E. Althaus Max-Plank-Institut fur Informatik

G. Calinescu Illinois Institute of Technology

I.I. Mandoiu UC San Diego

S. Prasad Georgia State University

N. Tchervenski Illinois Institute of Technology

A. Zelikovsky Georgia State University

Power Efficient Range Assignment in

Ad-hoc Wireless Networks

WCNC 2003 2

Outline

• Motivation• Previous work• Approximation results• Experimental Study

WCNC 2003 3

Ad Hoc Wireless Networks

• Applications in battlefield, disaster relief, etc• No wired infrastructure• Battery operated power conservation critical

WCNC 2003 4

Power Attenuation Model

• Signal power falls inversely proportional to dk, k[2,4]Transmission range radius ~ k-th root of power

• Omni-directional antennas• Uniform power attenuation coefficient k• Uniform transmission efficiency coefficients• Uniform receiving sensitivity thresholds

Transmission range = disk centered at the nodeSymmetric power requirements

Power(u,v) = Power(v,u)

WCNC 2003 5

Asymmetric Connectivity

Power ranges

b

a

c

d

g

f

e

Connectivity grapha

b

d

g

f

e

c

Multi-hop ACK!

a

b

d

g

f

e

c

WCNC 2003 6

Symmetric Connectivity

Per link acknowledgements

a2

3

11

b

d

g

f

e

c

1

1

1

Asymmetric Connectivity

Increase range of “b” by 1 Decrease range of “g” by 2

a 2

1

11

b

d

g

f

e

c

1

1

2

Symmetric Connectivity

WCNC 2003 7

• Given: set of nodes, coefficient k• Find: power levels for each node s.t.

– Symmetrically connected path between any two nodes– Total power is minimized

Problem Formulation

WCNC 2003 8

a

b

d

g

f

e

c

h

Power-cost of a Tree

Node power = power required by longest edge

Tree power-cost = sum of node powers

WCNC 2003 9

• Given: set of nodes, coefficient k• Find: spanning tree with minimum power-cost

Reformulation of Min-power Problem

WCNC 2003 10

Previous Work

d

• Max power objective– MST is optimal [Lloyd et al. 02]

• Total power objective– NP-hardness [Clementi,Penna,Silvestri 00] – MST gives factor 2 approximation [Kirousis et al. 00]– 1+ln2 1.69 approximation [Calinescu,M,Zelikovsky 02]

WCNC 2003 11

Our results

• 5/3 approximation factor– NP-hard to approximate within log(#nodes) for asymmetric

power requirements

• Optimum branch-and-cut algorithm – practical up to 35-40 nodes

• New heuristics + experimental study

WCNC 2003 12

MST Algorithm

Power cost of the MST is at most 2 OPT

(1) power cost of any tree is at most twice its cost

p(T) = u maxv~uc(uv) u v~u c(uv) = 2 c(T)

(2) power cost of any tree is at least its cost

(1) (2)p(MST) 2 c(MST) 2 c(OPT) 2 p(OPT)

WCNC 2003 13

Tight Example

1+ 1+ 1+

1 11

Power cost of MST is n

Power cost of OPT is n/2 (1+ ) + n/2 n/2

n points

WCNC 2003 14

Gain of a Fork

• Fork = pair of edges sharing an endpoint• Gain of fork F = decrease in power cost obtained by

– adding F’s edges to T– deleting longest edges from the two cycles of T+F

Gain = 10-3-1-3=3

a

b

d

g

f

e

c

12

2

h2

8

2

1013

1010

10

12 10

12

8

2

2

13

a

b

d

g

f

e

c

12

2

h2

8

2

1013

1013(+3)

10

13 (+1) 13 (+3)

2(-10)

8

2

2

13

WCNC 2003 15

Approximation Algorithms

• Every tree can be decomposed into a union of forks s.t. sum of power-costs = at most 5/3 x tree power-cost

Min-Power Symmetric connectivity can be approximated within a factor of 5/3 + for every >0

WCNC 2003 16

Experimental Setting

• Random instances with up to 100 points• Compared algorithms

– Edge switching

WCNC 2003 17

Edge Switching Heuristic

a

b

d

g

f

e

c

12

2

h

2

4

2

15

1010

12

13

12

4

2

2

WCNC 2003 18

Edge Switching Heuristic

• Delete edge

a

b

d

g

f

e

c

12

2

h

2

4

2

13

4

13

13

12

12

4

2

2

2

WCNC 2003 19

Edge Switching Heuristic

• Delete edge• Reconnect with min increase in power-cost

2

a

b

d

g

f

e

c

12

2

h

2

4

2

1315

13

1515

4

12

4

2

2

WCNC 2003 20

Experimental Setting

• Random instances with up to 100 points• Compared algorithms

– Edge switching– Distributed edge switching– Edge + fork switching– Incremental power-cost Kruskal – Branch and cut– Greedy fork-contraction

WCNC 2003 21

Greedy Fork Contraction Algorithm

• Start with MST• Find fork with max gain• Contract fork• Repeat

WCNC 2003 22

Percent Improvement Over MST

WCNC 2003 23

Percent Improvement Over MST

WCNC 2003 24

Runtime (CPU seconds)

WCNC 2003 25

Summary

• Efficient algorithms that reduce power consumption compared to MST algorithm

• Can be modified to handle obstacles, power level upper-bounds, etc.

• Ongoing research- Improved approximations / hardness results- Multicast- Dynamic version of the problem (still constant factor)