Symmetric Minimum Power Connectivity in Radio Networks

1
Symmetric Minimum Power Connectivity in Radio
Networks

• A. Zelikovsky (GSU)
• httpwww.cs.gsu.edu/cscazz
• Joint work with
• G. Calinescu, (Illinois
  IT)
• I. I. Mandoiu (UCSD)

2
Overview

• Connectivity in Radio Networks
• Symmetric Connectivity in Radio Networks
• Symmetric Minimum Power Problem (SPP)
• Graph Formulation of SPP
• Minimum Spanning Tree Algorithm
• Edge Swapping Heuristic
• Gain of Forks
• Greedy Algorithm
• Approximation Ratios
• Implementation Results

3
Connectivity in Radio Networks
1
1
1
1
3
1
Ranges
2
Nodes are 2-connected
1
1
1
1
3
1
Nodes transmit messages within a range depending
on their battery power. i.e., agb cgb,d
ggf,e,d,a
2
message from a to b has multi-hop
acknowledgement route.
Acknowledgement Problem
4
Symmetric Connectivity in Radio Networks

• Symmetric Connection ? 1 hop acknowledgement
• Two points are symmetrically connected
• ? they are in the range of each other

Asymmetric Connectivity
Symmetric Connectivity
1
1
1
1
1
1
1
1
3
1
1
2
2
2
Node a cannot get acknowledgement directly from
b
Increase range on b by 1 and decrease g by 2.
5
Symmetric Minimum Power Problem (SMPP)

• Range is proportional to the square root of power
• Power to connect (x1,y1 ) to (x2,y2) is ?
  (x2-x1)2(y2-y1)2
• Symmetric Minimum Power Problem (SMPP)
• Given a set S of points in Euclidean plane
• Find assignments of powers to each point such
  that
• set S becomes symmetrically connected
• total power is minimized

powers
16
d
distances
To support connectivity tree we should assign
the total power of p(T) 257 The power
assigned to node should cover the longest
incident edge!
4
4
f
2
10
c
2
100
g
100
16
b
1
2
4
16
a
1
h
e
4
6
Graph Formulation of SMPP

• Power cost of a node is the maximum cost of the
  incident edge
• Power cost of a tree is the sum of power costs of
  its nodes
• Symmetric Minimum Power Problem in graphs
• Given a set of points in a graph G(V,E,c),
  where c(e) is the power necessary to cover the
  length of the edge e
• Find a spanning tree in the graph with a minimum
  power cost.

d
4
4
2
f
10
2
10
c
2
g
13
12
b
13
12
2
12
a
h
13
e
2
Power costs of nodes are blue Total cost of the
tree is 68
7
MST Algorithm

• Find the minimum spanning tree (MST) of G.
• Implement using Prims Algorithm
• Theorem The power cost of the MST is at most 2
  OPT
• Proof
• power cost of optimal spanning tree gt its cost
• power cost of a tree is at most twice its cost
• Worst- case example

n points
?
?
?
1
1
1
1 ?
1 ?
1 ?
Power cost of blue MST is n Power cost of red
OPT tree is n/2 (1 ?) n/2 ? ? n/2
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Edge Swapping Heuristic

• For each edge do
• Delete an edge
• Connect with min increase in power-cost
• Undo previous steps if no gain

d
4
4
2
f
4
2
c
2
g
13
12
b
13
12
2
12
a
13
h
2
e
Remove edge 10 power cost decrease -6
4
d
4
f
2
2
4
c
2
g
13
12
b
13
15
15
2
12
15
a
h
2
e
Reconnect components with min increase in
power-cost 5
9
Gain of Forks

• A fork F is a pair of edges sharing an endpoint
• A gain of a fork w.r.t. a given tree T is the
  decrease in power cost obtained by
• adding fork edges F
• deleting two longest edges in two cycles of TF

8
d
2
8
f
13(3)
2
10
c
2
2(-10)
g
10
13
b
10
13
2
12
a
h
13 (3)
13 (1)
2
e
Fork with center a decreases the power-cost by
the gain 10-3-1-33
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Greedy Algorithm

• Input Graph G(V,E,cost) with edge costs
• Output Low power-cost tree all vertices V
• TfMST(G)
• HfGRepeat forever
• Find fork F with maximum rgainT(F)
• If r is non-positive, exit loop
• HfH U F
• VfV/F
• Output Union of remaining MST and H

11
Approximation Ratios

• Symmetric Minimum Power Problem in graphs is
  equivalent to Steiner Tree Problem in graphs
• Theorem
• all forks have non-positive gain w.r.t. to a
  tree T ?
• power-cost (T) ? 5/3 OPT
• Theorem
• The approximation ratio of greedy algorithm
  is at most 11/6
• Theorem
• There is an algorithm with approximation
  ratio at most 1.64

12
Implementation Results

• For random instances up to 100 points
• The average loss in power cost of MST w.r.t. OPT
• 19
• The average improvement over the MST algorithm is
• 2 for greedy algorithm
• 6.5 for edge swapping heuristic
• 8 for edge swapping heuristic followed by greedy