Throughput Optimization in Mobile Backbone Networks

1
Throughput Optimization inMobile Backbone
Networks

• Emily Craparo, Jonathan P. How and Eytan Modiano
• SIAM Conference on Optimization
• May 11, 2008

2
Cooperative Sensing

• Cooperative sensing tasks often involve spatial
  distribution of agents.
• Cooperative exploration/coverage
• Sensor networks
• Spatial distribution of sensing platforms imposes
  communication constraints.
• Power limitations are particularly relevant in
  multi-agent systems.
• One possible solution to this problem is use of a
  mobile backbone network architecture.
• Backbone nodes are dedicated to providing
  end-to-end communication infrastructure.

3
What is a Mobile Backbone Network?

• A mobile backbone network is a hierarchical
  communication framework in which some agents
  provide communication support for other agents.
• Regular nodes (RNs) have limited mobility and
  communication capability, but are able to sense
  the environment.
• Mobile backbone nodes (MBNs) have superior
  mobility and communication capability.

4
What is a Mobile Backbone Network?

• Examples include
• Air support of ground vehicles in cluttered
  environments.
• Data collection in a sensor network.
• Problem How to place K mobile backbone nodes
  (MBNs) and assign N regular nodes (RNs) to them
  in order to maximize the effectiveness of the
  resulting network?
• Our goal maximize the number of RNs that achieve
  throughput of at least .
• An RNs throughput is a function of the distance
  of the RN from the MBN to which it is assigned
  and the number of other RNs assigned to that MBN
• This is a subproblem in the maximum fair
  placement and assignment (MFPA) problem (Srinivas
  Modiano 2007).

5
Solution Strategy

• Key insight in an optimal solution, each MBN can
  be placed at the 1-center of its assigned RNs.
• The 1-center location minimizes the maximum
  distance from the MBN to any of its assigned RNs.
• There are a limited number (O(N3)) of such
  locations
• Single-RN location
• Diameter-type locations
• Circumcenter-type locations

Regular node (RN)
Mobile backbone node (MBN)
Communication radius
6
Previous Work

• Solution approach taken by Srinivas Modiano
  (2007) search over all possible MBN placements.
• Maximize the number of RNs assigned for each
  placement.
• Each assignment maximization is posed as a
  maximum flow problem
• Time required for this algorithm scales
  polynomially with the number of RNs,
  exponentially with the number of MBNs.
• Computation time surpasses 30 minutes with 10 RNs
  and 5 MBNs.

MBNs
RNs
7
Network Design Approach

• The exhaustive search over all MBN placements can
  be recast as a network design problem

• All possible MBN locations occur in graph.
• yj binary indicator variable indicates that an
  MBN is placed at location j.
• cj constant indicates the maximum number of RNs
  that can be assigned to the MBN at location j
  while achieving throughput t.
• Flow from node i to node Nj indicates assignment
  of RN i to MBN at location j.

Potential MBN locations

• The network design problem can be solved as a
  mixed-integer linear programming (MILP) problem.

8
Computation Time

• Computation time of the MILP-based approach
  compares favorably with the search-based approach
  of previous work.
• Nnumber of RNs, Knumber of MBNs.
• Conclusion the MILP approach is a good way to
  obtain an exact solution for problems of moderate
  size.

9
Approximate Solution

• Formulation as a single optimization problem
  facilitates the use of approximation algorithms.
• This problem has special structure that can be
  leveraged
• Maximum flow is a submodular function of the set
  of arcs/mobile backbone node locations selected.

N1
1
y1c1
2

• All binary variables occur on arcs incident to
  the sink.
• Selecting arc i cannot increase flow through
  arc j.

1

1
s
t
i
yjcj
1
Nj
1

1
yMcM

N
NM
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Submodularity Proof Sketch

• Transform the maximum flow problem into an
  equivalent bipartite matching problem

1
1
N1
N1
2
c11
2
N2
c23
s
N2
t
3
3
c32
N3
N3
N4
N4
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Submodularity Proof Sketch

• Prove submodularity condition using bipartite
  graphs

1
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c1
c1
c1
c1
2
2
2
2
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S
S
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cT
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cj
cj
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ici
ici
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MNs
MNs NiNj
MNs Nj
MNs Ni
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Approximation Algorithm

• Solve O(KN3) maximum flow problems to get
  provably good solution to the maximum assignment
  problem (performance guarantee of ).
• Maximum flow problems on bipartite graphs are
  particularly easy to solve.
• Further computational efficiency is achieved
  using max flow/min cut duality.

13
Performance of Approximation Alg.
The approximation algorithm achieves nearly the
same level of performance as exact algorithm for
all problem sizes considered (Nnumber of RNs,
Knumber of MBNs).
14
Computation Time of Approx. Alg.
Computation time required for the approximation
algorithm scales gracefully with problem size in
computational experiments.
15
Mobile RNs and MBNs

• Existing techniques allow optimal MBN placement
  and assignment for static or uncontrolled RNs.
• What if RNs can be controlled, e.g. mobile sensor
  platforms?
• Network design approach accommodates RN motion,
  enabling simultaneous placement and assignment of
  both RNs and MBNs

• Intermediate nodes are added to represent
  possible locations for RNs.
• Node i is connected to node Nj iff RN i can
  reach location j under its mobility constraint.
• Remainder of graph is unchanged.
• Problem can again be solved using MILP.

1
1
16
Example
Radius of motion
Unoccupied location
Communication radius
MBN
Location occupied by RN
Regular node motion

• RNs and MBNs have been placed in order to
  maximize the number of RNs achieving throughput
  t.

Initially, each RN can reach a subset of the
potential locations.
17
Computation Time

• Nnumber of RNs
• Knumber of MBNs
• Lnumber of candidate locations for RNs.

• MILP approach is again good for problems of
  moderate size.
• Greedy approximation technique is still
  applicable in the case of mobile RNs.

18
Cooperative Exploration

• Minimize the time required for N RNs and K MBNs
  to visit each of L sensing locations, where
  location l is visited at time t if
• Location l is occupied by RN n at time t, and
• RN n is assigned to MBN k at time t.
• Problem can be framed as a MILP, but number of
  binary variables is large ? computation time is
  high.
• For N2, K1, L7, computation time 20 minutes.
• Two logical approaches for decomposition
• Sequential placement of RNs and MBNs at each time
  step
• RNs are greedily placed at unvisited locations,
  then MBNs are optimally placed and assigned.
• Simultaneous placement and assignment of RNs and
  MBNs at each time step.
• RN and MBN placement and assignment are jointly
  optimized at each time step.

19
Performance of Greedy Approach
Simultaneous placement and assignment of RNs
and MBNs enables more efficient cooperative
exploration than sequential placement.
20
Time-Discounted Reward
Even the approximate version of the
simultaneous placement and assignment algorithm
achieves a significant increase in
time-discounted reward over sequential placement
and assignment.
21
Performance Bound on Exploration

• Isolate the effect of communication constraint on
  efficiency of exploration ? assume that RNs are
  unrestricted in movement.
• Number of locations visited is a submodular
  function of the sequence of MBN/RN configurations
  chosen.
• Simultaneous placement and assignment algorithm
  visits at least ( )L locations in T
  time steps
• At least K locations are visited at each time
  step thereafter.
• Total time to visit all locations is at most T,
  where

22
Summary

• Examined mobile backbone network optimization in
  the context of cooperative sensing (static and
  mobile).
• Posed a network design formulation of mobile
  backbone network optimization.
• Developed exact and polynomial-time approximation
  algorithms for maximizing the number of RNs that
  achieve a given minimum throughput level.
• Developed improved algorithms for the MFPA
  problem.
• New exact algorithms computation time is 2-3
  orders of magnitude less than existing
  techniques for problems of practical scale.
• Can accommodate mobile regular nodes.
• Developed a greedy (in time) algorithm for
  cooperative exploration with a performance
  guarantee (in some cases) and good empirical
  performance (in the general case).

23
Further Reading

• E. Craparo, J. How and E. Modiano, Optimization
  of Mobile Backbone Networks Improved Algorithms
  and Approximation, ACC 2008.
• E. Craparo, J. How and E. Modiano, Cooperative
  Exploration in Mobile Backbone Networks,
  submitted to CDC 2008.
• I. Rubin, A. Behzadm R. Zhang, H. Luo, and E.
  Caballero, TBONE a Mobile-Backbone Protocol for
  Ad Hoc Wireless Networks, Proc. IEEE Aerospace
  Conference, 6, 2002.
• A. Srinivas and E. Modiano, "Joint node placement
  and assignment for throughput optimization in
  mobile backbone networks, in 26th Annual IEEE
  Conference on Computer Communications, Anchorge,
  AK, 2007.
• A. Srinivas, G. Sussmann and E. Modiano, Mobile
  Backbone Networks Construction and Maintenance,
  ACM MOBIHOC 2006, May 2006.
• K. Xu, X. Hong, and M. Gerla, Landmark Routing
  in Ad Hoc Networks with Mobile Backbones,
  Journal of Parallel and Distributed Computing,
  63, 2, pp 110-122, 2003.

24
Possible Further Work

• RNs with data storage capacity
• Communication is easy to optimize for static RNs
  how to incorporate motion and measurement into
  optimization?
• Aggregate throughput optimization rather than
  maximizing the number of RNs that achieve
  throughput t, maximize the sum of all RNs
  throughputs
• If received power is equalized, MBNs can again be
  restricted to 1-center locations without
  compromising optimality of the solution, and
  assignment can be solved as a minimum cost
  maximum flow problem on a multigraph.
• If a TDMA (time division multiple access)
  protocol is adopted, MBNs can occupy arbitrary
  locations in an optimal solution, but
  assignment/scheduling can be solved as a maximum
  flow problem on a generalized network.
• How does discretization of the MBNs potential
  locations affect solution quality?

25
Submodular Function Maximization

• Previous work has been done on approximation
  algorithms for maximizing submodular functions
  via greedy selection.
• Greedy selection carries a performance guarantee
  of
• for nondecreasing submodular
  functions.
• For our problem, greedy selection is easy
• Each location selected requires solution of a
  polynomial number of maximum flow problems.