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Symmetric Connectivity With Minimum Power Consumption in Radio Networks

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### Symmetric Connectivity With Minimum Power Consumptionin 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

- 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

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Message from “a” to “b” has

multi-hop acknowledgement route

Asymmetric Connectivity1

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Range radii

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Strongly connected

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

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Asymmetric Connectivity

Symmetric Connectivity

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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

Power levels for k=2

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Distances

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

k=2 total power p(T)=257

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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

- 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]

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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

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Power costs of nodes are yellow

Total power cost of the tree is 68

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Graph FormulationPower 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

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n points

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Power cost of MST is n

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

MST AlgorithmTheorem: 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) uv~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)

n points

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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 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

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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

- 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

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

Output:Low power-cost tree spanning V

TfMST(G)

HfRepeat forever

Find fork F with maximum gain

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

HfH U F

TfT/F

OutputT H

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

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

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Remove edge 10

power cost decrease = -6

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

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

G. Calinescu (IL-IT)

I.I. Mandoiu (UCSD)

A. Zelikovsky (GSU)

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