Transcript Document
Onur G. Guleryuz & Ulas C.Kozat
DoCoMo USA Labs, San Jose, CA 95110
{guleryuz,kozat}@docomolabs-usa.com
Scenario
Phase 1: (NASA, Virgin Galactic,...) Phase 2: (DoCoMo) Phenomenon Central node Low power, low complexity, wireless sensor node (Less exciting applications possible)
Overview
• This paper is about information theoretic (rate-distortion based) clustering of sensor networks.
• How should information emanate from a sensor network?
• How should bandwidth/power be allocated to sensor nodes? • If the final application is detection/classification based on the received data, how should the above change?
• Based on looking just at the topology of an optimized network, can we tell something about what the network measures?
Information Setup
: Central node - Node
0 Task
: The final collector of all information, interested in: Case 1 : Each piece of data (does sensor i think there is an alien around?) Case 2 : The average or sum statistic of data (what is the total number of aliens? – we can also handle other linear combinations, several statistics, etc.) : Sensor node – Node
i (i=1,...,N
)
1.
x
(
E
[
x
i
x
j
]
i
2
i
,
j
)
(
i
)
: Set of nodes that have sent information to node
i
2.
Compress (quantize + entropy code) and communicate: Case 1 : Case 2 :
{
x i
}
x i
{
x
ˆ
j j
(
i
|
)
x
ˆ
j j
(
i
)}
x
ˆ (Re-compression is allowed)
x
Wireless Network Setup
C i
,
j
(
i
,
j
0 ,...,
N
)
C i
,
j i,j
. • Communication happens during well defined time intervals.
• We assume the capacity matrix remains unchanged over a reasonable duration (>> one time interval).
• Bandwidth is constrained: Node
i C i
,
j j
inside a time interval.
• Routing is constrained: Every node i (i=1,...,N) can transmit to at most one other node inside a time interval (
fan out = 1
).
(We can also operate under more general frameworks, under some capacity scaling constraints – please see the routing over a depth two tree example.)
Routing is over a tree
Routing Setup
0 4 5 depth of routing tree 2 1 3 6 (Nodes 1,3 send their r.v.’s to node 4, which combines the received information with its r.v. (case 1 or case 2), and sends everything to node 0. ...)
Problem Statement: Find the Optimal Information Flow
Find the
jointly optimal
compression, detection (case 1, case 2), and routing strategy for the given: •
C i
,
j
(
i
,
j
0 ,...,
N
)
•
i
2 (
E
[
x
i
x
j
]
i
2
i
,
j
, i.e., minimize the total distortion at node
0,
D
T
subject to constraints.
i
,
j
1 ,...,
N
)
j N
1
E
[(
x
j
x
ˆ
j
) 2 ] ,
D T
, 1
D T
, 2 : total distortion observed for case 1.
: total distortion observed for case 2.
We will find optimal solutions for each case and compare them.
Deployed nodes
Example
Optimal Case 1 routing: Optimal Case 2 routing:
D T
, 1 ? (>,<,=)
D T
, 2
Mini FAQ
Q:
Don’t you need to know the distribution of the r.v. before you compress, do rate allocation, etc.?
A:
No, we use a good upper bound. Practical (achievable) distortion D for
x
i
i
2
D
(
C
,
i
2 )
i
2 2 2
C
<= D <=
const
D
(
C
,
i
2 ) Using this bound, optimal rate allocation can be done using the “reverse water filling theorem”.
Q:
If case 2, shouldn’t the sensor network always send the linear combination since
entropy
(
N
x
)
j j
1 i.e., isn’t the routing problem trivial?
j N
1
entropy
(
x
j
)
A:
No. There is a penalty for collecting information within the sensor network due to capacity constraints. The routing problem is combinatorial in the general case.
2
Toy Scenario (given routing)
0 Setup:
C i
1
C
,
C i
0 0 (
i=2,..N
),
C
10 0
C
10
C
… 1
C
…
C
i N (a)
C
~
C
10 ,
i
2 2 ( (
D D T T
, 1 , 2 ) ) 2
N N
1 exp( 2
C
10 (
N N
1 ) ) Intra-network bandwidth is sufficient to achieve
exponential
improvements.
(b)
C
C
10
N
,
i
2 2
N N
1 exp( exp( 2
C
) 2
C
) exp( 2
C
10 ) ~
N N
1 Intra-network bandwidth is the bottleneck.
(Skipping many details, reverse water filling, dropping of coefficients, etc.)
Optimal Clustering: Harder Scenario
Arbitrary routing tree of depth
two
, with a fan-in constraint. 0 cluster 1 cluster L ...
C (
1
) C (L)
...
...
N(1)
nodes
N(L)
nodes
C (i)
W/N(i) N(i
))
C i
0
i
How many clusters?
N(i)
?
Which nodes are the cluster heads?
Dynamic Programming ~
O
(
N
3 )
log(
Harder Scenario contd.
(
W=2.5
) Range of exponential gains for
case 2
.
(Beyond this range little penalty for case1 optimal routing even if the actual scenario is case 2.)
Optimal Clustering: Hardest Scenario
Arbitrary
C i
,
j
,
i
2 Heuristic, steepest descent algorithm ( Central node is at the center)
Hardest Scenario (contd.)
( Central node is at the center)
Conclusion
• Optimal clustering of capacity constrained wireless sensor networks.
• Intra-network bandwidth is very important. Without sufficient intra-network bandwidth, no gains for sending statistics instead of the individual data in case 2.
• We can solve a dual problem of network lifetime maximization under the constant fidelity.
• We can comply with “scaling laws” and find optimal clusters.
• Based on looking just at the topology of an optimized network, can we tell something about what the network does?
(image from http://www.sruweb.com/~walsh/neuron.jpg)