Geometric Optics, Duality, and Congestion in Sensornets Christos H. Papadimitriou UC Berkeley...
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Transcript of Geometric Optics, Duality, and Congestion in Sensornets Christos H. Papadimitriou UC Berkeley...
Geometric Optics, Duality,and Congestion in Sensornets
Christos H. Papadimitriou
UC Berkeley
“christos”
CSNDSP: Patra, July 21 2006 2
Joint work with:
• Dick Karp
• Lucian Popa
• Afshin Rostami
• Ion Stoica
CSNDSP: Patra, July 21 2006 3
Sensornets
•Small nodes•Communicating by wireless•Power limitation
CSNDSP: Patra, July 21 2006 4
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
CSNDSP: Patra, July 21 2006 5
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
CSNDSP: Patra, July 21 2006 6
In large networks:Greedy routing Straight-line
routing
CSNDSP: Patra, July 21 2006 7
Assume circular region,uniform distribution
Routing affects congestion:•Average•Maximum
CSNDSP: Patra, July 21 2006 8
Calculating the congestion at r
ab = c 2 (1 – r 2 )
1 – r 2 cos x dxr
a
bc
c
CSNDSP: Patra, July 21 2006 9
Plotting the congestion
congestion
r
1
max = 1ave = .461
CSNDSP: Patra, July 21 2006 10
Average congestion
.46 = (the ave of straight-line routing)
(the ave of any routing scheme)
(the max of any routing scheme)
CSNDSP: Patra, July 21 2006 11
Route to minimize max congestion?
? 1
.46
?
CSNDSP: Patra, July 21 2006 12
Min max congestion: Our results
congestion
r1
1
curveball routing(max = .56)
min max
CSNDSP: Patra, July 21 2006 13
First attempt, metropolitan routing
•Follow circular arc•Jump to target radius•Finish by circular arc•Optimize when to jump
Not a very good idea…
CSNDSP: Patra, July 21 2006 14
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…
CSNDSP: Patra, July 21 2006 15
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”
CSNDSP: Patra, July 21 2006 16
Curveball routing:a different projection works better
N
CSNDSP: Patra, July 21 2006 17
congestion
r
1
(max = .56)
(simulation results validated on the Intel testbed)
CSNDSP: Patra, July 21 2006 18
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
CSNDSP: Patra, July 21 2006 19
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
CSNDSP: Patra, July 21 2006 20
Remember Snell’s law
1
2
=sin 1
sin 2
c1
c2
CSNDSP: Patra, July 21 2006 21
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.
CSNDSP: Patra, July 21 2006 22
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
CSNDSP: Patra, July 21 2006 23
1/(r)
r
Experimentally, the optimum seems to be…
CSNDSP: Patra, July 21 2006 24
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?
CSNDSP: Patra, July 21 2006 25
thank you!