interference.
• Single-Path Flows.
• Simulations.
• Conclusion Future Work.

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Single-Path Flows

• Traffic for each source-destination pair is
  restricted to a single path.
• Simplified network configuration.
• Some problems (such as packet re-ordering) dont
  exist.
• We use randomized rounding technique to ensure
  given solutions are close to optimal with high
  probability.

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Approach Overview

• LP formulation.
• Path stripping.
• Randomized rounding.
• Scaling-down LP solution.
• Use greedy placement for scheduling.

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Pairwise Interference

• Path stripping
• Using DFS, Fi is divided into a set of paths each
  with fractional flow of xij s.t. Sj xij1.
• Randomized rounding
• For each i, exactly one xij is set to 1 (with
  probability xij) and the rest are set to 0.
• Flow routed unsplit on the jth path (with rounded
  xij1).
• May lead to violations of the interface and
  interference constraints.

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Pairwise Interference

• Scaling-down LP solution
• Need to make sure no interface or interference
  constraint is violated.
• Define r.v. to represent interference in C(e) due
  to is single-path flow.
• Use Hoeffdings inequality to bound their sum.
• Similarly, define r.v. to represent interference
  in N(u) due to is single-path flow, and bound
  their sum.
• Resulting scaling-down factor is (c 2)ßmax with
  probability (1 e), whereD max edges of
  C(e) used by a single path.m of
  source-destination pairs.

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Generalizations

• Above techniques can be easily generalized for
• Directional antenna.
• Non-uniform link capacities.
• Varying power transmissions.
• Incorporating other constraints.
• Other objective functions.

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Physical Interference

• Length-class-based approach can be extended to
  single-paths with minimal changes
• Apx factor is 4(q 2)g(L)ßmax with probability
  (1 e).
• Weight-based approach requires more involved
  analysis
• Still yields (C 3)ßmax-apx factor with
  probability (1 e).

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Outline

• Introduction Related Work.
• JRCAS with Pairwise Interference.
• JRCAS with Physical Interference.
• TDMA Link Scheduling with Physical Interference.
• Single-Path Flows.
• Simulations.
• Conclusion Future Work.

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Simulations
Pairwise Interference
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Simulations
Physical Interference with multi-path routing
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Simulations
Physical Interference with single-path routing
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Conclusion

• Presented apx algorithms for several problems
• JRCAS problem
• with Pairwise Interference.
• with Physical Interference.
• TDMA Link Scheduling with Physical Interference.
• Single-path JRCAS problem
• with Pairwise Interference.
• with Physical Interference.

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Future Work

• Better apx bounds for Physical interference.
• Distributed algorithms.
• Handle other settings
• Non-orthogonal channels.
• More realistic models of Physical Interference.

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Questions?