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

• TCS’s 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 networks:Greedy 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 dxr

a

bc

c

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Plotting the congestion

congestion

r

1

max = 1ave = .461

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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

r1

1

curveball routing(max = .56)

min max

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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 ar

f(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

N

“Tokyo”

“Rabat”

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Curveball routing:a different projection works better

N

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congestion

r

1

(max = .56)

(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 tAx = 1Bx tx 0 limit on

congestion,one constraintper “ring”

one variableper path, acontinuum

of variables…

Dual LP:min + t AT + BT 0

0

“speed of light”in each ring

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Remember Snell’s law

1

2

=sin 1

sin 2

c1

c2

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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) > 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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1/(r)

r

Experimentally, the optimum seems to be…

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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!