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Symmetric Connectivity With Minimum Power Consumption in Radio Networks. G. Calinescu (IL-IT) I.I. Mandoiu (UCSD) A. Zelikovsky (GSU). Ad Hoc Wireless Networks. Applications in battlefield, disaster relief, etc. No wired infrastructure Battery operated  power conservation critical

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Symmetric connectivity with minimum power consumption in radio networks
Symmetric Connectivity With Minimum Power Consumptionin Radio Networks

G. Calinescu (IL-IT)

I.I. Mandoiu (UCSD)

A. Zelikovsky (GSU)


Ad hoc wireless networks
Ad Hoc Wireless Networks

  • Applications in battlefield, disaster relief, etc.

  • No wired infrastructure

  • Battery operated  power conservation critical

  • Omni-directional antennas + Uniform power detection thresholds

    Transmission range = disk centered at the node

  • Signal power falls inversely proportional to dk

    Transmission range radius = kth root of node power


Asymmetric connectivity

e

e

e

d

d

d

f

f

f

c

c

c

g

g

g

b

b

b

a

a

a

1

1

1

1

3

1

2

Message from “a” to “b” has

multi-hop acknowledgement route

Asymmetric Connectivity

1

1

1

1

3

1

Range radii

2

Strongly connected

Nodes transmit messages within a range depending on their battery power, e.g., agb cgb,d ggf,e,d,a


Symmetric connectivity

e

e

d

d

f

f

c

c

g

g

b

b

Asymmetric Connectivity

Symmetric Connectivity

a

a

1

1

1

1

1

1

1

1

3

1

1

2

2

2

Node “a” cannot get acknowledgement directly from “b”

Increase range of “b” by 1 and decrease “g” by 2

Symmetric Connectivity

  • Per link acknowledgements  symmetric connectivity

  • Two nodes are symmetrically connected iff they are within transmission range of each other


Min power symmetric connectivity problem

Power levels for k=2

16

d

Distances

Power assigned to a node = largest power requirement of incident edges

k=2 total power p(T)=257

4

4

f

2

10

c

2

100

g

16

100

b

1

2

4

16

a

1

h

e

4

Min-power Symmetric Connectivity Problem

  • Given: set Sof nodes (points in Euclidean plane), and coefficient k

  • Find: power levels for each node s.t.

    • There exist symmetrically connected paths between any two nodes of S

    • Total power is minimized


Previous results
Previous Results

  • 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]

d


Our results
Our results

  • General graph formulation

  • Similarity to Steiner tree problem

    • t-restricted decompositions

  • Improved approximation results

    • 1+ln2 +   1.69+ 

    • 15/8 for a practical greedy algorithm

  • Efficient exact algorithm for Min-Power Symmetric Unicast

  • Experimental study


Graph formulation

4

4

2

f

10

2

10

c

2

Power costs of nodes are yellow

Total power cost of the tree is 68

g

13

12

b

13

12

2

12

a

h

13

e

2

Graph Formulation

Power cost of a node = maximum cost of the incident edge

Power cost of a tree = sum of power costs of its nodes

Min-Power Symmetric Connectivity Problem in Graphs:

Given: edge-weighted graph G=(V,E,c), where c(e) is the power required to establish link e

Find:spanning tree with a minimum power cost

d


Mst algorithm

n points

1

1

1

1+ 

1+ 

1+ 

Power cost of MST is n

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

MST Algorithm

Theorem: The power cost of the MST is at most 2 OPT

Proof

  • 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)


Size restricted tree decompositions

n points

1

1

1

1+ 

1+ 

1+ 

p(Q) = 2c(T) = n (1+ )

p(T) = n/2 (1+ 2)

Size-restricted Tree Decompositions

  • A t-restricted decomposition Q of tree T is a partition into edge-disjoint sub-trees with at most t vertices

  • Power-cost of Q = sum of power costs of sub-trees

  • t = supT min {p(Q):Q t-restricted decomposition of T} / p(T)

  • E.g., 2 = 2


Size restricted tree decompositions1
Size-restricted Tree Decompositions

Theorem:For every T and t, there exists a 2t-restricted decomposition Q of T such that p(Q)  (1+1/t) p(T)

  • t  1 + 1 / log k

  • t  1 when t  

  • Theorem:For every T, there exists a 3-restricted decomposition Q of T such that p(Q)  7/4 p(T)

    • 3  7/4


Gain of a sub tree

8

8

d

d

2

2

8

8

f

f

10

13(+3)

2

2

10

10

c

c

2

2

2(-10)

g

g

12

10

10

13

13

b

b

10

10

13

13

2

2

12

12

a

a

h

h

10

13 (+3)

2

12

2

13 (+1)

e

e

Fork {ac,ab} decreases the power-cost by gain = 10-3-1-3=3

Gain of a Sub-tree

  • t-restricted decompositions are the analogue of t-restricted Steiner trees

  • Fork = sub-tree of size 2 = pair of edges sharing an endpoint

  • The gain of fork F w.r.t. a given tree T = decrease in power costobtained by

    • adding edges in fork F to T

    • deleting two longest edges in two cycles of T+F


Approximation algorithms
Approximation Algorithms

  • For a sub-tree H of G=(V,E) the gain w.r.t. spanning tree T is defined by

    gain(H) = 2 c(T) – 2 c(T/H) – p(H)

    where G/H = G with H contracted to a single vertex

  • [Camerini, Galbiati & Maffioli 92 / Promel & Steger 00]

    • 3 +  7/4 + approximation

  • t-restricted relative greedy algorithm [Zelikovsky 96]

    • 1+ln2 +   1.69 +  approximation

  • Greedy triple (=fork) contraction algorithm [Zelikovsky 93]

    • (2 + 3) / 2  15/8 approximation


Greedy fork contraction algorithm
Greedy Fork Contraction Algorithm

Input: Graph G=(V,E,cost) with edge costs

Output:Low power-cost tree spanning V

TfMST(G)

HfRepeat forever

Find fork F with maximum gain

If gain(F) is non-positive, exit loop

HfH U F

TfT/F

OutputT  H


Experimental study
Experimental Study

  • Random instances up to 100 points

  • Compared algorithms

    • branch and cut based on novel ILP formulation [Althaus et al. 02]

    • Greedy fork-contraction

    • Incremental power-cost Kruskal

    • Edge swapping

    • Delaunay graph versions of the above


Edge swapping heuristic
Edge Swapping Heuristic

  • For each edge do

    • Delete an edge

    • Connect with min increase in power-cost

    • Undo previous steps if no gain

4

d

4

2

4

f

d

4

2

c

2

2

4

g

12

13

f

10

b

2

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c

2

13

12

2

12

g

12

13

a

13

h

b

2

e

13

12

15

4

Remove edge 10

power cost decrease = -6

d

2

12

a

h

13

2

e

4

f

2

2

4

c

2

g

12

13

b

13

15

15

2

12

15

a

h

2

e

Reconnect components with min increase in power-cost = +5





Summary and ongoing research
Summary and Ongoing Research

  • Graph-based algorithms handle practical constraints

    • Obstacles, power level upper-bounds

  • Improved approximation algorithms based on similarity to Steiner tree problem in graphs

  • Ideas extend to Min-Power Symmetric Multicast

  • Ongoing research

    -- Every tree has 3-decomposition with at most 5/3 times larger power-cost

    • 5/3+ approximation using [Camerini et al. 92 / Promel & Steger 00]

    • 11/6 approximation factor for greedy fork-contraction algorithm


Symmetric connectivity with minimum power consumption in radio networks1
Symmetric Connectivity With Minimum Power Consumptionin Radio Networks

G. Calinescu (IL-IT)

I.I. Mandoiu (UCSD)

A. Zelikovsky (GSU)


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