1. © Yamacraw, 2002 Symmetric Minimum Power Connectivity in Radio Networks A. Zelikovsky (GSU) http: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)

2. © Yamacraw, 2002Overview 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

3. © Yamacraw, 2002 Connectivity in Radio Networks Nodes are 2-connected Nodes transmit messages within a range depending on their battery power. i.e., a  b c  b,d g  f,e,d,a a 1 2 3 1 1 1 1 b d g f e c b a c d g f e message from “a” to “b” has multi-hop acknowledgement route. Ranges a 2 3 1 1 b d g f e c 1 1 1 Acknowledgement Problem:

4. © Yamacraw, 2002 Symmetric 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. a 2 3 1 1 b d g f e c 1 1 1 a 2 1 1 1 b d g f e c 1 1 2 Asymmetric ConnectivitySymmetric Connectivity

5. © Yamacraw, 2002 Symmetric Minimum Power Problem (SMPP) Range is proportional to the square root of power Power to connect (x 1,y 1 ) to (x 2,y 2 ) is  (x 2 -x 1 ) 2 +(y 2 -y 1 ) 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)= 257 The power assigned to node should cover the longest incident edge! a b d g f e c 4 2 h 2 4 2 1 10 100 16 4 4 1 powers distances

6. © Yamacraw, 2002 Graph 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. a b d g f e c 12 2 h 2 4 2 13 10 13 12 13 12 4 2 2 Power costs of nodes are blue Total cost of the tree is 68

7. © Yamacraw, 2002 MST 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 1+  1   11  Power cost of blue MST is n Power cost of red OPT tree is n/2 (1+  ) + n/2   n/2 n points

8. © Yamacraw, 2002 Edge Swapping Heuristic a b d g f e c 12 2 h 2 4 2 13 Remove edge 10 power cost decrease = -6 Reconnect components with min increase in power-cost = +5 a b d g f e c 12 2 h 2 4 2 13 For each edge do Delete an edge Connect with min increase in power-cost Undo previous steps if no gain 15 4 13 15 4 12 4 4 2 2 2 a b d g f e c 2 h 2 4 2 13 15 10 13 12 13 12 4 2 2 2

9. © Yamacraw, 2002 Gain 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 a b d g f e c 12 2 h 2 8 2 10 13 10 12 10 12 8 2 2 13 a b d g f e c 12 2 h 2 8 2 10 13 10 13(+3) 10 13 (+1) 13 (+3) 2(-10) 8 2 2 13

10. © Yamacraw, 2002 Greedy Algorithm Input: Graph G=(V,E,cost) with edge costs Output: Low power-cost tree all vertices V T  MST(G) H  G Repeat forever Find fork F with maximum r=gain T (F) If r is non-positive, exit loop H  H U F V  V/F Output Union of remaining MST and H

11. © Yamacraw, 2002 Approximation 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

12. © Yamacraw, 2002 Implementation 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