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Symmetric Minimum Power Connectivity in Radio Networks

Symmetric Minimum Power Connectivity in Radio Networks. A. Zelikovsky (GSU) http:www.cs.gsu.edu/~cscazz Joint work with G. Calinescu, (Illinois IT) I. I. Mandoiu (UCSD). Overview. Connectivity in Radio Networks

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Symmetric Minimum Power Connectivity in Radio Networks

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  1. Symmetric Minimum Power Connectivity in Radio Networks A. Zelikovsky (GSU) http:www.cs.gsu.edu/~cscazz Joint work with G. Calinescu, (Illinois IT) I. I. Mandoiu (UCSD)

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

  4. e e d d f f c c g g b b a a Symmetric Connectivity in Radio Networks • Symmetric Connection  1 hop acknowledgement • Two points are symmetrically connected  they are in the range of each other Asymmetric Connectivity Symmetric Connectivity 1 1 1 1 1 1 1 1 3 1 1 2 2 2 Node “a” cannot get acknowledgement directly from “b” Increase range on “b” by 1 and decrease “g” by 2.

  5. 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 Sof points in Euclidean plane • Find assignments of powers to each point such that • set Sbecomes symmetrically connected • total power is minimized powers 16 d distances 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! 4 4 f 2 10 c 2 100 g 16 100 b 1 2 4 16 a 1 h e 4

  6. Graph Formulation of SMPP Power cost of a nodeis the maximum cost of the incident edge Power cost of a treeis 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. d 4 4 2 f 10 2 10 c 2 g 13 12 b 13 12 2 12 a h 13 e 2 Power costs of nodes are blue Total cost of the tree is 68

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

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

  9. 8 d 2 8 f 10 2 10 c 2 g 12 10 13 b 10 13 2 12 a h 10 2 12 e Gain of Forks • 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 costobtained by • adding fork edges F • deleting two longest edges in two cycles of T+F 8 d 2 8 f 13(+3) 2 10 c 2 2(-10) g 10 13 b 10 13 2 12 a h 13 (+3) 2 13 (+1) e Fork with center a decreases the power-cost by the gain = 10-3-1-3=3

  10. Greedy Algorithm Input: Graph G=(V,E,cost) with edge costs Output:Low power-cost tree all vertices V TfMST(G) HfGRepeat forever Find fork F with maximum r=gainT(F) If r is non-positive, exit loop HfH U F VfV/F Output Union of remaining MST and H

  11. Approximation Ratios • Symmetric Minimum Power Problem in graphsis 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. 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

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