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Power Efficient Range Assignment Power Efficient Range Assignment in Ad-hoc Wireless Networksin Ad-hoc Wireless Networks

E. Althous (MPI)E. Althous (MPI)

G. Calinescu (IL-IT)G. Calinescu (IL-IT)

I.I. Mandoiu (UCSD)I.I. Mandoiu (UCSD)

S. Prasad (GSU)S. Prasad (GSU)

N. Tchervinsky (IL-IT)N. Tchervinsky (IL-IT)

A. Zelikovsky (GSU)A. Zelikovsky (GSU)

ES0036ES0036

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Ad Hoc Wireless NetworksAd Hoc Wireless Networks

• Applications in battlefield, disaster relief, etc.• No wired infrastructure• Battery operated power conservation critical• Omni-directional antennas + Uniform power detection

thresholdsTransmission range = disk centered at the node

• Signal power falls inversely proportional to dk

Transmission range radius = kth root of node power

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Asymmetric ConnectivityAsymmetric Connectivity

Strongly connected

Nodes transmit messages within a range depending on their battery power, e.g., ab cb,d gf,e,d,a

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Symmetric ConnectivitySymmetric Connectivity

• Per link acknowledgements symmetric connectivity• Two nodes are symmetrically connected iff they are within transmission

range of each other

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

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

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Min-power Symmetric Connectivity ProblemMin-power Symmetric Connectivity Problem

• Given: set S of 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

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

k=2 total power p(T)=257a

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ResultsResults

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• 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]

• Our results – General graph formulation

– Improved approximation results• 5/3 + • 11/6 for a practical greedy algorithm

– New ILP formulation

– Several swapping heuristics

– Experimental study

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Graph FormulationGraph 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 costd

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Power costs of nodes arePower costs of nodes are yellow yellowTotal power cost of the tree isTotal power cost of the tree is 68 68

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MST AlgorithmMST Algorithm

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

Proof(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)

1+ 1+ 1+

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Power cost of MST is n

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

n points

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Greedy Fork Contraction AlgorithmGreedy Fork Contraction Algorithm

Fork F is the set of two adjacent edges

Gain of fork F, gain(F), is by how much inserting of F and removing other two edges improves the power cost

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

Output: Low power-cost tree spanning V

TMST(G)

HRepeat forever

Find fork F with maximum gain

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

HH U F

TT/F

Output T H

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Edge Swapping HeuristicEdge Swapping Heuristic

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Remove edge 10 Remove edge 10 power cost decrease = -6power cost decrease = -6

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

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

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© Yamacraw, Fall 2002

Integer Linear Program FormulationInteger Linear Program Formulation

yuv = range variable, =1 if for uv is maximum weight edge from u in tree T

xuv = tree variable, =1 if uv is in tree T

- choose a single power range

- power range connects endpoints

- connectivity requirement

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Experimental StudyExperimental 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

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Percent Improvement Over MSTPercent Improvement Over MST

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Runtime (CPU seconds)Runtime (CPU seconds)

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Percent Improvement Over MSTPercent Improvement Over MST