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Power-Efficient Range Assignment in Ad-Hoc Wireless Networks

This paper addresses the power-efficient range assignment problem in ad-hoc wireless networks, which are crucial in applications like battlefield communications and disaster relief where no wired infrastructure exists. The study emphasizes the importance of energy conservation, as nodes are battery-operated. It explores various connectivity models—both symmetric and asymmetric—and presents optimization results, algorithms, and experimental studies to minimize total power costs while maintaining connectivity.

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Power-Efficient Range Assignment in Ad-Hoc Wireless Networks

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  1. ES0036 Power Efficient Range Assignment in Ad-hoc Wireless Networks E. Althous (MPI) G. Calinescu (IL-IT) I.I. Mandoiu (UCSD) S. Prasad (GSU) N. Tchervinsky (IL-IT) A. Zelikovsky (GSU)

  2. 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

  3. e e e d d d f f f c c c g g g b b b a a a Asymmetric Connectivity 1 1 1 1 3 1 Range radii 2 Strongly connected 1 1 1 1 3 1 Nodes transmit messages within a range depending on their battery power, e.g., agb cgb,d ggf,e,d,a 2 Message from “a” to “b” has multi-hop acknowledgement route

  4. e e d d f f c c g g b b Asymmetric Connectivity Symmetric Connectivity a a 1 1 1 1 1 1 1 1 3 1 1 2 2 2 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

  5. Power levels for k=2 16 d Distances Power assigned to a node = largest power requirement of incident edges k=2 total power p(T)=257 4 4 f 2 10 c 2 100 g 16 100 b 1 2 4 16 a 1 h e 4 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

  6. Results • 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] • Our results • General graph formulation • Improved approximation results • 5/3 +  • 11/6 for a practical greedy algorithm • New ILP formulation • Several swapping heuristics • Experimental study d

  7. 4 4 2 f 10 2 10 c 2 Power costs of nodes are yellow Total power cost of the tree is 68 g 13 12 b 13 12 2 12 a h 13 e 2 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 d

  8. n points    1 1 1 1+  1+  1+  Power cost of MST is n Power cost of OPT is n/2 (1+ ) + n/2  n/2 MST Algorithm Theorem: 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) 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)

  9. Greedy Fork Contraction Algorithm Fork F is the set of two adjacent edges Gain of fork F, gain(F), is by how much inserting of F and removing other two edges improves the power cost Input: Graph G=(V,E,cost) with edge costs Output:Low power-cost tree spanning V TfMST(G) HfRepeat forever Find fork F with maximum gain If gain(F) is non-positive, exit loop HfH U F TfT/F OutputT  H

  10. 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 4 f d 4 2 c 2 2 4 g 12 13 f 10 b 2 10 c 2 13 12 2 12 g 12 13 a 13 h b 2 e 13 12 15 4 Remove edge 10 power cost decrease = -6 d 2 12 a h 13 2 e 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

  11. Integer Linear Program Formulation yuv = range variable, =1 if for uv is maximum weight edge from u in tree T xuv = tree variable, =1 if uv is in tree T - choose a single power range - power range connects endpoints - connectivity requirement

  12. 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

  13. Percent Improvement Over MST

  14. Runtime (CPU seconds)

  15. Percent Improvement Over MST

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