Symmetric Connectivity With Minimum Power Consumption

in Radio Networks

G. Calinescu (IL-IT)G. Calinescu (IL-IT)I.I. Mandoiu (UCSD)I.I. Mandoiu (UCSD)A. Zelikovsky (GSU)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 thresholdsTransmission range = disk centered at the node

• Signal power falls inversely proportional to dk

Transmission range radius = kth root of node power

Asymmetric 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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Range radii

Message from “a” to “b” has multi-hop acknowledgement route

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Symmetric 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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Asymmetric Connectivity

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

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Symmetric Connectivity

Min-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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Power levels for k=2

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Previous Results

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• 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• 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

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

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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

MST Algorithm

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

(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)

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

n points

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

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p(Q) = 2c(T) = n (1+ )p(T) = n/2 (1+ 2)

n points

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

• 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 cost

obtained by – adding edges in fork F to T– deleting two longest edges in two cycles of T+F

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

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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 VTMST(G)H

Repeat foreverFind fork F with maximum gainIf gain(F) is non-positive, exit loopHH U FTT/F

Output T 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

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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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Percent Improvement Over MST

Runtime (CPU seconds)

Percent Improvement Over MST

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 Networks

G. Calinescu (IL-IT)G. Calinescu (IL-IT)I.I. Mandoiu (UCSD)I.I. Mandoiu (UCSD)A. Zelikovsky (GSU)A. Zelikovsky (GSU)