Power Efficient Range Assignment in Ad-hoc Wireless Networks

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