Geometric Optics, Duality, and Congestion in Sensornets

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Geometric Optics, Duality,and Congestion in
Sensornets

• Christos H. Papadimitriou
• UC Berkeley
• christos

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Joint work with

• Dick Karp
• Lucian Popa
• Afshin Rostami
• Ion Stoica

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Sensornets

• Small nodes
• Communicating
• by wireless
• Power limitation

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The strange affinity betweenTheoretical CS and
Sensornets

• TCSs obsession with resource minimization finds
  a customer
• Open-ended scale
• Novel problems
• We were already working on the Internet
• Young field, fluid paradigms, open spirit

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Routing in sensornets

• IP envy
• Greedy routing (give the packet to your neighbor
  who is closest to the destination) may get stuck
• Fake coordinates help PRRSS03, PR05
• But greedy routing increases congestion

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In large networksGreedy routing ?
Straight-line routing
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Assume circular region,uniform distribution

• Routing affects
• congestion
• Average
• Maximum

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Calculating the congestion at r
ab c 2 (1 r 2 ) ???1 r 2 cos x dx
a
c
r
b
c
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Plotting the congestion
congestion
max 1 ave .46
1
r
1
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Average congestion

• .46 (the ave of straight-line routing)
• ? (the ave of any routing scheme)
• ? (the max of any routing scheme)

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Route to minimize max congestion?
?
1
?
.46
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Min max congestion Our results
congestion
1
curveball routing (max .56)
min max
r
1
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First attempt, metropolitan routing

• Follow circular arc
• Jump to target radius
• Finish by circular arc
• Optimize when to jump

Not a very good idea
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Fake coordinates

• Move to f(r)
• Intuitively, straight routes
• will curve in real space
• Optimum f?
• Assume f(r) ra
• Optimize a

f(r)
r
Not a very good idea
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Almost the right idea Airline routing

• Project to (northern)
• hemisphere
• Route by geodesic
• Intuitively, route will
• now avoid center
• Optimize z scale

Tokyo
N
Rabat
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Curveball routinga different projection works
better

• N

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congestion
1
(max .56)
r
(simulation results validated on the Intel
testbed)
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The optimum?

• Infinite-dimensional linear programming!
• Consider all admissible paths between a and b
• Optimum routing scheme will choose one of them
• Subdivide the disc into infinitely many rings
• Each path burdens each ring by some fixed amount
  of congestion

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Linear Programming!
min t Ax 1 Bx ? t x ? 0
Dual LP min ?? t ?? AT? BT? ? 0 ? ? 0
limit on congestion, one constraint per ring
speed of light in each ring
one variable per path, a continuum of variables
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Remember Snells law
c1
sin ?1

?1
sin ?2
c2
?2
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Characterization of the optimum

• Theorem There is a function ? 0,1 ? R
• such that the optimum routing scheme is a
  shortest path routing when the speed of light at
  radius r is ?(r). Furthermore, if ?(r) gt 0 then
  the congestion at radius r is maximum.

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Primal-dual algorithm!

• Subdivide the disc into finitely many rings
• Start with any set of speeds of light
• Calculate shortest paths, compute congestion
• Decrease speed of light where congestion is high,
  and repeat

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Experimentally, the optimum seems to be
1/?(r)
r
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Open problems

• Closed form of the optimum ?(r)?
• Are the optimum paths computable in a local way?
• Better practical algorithm than curveball?
• Extensions to other shapes and distributions?

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thank you!