Mario Čagalj

joint work with Jean-Pierre Hubaux and Christian Enz

Minimum-Energy Broadcast in All-Wireless Networks:

NP-Completeness and Distribution Issues

2

Numerous sensor devices equipped with

Modest wireless communication, processing, and memory capabilitiesForm ad hoc network (self-organized)Low mobility (static)

Distributed systemBroadcasting - important communication primitive

Wireless Sensor Networks

3

Find a minimum-energy broadcast tree

Design goals

j

i

l

o

m

n

kp

Minimize power consumption per packet

pjpk

pi

4

Wireless Multicast Advantage (WMA) [WieselthierNE00]

Nodes equipped with omnidirectional antennas

i transmits at and reaches both j and k

Energy expenditure

j

Peculiarities of wireless media

i

k

pik

pij

},max{ ikij pp},max{ ikijtot ppp

Wireless media is a node-based environment

5

Graph

ijc

ilcikc

ipc

Node-based network model

j

i

l

o

n

kp

),( EVG vip

s.t. are said to be covered by node

Captures the WMA property

Vx vip iixc

m

6

Formal problem definition

Minimum Broadcast Cover (MBC) problemGiven , , edge costs , the source and an assignment

Find a power assignment vector s.t. it induces graph , , in which there is a path from r to any node of V, and . is minimized

Special case: Geometric MBC (GMBC) MBC in two-dimensional Euclidean metric space

Edge costs are given by

),( EVG P R)(: GEcijVr PGVp v

i )(:][ 21

vV

vv ppp ),( '' EVG }:),{(' v

iij pcEjiE

Vi

vip

αijij kdc

7

COROLLARY 1. MBC cannot be approximate better than O(log ) if P NP.

SC to MBC transformation preserves approximation ratio achievable for SCNo polynomial-time algorithm approximates SC better than O(log ) if P NP [D.S. Hochbaum]

Complexity issuesTHEOREM 1. Minimum Broadcast Cover (MBC) is NP-complete.

Set Cover (SC) is NP-complete [M. Garey and D. Johnson]MBCNP and SC MBC ( - polynomial transformation)

THEOREM 2. Geometric MBC (GMBC) is NP-complete.

Planar 3-SAT (P3SAT) is NP-complete [M. Garey and D. Johnson]GMBCNP and P3SAT GMBC

8

Broadcast Incremental Power (BIP) [WieselthierNE00]

pb=8

8

2

pa=2

Heuristics based approach

d

i

h

c

ab

f

jg

e

1

pc=55

5

5

5pe=4

4

pd=4

4

Embedded Wireless Multicast Advantage (EWMA)Begin with an initial feasible solution (minimum spanning tree)

Improve the initial solution by embedding the WMA property while preserving the feasibility of the solution

[WanCLF01])(12)( OPTpBIPp tottot

,αijij dc 2

9

EWMA by example (1/2),α

ijij dc 2

11}{max},{

aax

ihx

da pcp

Calculate gains6 d

acbdda ppppg

5 eadcbe

ea pppppg

2 cac

ca ppg

6 badcb

ba ppppg

Calculate new transmission power

0},{maxarg

xa

xaa

newa

xa

ggppp

pb=8

8

2

pa=2

d

i

h

c

ab

f

jg

e

1

pc=55

55pe=4

4

pd=4

4

4

10

EWMA by example (2/2)

d

i

h

c

ab

f

jg

e

1

10pe=4

4

13

9

13

13

9

pa=13

17)( ea ppEWMAptot

pb=8

8

2

pa=2

d

i

h

c

ab

f

jg

e

1

pc=55

5

5

5pe=4

4

pd=4

4

23

)()(

edcba ppppp

MSTpBIPp tottot

[WanCLF01])(12)( OPTpMSTp tottot )(12)()( OPTpMSTpEWMAp tottottot

11

Phase 1: Distributed-MST [GallagerHS83]

Distributed EWMA (1/2)

j

i

l

o

n

kp

m

Node x waits before it potentially becomes a forwarder x

wT

xx

xx

xx

egge

ggT 2

max

1

max2

maxmax

1

0

0 if

if xw

Phase 2: Local EWMA

maxlg

maxjgmaxpg

12

Distributed EWMA (2/2)Lack of global information

Propagate information about forwarding nodes along the transmission chainPhase 2 organized in rounds of duration Probation , correction and active periods

j

i

l

o

n

kp

m

maxTcorrT actTprobT

probT corrT actT t

13

Performance evaluationSimulation setup

100 instances x 10,30,50,100-nodes networks

Spatial Poisson distribution of nodes

Cost of links

The performance metric is a normalized power For k-nodes network

]4..2[, ijij dc

},,{},100..1{

)},({min

)()(

,

min

min0

EWMABIPMSTim

mpp

p

mpmp

itot

mitot

tot

itoti

)(0 mp i

14

Performance evaluation (1/2) Avg. Normalized Power vs. Network Size

= 4 = 3

15

Performance evaluation (2/2)Avg. Normalized Power vs. Network Size

= 2 = 2

16

ConclusionAchievements

Proved that building the minimum-energy broadcast tree in wireless networks is NP-complete

Devised a new algorithm:Embedded Wireless Multicast Advantage (EWMA)

Shown that it can be distributed

Future work Extend the network model (i.e. mobility, interference)

Take into account the battery lifetime

Consider the minimum-energy multicast problem

http://www.terminodes.org

17

Synchronization of DEWMA

Necessary conditions

Duration of the second phase is bounded by

actj

rj

riract

ja

ir

ja TTT

TTTT

TTT

and 0

max

max

maxTF

}{max}{max 2 jFj

jr

Fjact eTT

m axT

pro bT co rrT ac tT

i j j

jaT

jrT

irT

nro u n d 1nro u n d 2nro u n d