Iso-Contour Queries and Gradient Descent with Guaranteed Delivery in Sensor Networks

Rik Sarkar, Xianjin Zhu, Jie Gao, Joseph S. B. Micchell , Leonidas J. Guibas

IEEE INFOCOM 2008

Speaker: Chen-Yan Wu

Outline

Introduction Proposed Mechanism Simulation Conclusion

Introduction Wireless sensor networks have shown great potential for providing dense

monitoring and sensing capabilities with modest cost and management effort.

In the application of environmental monitoring, sensors measure readings of the physical space, such as chemical concentration.

An iso-contour at an value x is the collection of points with value equal to x. iso-contour

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Introduction – Scenario

The iso-contours encode spatial structures of the signal field, such as boundaries of the ‘hot’ regions.

Users with hand-held devices communicate with nearby sensors to obtain directions to places indicated by the sensor data being within a specified range. Iso-contour query Value-restricted routing

q

Dest.

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Introduction – Challenge

Simple gradient descending/ascending routing can typically lead the query message to one iso-contour, unless the query message reaches a local minimum or local maximum, in which case the query gets stuck.

q

Dest.

Overview

Preprocessing Local identification for local maximum and local minimum Sweep for saddle point Distributed construction of contour tree

+ ˙-local maximum local minimum saddle point

++

---

˙

˙

a

b

c

e

d

fg

+

+

-

--

˙

a

bc

e

f ˙d

g

Definition Given a continuous signal field F , and a node id q with value F(q) = x

F(q) =70

q

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the highest temperature of interior

the lowest temperature of interior

the highest temperature of exterior

the lowest temperature of exterior

Preprocessing

Local identification for local maximum and local minimum A node identifies itself as a local maximum if it discovers that all its

1-hop neighbors have value no greater than itself. It then initiates a sweep top down. And vise versa.

+

-local maximum

local minimum+

+

-

--

a

b

e

f g

++i

h

Preprocessing

Sweep Each sweep is initiated and labeled by a critical node (a maximum, min

imum or a saddle node). In the sweep initiated by a local maximum a, the sweep message carrie

s the tuple (a, F(a)).

+

-

local maximum

local minimum+

+

-

--

a

b

e

f g

++i

h

Preprocessing

Sweep – saddle points If a node gets two sweep messages from different local maximum

(minimum), this indicates that two contour components start to merge. Thus a saddle point should be identified.

+

-̇

local maximum

saddle point

local minimum+

+

-

--

a

b

e

f g

++i

h

˙c

˙d

(a, F(a)) (b, F(b))

Preprocessing

Sweep – saddle point The sweep messages, the tuple (a, F(a)) and (b, F(b)), encounter at nod

e c. The sweep message will be changed as (c, F(c), M(a,b)). The sweep messages, the tuple (f, F(f)) and (g, F(g)), encounter at node

c. The sweep message will be changed as (d, F(d), S(f,g)).

++

+

-̇

-

--

˙

a

bc

e

f ˙d

g

local maximum

saddle point

local minimum

(c, F(c), M(a,b))

++h

i

Preprocessing

Distributed construction of contour tree a node q with value F(q)= x After node a, b broadcast. After node c broadcasts. After node e, f, g broadcast. After node d broadcasts.

++

+

-̇

-

--

˙

a

bc

e

f ˙d

g

local maximum

saddle point

local minimumq

˙ c

- --e fg

++

---

˙

˙

a

b

c

e

d

fg

+ +a b

˙ d

Proposed Mechanism

Iso-contour queries Gradient descent routing for iso-contour queries

Users at node q want to find value x

x=40

++

---

˙

˙

a=80

b=60

c=30

e=20

d=30

f=20g=20

+

+

-

--

˙

a

bc

e

f ˙d

g

q

t

s

F(s)=30

Proposed Mechanism

Value restricted routing Given a source s and destination t, find a path P from s to t such that at

every node x on P, y ≤ F(x) ≤ z.

z=40

++

---

˙

˙

a=80

b=60

c=30

e=20

d=30

f=20g=20y=20

+

+

-

--

˙

a

bc

e

f ˙d

g

q

t

s

F(s)=30

Simulation

Parameters 1600 nodes 16 by 16 units square region with unit disk graph as the

communication model The average number of neighbors per node is about 21. The sensors sample from a continuous signal field shown below.

Elevation map of West Reno (obtained from usgs.gov) and its sampling

The message complexity of contour tree construction

The CDF of the node load distribution

Conclusion

Proposed the distributed construction of a contour tree and its application in iso-contour queries by gradient routing with guaranteed delivery.