© Yamacraw, 2002

Symmetric Minimum Power Connectivity Symmetric Minimum Power Connectivity in Radio Networksin Radio Networks

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

http:www.cs.gsu.edu/~cscazzhttp:www.cs.gsu.edu/~cscazz

Joint work with Joint work with

G. Calinescu, (Illinois IT)G. Calinescu, (Illinois IT)

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

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OverviewOverview• 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

© Yamacraw, 2002

Connectivity in Radio NetworksConnectivity in Radio Networks

Nodes are 2-connected

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

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

Ranges

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Acknowledgement Problem:

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Symmetric Connectivity in Radio NetworksSymmetric Connectivity in Radio Networks

• Symmetric Connection 1 hop acknowledgement• Two points are symmetrically connected

they are in the range of each other

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

Increase range on “b” by 1 and decrease “g” by 2.

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Asymmetric Connectivity Symmetric Connectivity

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Symmetric Minimum Power Problem (SMPP)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

To support connectivity tree we should assign the total power of p(T)= 257The power assigned to node should cover the longest incident edge!

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Graph Formulation of SMPPGraph 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.

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Power costs of nodes are Power costs of nodes are blueblueTotal cost of the tree is Total cost of the tree is 68 68

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MST AlgorithmMST Algorithm

• Find the minimum spanning tree (MST) of G.• Implement using Prim’s Algorithm

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

• Proof: – power cost of optimal spanning tree > its cost

– power cost of a tree is at most twice its cost

• Worst- case example

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Power cost of blue MST is n

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

n points

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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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Gain of ForksGain of Forks

Fork with center a decreases the power-cost by the Fork with center a decreases the power-cost by the gain = 10-3-1-3=3

• 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 T+F

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Greedy AlgorithmGreedy Algorithm

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

Output: Low power-cost tree all vertices V

TMST(G)

HGRepeat forever

Find fork F with maximum r=gainT(F)

If r is non-positive, exit loop

HH U F

VV/F

Output Union of remaining MST and H

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Approximation RatiosApproximation 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

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Implementation ResultsImplementation 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