Minimumenergy broadcasting in static ad hoc wireless networks

1
Minimum-energy broadcasting in static ad hoc
wireless networks

• P.J. Wan, G. Calinescu, X.Y.Li and O. Frieder
• Wireless Networks 8, 607-617, 2002
• Advisor Rong Homg Jan
• Speaker An kai Jeng

2
Outline

• Introduction
• Lower bound on the approximation ratio of SPT
• Lower bound on the approximation ratio of BAIP
• Lower bound on the approximation ratio of MST
• Lower bound on the approximation ratio of BIP
• Upper bound on the approximation ratio of MST and
  BIP
• Conclusion

3
Introduction

• Energy conservation is a critical issue in ad hoc
  network.
• One major approach for energy conservation is to
  route a communication session along the route
  which requires the lowest total energy
  consumption.
• the power required to support a link between two
  nodes separated by a distance r is rk

4

• Relaying a signal between two nodes may result in
  lower total transmission power.

When k 2, and ?p1p3p2 is obtuse, p1p32
p3p22 p1p22
5

• Unicast routing problem is polynomial solvable
• G(k)

2
p2p4k
p1p2k
p4p5k
4
5
1
p1p3k
p3p4k
3
Can be solved by applying any shortest-path
algorithm
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• The broadcast routing problem is NP-hard
• Three greedy heuristics were proposed in
  literatures
• MST (minimum spanning tree)
• SPT (shortest-path tree)
• BIP (broadcasting incremental power )

7

• They have been evaluated through simulations, but
  little is known about their analytical
  performance.
• Some heuristics may perform quite well but very
  poorly in other situations.
• The main issue of this paper is to find their
  approximation ratios. (i.e. r worst(P,A)
  /opt(P) C)
• In the general graph version, there exists no
  sub-logarithmic approximation. (i.e.O(r) lt logn)
• Its geometric version can be approximated within
  a constant factor

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Lower bound on the approximation ratio of SPT

• SPT

Shortest path of G(k)
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The hard instance of STP
STP
OPT
1-?
?
1-?
?
TPOPT 1
TPSTP ?kn/2(1-?)k
When ? ? 0 , TPSTP / TPOPT ? n/2
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Lower bound on the approximation ratio of BAIP

• BIP

11

• BAIP

12

• The hard instance of BAIP

1
2
i
n-1
n
When ? ? 0 , TPSTP / TPBAIP ?
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Lower bound on the approximation ratio of MST
TPOPT (1?)2
TPMST 6126
When ? ? 0 , TPSTP / TPOPT ? 6
14
Lower bound on the approximation ratio of BIP
TPOPT 1
When ? ? 0 , TPBAP / TPOPT ? 13/3
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Upper bound on the approximation ratio of MST
and BAIP

• Upper bounds on the approximation ratios of these
  heuristics need to be analyzed for all possible
  instances.
• Rely on the geometric structures of Euclidean
  MSTs.
• Worst(A,P)/OPT(P) C
• UB(Worst(A,P))/LB(OPT(P)) C
• Worst(A,P)/OPT(P) UB(Worst(A,P))/LB(OPT(P))

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• The Radius of a point set P

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• Theorem 3

MST
Radius1
18

• Lemma 4

r5
r3
r2
TPOPT
r1
r4
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Theorem 3
Lemma 4
Theorem 6
Lemma 6
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Conclusion

• The approximation ratio of MST is between 6 and
  12
• The approximation ratio of BIP is between 13/3
  and 12
• The approximation ratio of SPT is at least n/2
• The approximation ratio of BAIP is at least
  4n/lnn-o(1)
• Future works
• Find tight upper bounds
• Construct harder instances that can lead to
  better lower bounds of MST and BIP
• Distributed algorithms

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The End