1. Introduction Wireless Ad-Hoc Network
Set of transceivers communicating by radio

2. Introduction Wireless Ad-Hoc Network
Each transceiver has a transmission power
which results in a transmission range

3. Introduction Wireless Ad-Hoc Network
Transceiver receives transmission from only if

4. Introduction Wireless Ad-Hoc Network
As a result a directed communication graph is
induced

5. Model & Problems
Definition
A set of transceivers

6. Model & Problems Definition A set of transceivers
is the power assignment

7. Model & Problems Definition A set of transceivers
is the power assignment

8. Model & Problems Definitions A set of transceivers
is the power assignment
is the communication graph

9. Model & Problems Definitions A set of transceivers
is the power assignment
is the communication graph
is the cost of the assignment

10. Outline Connectivity problems Bounded hop broadcast Spanners
Interference-free broadcast

11. Connectivity Definitions A graph is k-vertex-connected if for
any two nodes there exist k-vertex-disjoint
paths connecting to
2-vertex-connected

12. Connectivity Definitions For graph , a subset is a
connected backbone if restricted to is
strongly connected and for each
there exists so that
Connected backbone

13. Connectivity Problem 1 (k-vertex-connectivity)
Input: A set of transceivers, and a parameter
Output: A power assignment with minimal possible cost , where is k-vertex connected

14. Connectivity Problem 1 (k-vertex-connectivity)
Input: A set of transceivers, and a parameter
Output: A power assignment with minimal possible cost , where is k-vertex connected
-approximation algorithm

15. Connectivity Problem 2 (connected backbone)
Input: A set of transceivers
Output: A subset of and a power assignment with minimal possible cost , where (restricted to ) is strongly connected, and for each , there exists , such that

16. Connectivity Problem 2 (connected backbone)
Input: A set of transceivers
Output: A subset of and a power assignment with minimal possible cost , where (restricted to ) is strongly connected, and for each , there exists , such that
Constant-factor approximation algorithm in time

17. Fault-Tolerant Power Assignment
Definitions
For each , let be a set of closest
nodes to

18. Fault-Tolerant Power Assignment
Definitions
For each , let be a set of closest
nodes to

19. Fault-Tolerant Power Assignment
Definitions
For each , let be a set of closest
nodes to
Let

20. Fault-Tolerant Power Assignment
The algorithm
Assign each the range (denote )
Compute an of

21. Fault-Tolerant Power Assignment
The algorithm
Assign each the range (denote )
Compute an of

22. Fault-Tolerant Power Assignment
The algorithm
For each edge of increase the range
of the nodes in such that each node
can reach all nodes in , and vice versa (denote )

23. Fault-Tolerant Power Assignment
The algorithm
For each edge of increase the range
of the nodes in such that each node
can reach all nodes in , and vice versa (denote )

24. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Case 1:

25. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Case 1:

26. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Case 2:

27. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Case 2:

28. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Easy to see

29. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Easy to see
Kirousis et al. proved

30. Fault-Tolerant Power Assignment
Proof sketch
Let
In each is assigned at most
Easy to see
Kirousis et al. proved
As a result and since degree of MST is constant

31. Connected Backbone Power Assignment
Definitions
Given the of , for any node , let
be the size of the longest edge adjacent to

32. Connected Backbone Power Assignment
Definitions
Given the of , for any node , let
be the size of the longest edge adjacent to

33. Connected Backbone Power Assignment
The algorithm
Compute an of

34. Connected Backbone Power Assignment
The algorithm
Compute an of

35. Connected Backbone Power Assignment
The algorithm
Compute an of
Let be the set of all internal nodes of

36. Connected Backbone Power Assignment
The algorithm
Compute an of
Let be the set of all internal nodes of
Assign each with (denote )

37. Connected Backbone Power Assignment
The algorithm
Compute an of
Let be the set of all internal nodes of
Assign each with (denote )

38. Connected Backbone Power Assignment
The algorithm
Compute an of
Let be the set of all internal nodes of
Assign each with (denote )

39. Connected Backbone Power Assignment
Proof sketch
Construct a power assignment for which
it holds and ,
as a result obtaining
is derived from

40. Connected Backbone Power Assignment
Proof sketch
Let be the connected backbone in
For each node let be the transmission
range of in

41. Connected Backbone Power Assignment
Proof sketch
For each node let be all the nodes
within distance from

42. Connected Backbone Power Assignment
Proof sketch
For each node let be all the nodes
within distance from

43. Connected Backbone Power Assignment
Proof sketch
For each node let be all the nodes
within distance from
For each node compute of

44. Connected Backbone Power Assignment
Proof sketch
For each node let be all the nodes
within distance from
For each node compute of

45. Connected Backbone Power Assignment
Proof sketch
In :
Each node is assigned

46. Connected Backbone Power Assignment
Proof sketch
In :
Each node is assigned

47. Connected Backbone Power Assignment
Proof sketch
In :
Each node is assigned
Each node is assigned

48. Connected Backbone Power Assignment
Proof sketch
Carmi et al. showed that

49. Connected Backbone Power Assignment
Proof sketch
Carmi et al. showed that

50. Connected Backbone Power Assignment
Proof sketch
Carmi et al. showed that

51. Connected Backbone Power Assignment
Proof sketch
Carmi et al. showed that + + +

52. Connected Backbone Power Assignment
Proof sketch
Carmi et al. showed that
Using this and is at least longest edge in we obtain

53. Connected Backbone Power Assignment
is at least longest edge in and
Thus (summing over all v),

54. Connected Backbone Power Assignment
Proof sketch
Kirousis et al. proved that given an
assigning each node with yields
a 2-factor approximation for strong-connectivity
(denote )

55. Connected Backbone Power Assignment
Proof sketch
Kirousis et al. proved that given an
assigning each node with yields
a 2-factor approximation for strong-connectivity
(denote )
Using this fact and that B gives us strong connectivity, we obtain

56. Connected Backbone Power Assignment
Proof sketch
Therefore,

57. Broadcast A graph is a broadcast graph rooted
at if there is a path from to any

58. Broadcast A graph is a broadcast graph rooted
at if there is a path from to any

59. Broadcast A graph is a h-bounded-hop broadcast
graph rooted at if there is a path from to any
and the number of hops is limited by
4-bounded-hop
broadcast

60. Broadcast A graph is a k-h-broadcast graph if
it remains h-bounded-hop broadcast graph
even with the removal of up to nodes
2-4-bounded-hop
broadcast

61. Broadcast A graph is a k-h-broadcast graph if
it remains h-bounded-hop broadcast graph
even with the removal of up to nodes
2-vertex disjoint
paths under 4 hops

62. Broadcast A graph is a k-h-broadcast graph if
it remains h-bounded-hop broadcast graph
even with the removal of up to nodes
2-vertex disjoint
paths under 4 hops

63. Problem 3 (k-h-bounded broadcast)
Input: A set of transceivers in ,
root node and parameters
Output: A power assignment so that
is k-h-broadcast and is minimized

64. Planar Case The Algorithm Take a power assignment so that
is 1-h-bounded hop graph

65. Planar Case The Algorithm Take a power assignment so that
is 1-h-bounded hop graph
Let be a directed spanning tree of
Max distance –
h hops

66. Planar Case The Algorithm Take a power assignment so that
is 1-h-bounded hop graph
Let be a directed spanning tree of
Max distance –
h hops

67. Planar Case
The Algorithm
Add edges from to its grandchildren

68. Planar Case The Algorithm Add edges from to its grandchildren
Remove edges from the children of

69. Planar Case The Algorithm Add edges from to its grandchildren
Remove edges from the children of
Denote the resulting tree
Max distance –
h-1 hops

70. Planar Case The Algorithm No power is assigned yet!
We have a skeleton with a bounded cost

71. Planar Case
The Algorithm
Assign

72. Planar Case
The Algorithm
Assign to reach k closest neighbors.

73. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

74. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

75. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

76. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

77. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

78. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

79. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

80. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

81. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

82. Planar Case The Algorithm For each directed edge in
increase the range of all nodes in to
reach all nodes in

83. Planar Case
The Algorithm
Denote the resulting power assignment

84. Planar Case The Algorithm Denote the resulting power assignment
Along each path in there are
vertex-disjoint paths in of at most hops

85. Analysis
For a single edge in the power
increase of is bounded by:

86. Analysis
For a single edge in the power
increase of is bounded by:

87. Analysis For a single edge in the power increase of is bounded by:
Power assignment in

88. Planar Case Analysis For a single edge in the power
increase of is bounded by:
Node can be in many -s

89. Planar Case Analysis For a single edge in the power
increase of is bounded by:
Node can be in many -s, with many edges

90. Planar Case Analysis For a single edge in the power
increase of is bounded by:
Node can be in many -s, with many edges
But eventually only one ‘dominates’ the bound

91. Planar Case Analysis A node can be dominated only by the
outgoing edges of in

92. Planar Case Analysis A node can be dominated only by the
outgoing edges of in
A single edge can dominate at most
nodes (those in )

93. Analysis A node can be dominated only by the outgoing edges of in
A single edge can dominate at most
nodes (those in )
Recall,

94. Analysis A node can be dominated only by the outgoing edges of in
A single edge can dominate at most
nodes (those in )
As a result,

95. Analysis

96. Analysis
Due to

97. Analysis
PTAS due to Funke and Laue [24]

98. Analysis Let be the optimal power assignment
for the k-h-broadcast problem
From ,

99. Analysis Let be the optimal power assignment
for the k-h-broadcast problem
From ,
We need to bound

100. Analysis Let be a power assignment so that each
node has at least neighbors
Clearly,

101. Analysis - Hamiltonian cycle based power
assignment for the k-(n-1)-broadcast
problem, so that

102. Analysis - Hamiltonian cycle based power
assignment for the k-(n-1)-broadcast
problem, so that
In each node has at least neighbors

103. Analysis – Hamiltonian cycle based power
assignment for the k-(n-1)-broadcast
problem, so that
In each node has at least neighbors
From ,

104. k-(n-1)-broadcast
The Algorithm

105. k-(n-1)-broadcast
The Algorithm
Compute an MST of

106. k-(n-1)-broadcast The Algorithm Compute an MST of
Construct a Hamiltonian cycle with cost

107. k-(n-1)-broadcast The Algorithm Compute an MST of
Construct a Hamiltonian cycle with cost
Assign each node to reach nodes
in both directions of the cycle 
Example: k=4

108. k-(n-1)-broadcast The Algorithm Compute an MST of
Construct a Hamiltonian cycle with cost
Assign each node to reach nodes
in both directions of the cycle 
As a result,

109. k-(n-1)-broadcast
Hamiltonian Cycle Stage
Compute an MST of

110. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)

111. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)

112. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)

113. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)

114. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)

115. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)

116. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)
2-strongly connected
undirected graph

117. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)
Apply TSP-Approx (Bender and Checkuri)
Square of every
biconnected graph
is Hamiltonian (Fleischner)

118. k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of
Apply MST-Augmentation (Calinescu and Wan)
Apply TSP-Approx (Bender and Checkuri)
As a result,
The cost of the
Hamiltonian cycle

119. Back to k-h-broadcast Analysis - A simple approximation due to:
For any it holds:

120. Back to k-h-broadcast
Analysis -
Take as before

121. Back to k-h-broadcast Analysis - Take as before
The most distant node at most hops away

122. Back to k-h-broadcast Analysis - Take as before
The most distant node at most hops away
Assign the root to reach all!

123. Spanners What is a spanner? A spanning subgraph that approximates
some measure of the original graph

124. Spanners What is a spanner? A spanning subgraph that approximates
some measure of the original graph
E.g., Euclidean distance

125. Spanners What is a spanner? A spanning subgraph that approximates
some measure of the original graph
E.g., Euclidean distance

126. Spanners What is a spanner? A spanning subgraph that approximates
some measure of the original graph
E.g., Euclidean distance
Shortest path is at most
times longer than in

127. Spanners What is a spanner? A spanning subgraph that approximates
some measure of the original graph
E.g., Euclidean distance
Shortest path is at most
times longer than in
stretch factor

128. Spanners We propose two spanner optimization measures
Distance – reducing transmission latency
Energy – increasing network lifetime

129. Spanner optimization measures
The original graph
Let be the wireless nodes in the plane
Let be a weighted complete
graph
Weight function:
The Euclidean distance

130. Spanner optimization measures
The original graph
Let be the wireless nodes in the plane
Let be a weighted complete
graph
Weight function:
The Euclidean distance
Proportional to the energy required
to transmit from to

131. Spanner optimization measures
The original graph
Let be the wireless nodes in the plane
Let be a weighted complete
graph
Weight function:

132. Spanner optimization measures
The spanner
Let p be a power assignment

133. Spanner optimization measures
The spanner
Let p be a power assignment
is an induced directed graph,
where

134. Spanner optimization measures
The spanner
Let p be a power assignment
is an induced directed graph,
where
The cost:

135. Spanner optimization measures
Energy measure (stretch factor)
The energy of some path is its weight

136. Spanner optimization measures
Energy measure (stretch factor)
The energy of some path is its weight
The minimum energy from to in

137. Spanner optimization measures
Energy measure (stretch factor)
The energy of some path is its weight
The minimum energy from to in
The minimum energy from to in

138. Spanner optimization measures
Energy measure (stretch factor)
The energy stretch factor of

139. Spanner optimization measures
Energy measure (stretch factor)
The energy stretch factor of
We aim to minimize both and

140. Spanner optimization measures
Energy measure (stretch factor)
The energy stretch factor of
We aim to minimize both and
Clear benefits
Prolonged network lifetime
Low cost
Low interference…

141. Spanner optimization measures
Distance measure (stretch factor)
The distance of some path

142. Spanner optimization measures
Distance measure (stretch factor)
The distance of some path
The minimum distance from to in

143. Spanner optimization measures
Distance measure (stretch factor)
The distance stretch factor of

144. Spanner optimization measures
Distance measure (stretch factor)
The distance stretch factor of
We aim to minimize both and

145. Spanner optimization measures
Distance measure (stretch factor)
The distance stretch factor of
We aim to minimize both and
Clear benefits
Low delay in message delivery
Low cost

146. Main results Preliminaries We consider a random, independent, and
uniform node distribution in a unit square
The probability of our results converges to 1
as the number of nodes, n, increases

147. Main results Preliminaries We consider a random, independent, and
uniform node distribution in a unit square
Spanners make sense only if the induced
graph is strongly connected

148. Main results Preliminaries We consider a random, independent, and
uniform node distribution in a unit square
Spanners make sense only if the induced
graph is strongly connected
Otherwise, the stretch factor is infinity
Path does not exist

149. Main results Preliminaries We consider a random, independent, and
uniform node distribution in a unit square
Spanners make sense only if the induced
graph is strongly connected
The cost of any spanner is at least
the minimum cost of strong connectivity

150. Main results Preliminaries We consider a random, independent, and
uniform node distribution in a unit square
Spanners make sense only if the induced
graph is strongly connected
The cost of any spanner is at least
the minimum cost of strong connectivity
(denote this cost )

151. Main results
Energy spanner
Develop power assignment so that
where , ,

152. Main results Distance spanner Develop a power assignment so that

153. Lower bound on the cost of any spanner
Technical details
Some bounds…
Using [Zhang and Hou ‘05]
Lower bound on the cost of any spanner

154. Minimum spanning tree of G
Technical details
Some bounds…
Using [Zhang and Hou ‘05]
From [Kirousis et al. ‘00]
Minimum spanning tree of G
The weight of the tree

155. Technical details Some bounds… Using [Zhang and Hou ‘05]
From [Kirousis et al. ‘00]
Using [Berend et al. ‘08] & [Penrose ‘97]
Maximum length edge of MST

156. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]

157. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)

158. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U

159. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter

160. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter
Add the th node to U

161. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter
Add the th node to U
Remove first nodes from the diameter

162. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter
Add the th node to U
Remove first nodes from the diameter

163. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter
Add the th node to U
Remove first nodes from the diameter

164. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter
Add the th node to U
Remove first nodes from the diameter

165. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Take diameter
Add the th node to U
Remove first nodes from the diameter

166. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U

167. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Find the minimum spanning tree (MST)
Lemma: We can find nodes
so that any node is within
hops from some node in U
Let be a LAST rooted at
LAST [Khuller et al. ’93] is a spanning tree T of G, rooted at some
so that and

168. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Define the power assignment p so that

169. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Define the power assignment p so that
Let

170. Energy spanner [power assignment]
Technical details
Energy spanner [power assignment]
Define the power assignment p so that
Let
Finally,
For technical reasons

171. Energy spanner [cost analysis]
Technical details
Energy spanner [cost analysis]

172. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
If , there is a path P in G, so that
and

173. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
If , there is a path P in G, so that
and
Therefore, since for every u,
path P also exists in

174. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
Otherwise,

175. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
For any two nodes, s and t, the path in
first arrives at some LAST origin
by using the MST edges (denote P’)

176. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
For any two nodes, s and t, the path in
first arrives at some LAST origin
by using the MST edges (denote P’)

177. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
For any two nodes, s and t, the path in
first arrives at some LAST origin
by using the MST edges (denote P’)
second travels through the edges of
from to t (denote P’’)

178. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
For any two nodes, s and t, the path in
first arrives at some LAST origin
by using the MST edges (denote P’)
second travels through the edges of
from to t (denote P’’)

179. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
Otherwise,
We bound the weight of P’ and P’’
Lemma
Maximum edge of MST

180. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
Otherwise,
We bound the weight of P’ and P’’
A possible path goes through s

181. Energy spanner [stretch analysis]
Technical details
Energy spanner [stretch analysis]
Otherwise,
We bound the weight of P’ and P’’
Eventually,

182. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
The general idea is that for uniformly
distributed nodes, we can always find
“good” relays between any pair of nodes

183. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
To find these relays, for any pair of nodes,
s and t, we start a recursive process

184. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
To find these relays, for any pair of nodes,
s and t, we start a recursive process
At step i, we place adjacent disks along
the edge
The diameter of a disk at step i is

185. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
To find these relays, for any pair of nodes,
s and t, we start a recursive process
At step i, we place adjacent disks along
the edge
The diameter of a disk at step i is

186. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
To find these relays, for any pair of nodes,
s and t, we start a recursive process
At step i, we place adjacent disks along
the edge
The diameter of a disk at step i is

187. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
To find these relays, for any pair of nodes,
s and t, we start a recursive process
At step i, we place adjacent disks along
the edge
The diameter of a disk at step i is
The process ends when one of the disks has no relay nodes

188. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
To find these relays, for any pair of nodes,
s and t, we start a recursive process
At step i, we place adjacent disks along
the edge
Finally, we use relay nodes to obtain a path
We use an arbitrary node in each disk at the last non-empty step

189. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
The power assignment p is obtained by
ensuring that all paths are in

190. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
The power assignment p is obtained by
ensuring that all paths are in
Let be the constructed path from s to t

191. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
The power assignment p is obtained by
ensuring that all paths are in
Let be the constructed path from s to t
And be all
the edges from u in all the paths

192. Distance spanner [power assignment]
Technical details
Distance spanner [power assignment]
The power assignment p is obtained by
ensuring that all paths are in
Let be the constructed path from s to t
And be all
the edges from u in all the paths
Finally,

193. Distance spanner [analysis]
Technical details
Distance spanner [analysis]
Lemma: Let D be the maximum radius
disk which can be placed inside the unit
square, so there are no nodes in D
Let r be the radius of D

194. Distance spanner [analysis]
Technical details
Distance spanner [analysis]
Lemma: Let D be the maximum radius
disk which can be placed inside the unit
square, so there are no nodes in D
Then,
Let r be the radius of D

195. Distance spanner [analysis]
Technical details
Distance spanner [analysis]
From Lemma,

196. Distance spanner [analysis]
Technical details
Distance spanner [analysis]
From Lemma,
Clearly,

197. Extended wireless network model
Power assignment
Nodes have no fixed power supply
Each node has an initial battery charge b(v)
The lifetime of node v is
The network lifetime is

198. Wireless network model
Power assignment
Interference is a direct consequence of
a power assignment p ?

199. Wireless network model
Power assignment
Interference is a direct consequence of
a power assignment p
Several interference models exist
Number of nodes affected by transmission
Number of edges affected by transmission

200. Wireless network model
Power assignment
Interference is a direct consequence of
a power assignment p
Several interference models exist
Number of nodes affected by transmission
Number of edges affected by transmission
We combine several common models by
defining the interference to be

201. Main results Contribution We develop power assignment:
can be computed in time
where n is the number of nodes and

202. Technical details The construction The power assignment is computed
by dividing the unit square into k grid cells

203. Technical details The construction The power assignment is computed
by dividing the unit square into k grid cells
Then we compute a k shortest path trees rooted
at an arbitrary node in each cell

204. Technical details The construction The power assignment is computed
by dividing the unit square into k grid cells
Then we compute a k shortest path trees rooted
at an arbitrary node in each cell
The power assignment of nodes is increased to
assure all these k trees are included

205. Technical details The construction The power assignment is computed
by dividing the unit square into k grid cells
Then we compute a k shortest path trees rooted
at an arbitrary node in each cell
The power assignment of nodes is increased to
assure all these k trees are included
The power assignment of nodes is increased again
to be at least