Efficient and Robust Query Processing in DynamicEnvironments Using Random Walk Techniques

Chen Avin

Carlos Brito

IPSN04 - Berkeley - 04/27/2004 2

Outline

Motivation Random Walk and Partial Cover Time Efficiency Robustness Quality Load Balancing, Scalability and Latency Discussion

IPSN04 - Berkeley - 04/27/2004 3

Motivation

Sensor Network as large, dense and dynamic networks

Task: Query the network Common systems depend on state information

stored in the nodes for proper operation and control (i.e. spanning trees, cluster heads)

Critical points of failure lead to recovery mechanism Explore the properties of uncontrolled scheme like

random walk Simple process, no critical point of failure, all nodes

are equally unimportant at all times

IPSN04 - Berkeley - 04/27/2004 4

Random Walk

Visiting the nodes of the graph in a random order

At each step, a token moves to a neighbor with some distribution(simple = uniform)

IPSN04 - Berkeley - 04/27/2004 5

Random Walk for Sensor Nets Easily implemented in sensor networks: base

station issues a token with a query (almost) Assumption free method, the

protocol does not require knowledge of: Location Neighbors Transmission range Symmetric connection

High density and redundancy are advantage

IPSN04 - Berkeley - 04/27/2004 6

Cover Time

Cover Time: the expected time to visit all the nodes in a random walk (starting at the worst case node)

How efficient is the process ? hij : the expected time to go from node i to j

hmax: max (hij | all nodes in the graph)

Matthew’s Bound: C ≤ hmax·log(n)

IPSN04 - Berkeley - 04/27/2004 7

Cover Time

Known results: Worst cases: O(n3)

Lollipop graph Line: O(n2)

Best cases: O(n·log(n)) Star Complete Graph Hypercube

Grid: O(n·log2(n)) Random sensor networks ?

IPSN04 - Berkeley - 04/27/2004 8

Partial Cover Time (PCT) In sensor network we don’t need to consult every

node How efficient is to visit 80% of the nodes ?

Lemma:PCT(c) ≤ O(hmax )

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Unvisited nodes Visited nodes

O(n) in Hypercube O(n·log(n)) in Grid

IPSN04 - Berkeley - 04/27/2004 10

Outline

Overview of our approach Random Walk and Partial Cover Time Efficiency Robustness Quality Load Balancing, Scalability and Latency Discussion

IPSN04 - Berkeley - 04/27/2004 11

Efficiency – Simple Walk

0

2

4

6

8

10

12

14

16

10 20 30 40 50 60 70 80 90 100

Num

ber

of s

teps

nor

mal

ize

to n

% of Cover

3.12

GridRandom 15Random 19Hyper Cube

IPSN04 - Berkeley - 04/27/2004 12

Biased Random Walk

Can we improve this results? Give priority to unvisited nodes Define bias parameter: 0 ≤ bias ≤ 1 Visited neighbor selected with probability

(1- bias) / d Unvisited with

(1- bias) / d + bias / du

The protocol remain (almost) the same

IPSN04 - Berkeley - 04/27/2004 13

Biased Random Walk

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90 100

Nu

mb

er

of

ste

ps

no

rma

lize

to

n

% of Cover

Bias = 0Bias = 0.1Bias = 0.2Bias = 0.4Bias = 0.6Bias = 0.8

Bias = 1

3.12

IPSN04 - Berkeley - 04/27/2004 14

Comparison with Clustering

1

2

3

4

5

6

7

8

9

10

100 1000 10000 100000 1e+06

Exp

ect

ed

nu

mb

er

of m

ess

ag

es

norm

aliz

e t

o n

Network size

10 X 10 25 X 25 50 X 5075 X 75

100 X 100

250 X 250500 X 500

1000 X 1000

80% cover walk on grid with bias 0.580% cover walk on grid with no bias

Optimal Cluster head protocol for grid

Analytical result for Cluster Head scheme shows that the number of messages for optimal protocol on grid require ≈ 0.945n7/6

The efficiency ofboth systems issimilar

IPSN04 - Berkeley - 04/27/2004 15

Outline

Overview of our approach Random Walk and Partial Cover Time Efficiency Robustness Quality Load Balancing, Scalability and Latency Discussion

IPSN04 - Berkeley - 04/27/2004 16

Robustness to Dynamics The probability that a node will fail when it

has the token is negligible No critical point of failure (but do need

reliable token passing) All we require is connectivity in the token

neighborhood Robust to independent and dependent

failures (disaster areas)

IPSN04 - Berkeley - 04/27/2004 17

Spanning tree in dynamic env. Nodes close to the root are more important When a node fails all nodes in the sub-tree

are disconnected from the root and must participate in recovery mechanism

Assuming independent failure (or duty cycle) probability p, (q=1-p) the expected number of nodes to report is O(qh)

Since R << network area, h is large p=0.1. h=10 65% will not report to the root.

IPSN04 - Berkeley - 04/27/2004 18

Outline

Overview of our approach Random Walk and Partial Cover Time Efficiency Robustness Quality Load Balancing, Scalability and Latency Discussion

IPSN04 - Berkeley - 04/27/2004 19

How far are the unvisited nodes from visited ones ?

90% are atmost 2 hops

Expected random walkwill not leavelarge area uncovered

Quality of Partial Cover - 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Random Network Grid

Network Type

Exp

ecte

d p

ort

ion

fro

m u

nco

vere

d n

od

es 12

11

10

9

8

7

6

5

4

3

2

1

IPSN04 - Berkeley - 04/27/2004 20

Quality of Partial Cover - 2

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Expected Walk on Random Graph

Net

wo

k p

ort

ion

4 Walks

3 Walks

2 Walks

1 Walk

0 Walks

How long must a node wait before a walk will visit its neighborhood?

85% are visitedat most every other run

At most will need to wait4 runs

IPSN04 - Berkeley - 04/27/2004 21

Application Example

Nodes temperature 30 25 20 15 10 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X Location 0

0.1 0.2

0.3 0.4

0.5 0.6

0.7 0.8

0.9 1

Y Location

0 5

10 15 20 25 30 35

Temp

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

Netw

ork

port

ion

21 Bar histogram between min and max temperature

80% random walkReal data

Find the histogram of the data in the network Assume non uniform distribution Token report after seeing 80% of the nodes

IPSN04 - Berkeley - 04/27/2004 22

Outline

Overview of our approach Random Walk and Partial Cover Time Efficiency Robustness Quality Load Balancing, Scalability and Latency Discussion

IPSN04 - Berkeley - 04/27/2004 23

Load Balancing

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

Netw

ort

port

ion

Expected number of visits to a node in 80% random walk

Expected PCT of 80%

The stationary distribution of the Markov chain π = (π1, …, πn) is πi=di/2m

In regular graphsπ is uniform,but this only afterlong walks

Here we issue many“short” walks

IPSN04 - Berkeley - 04/27/2004 24

Scalability

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70 80 90 100

Exp

ect

ed

nu

mb

er

of

ste

ps

no

rma

lize

to

n

% of Cover

1024 Random Network R = 0.082048 Random Network R = 0.05656

4096 Random Network R = 0.048192 Random Network R = 0.02828

16384 Random Network R = 0.02

2.92n

3.37n

X 16

IPSN04 - Berkeley - 04/27/2004 25

Latency

Random walk is sequential process The latency is proportional to the number of

steps to accomplish the task Reduce the range of applicability Future work: combine result from few parallel

random walks in the network

IPSN04 - Berkeley - 04/27/2004 26

Discussion

Achieving control in highly dynamic env. is problematic, and in many cases not energy efficient do to recovery mechanism

How do we do with uncontrolled process such as random walk? Not Bad !

Not applicable in all cases, but, When applicable provides an elegant, simple

and efficient solution