Symmetric Connectivity With Minimum Power Consumption in Radio Networks

1. Symmetric Connectivity With Minimum Power Consumptionin Radio Networks G. Calinescu (IL-IT) I.I. Mandoiu (UCSD) 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 1 1 1 1 3 1 2 Message from “a” to “b” has multi-hop acknowledgement route Asymmetric Connectivity 1 1 1 1 3 1 Range radii 2 Strongly connected Nodes transmit messages within a range depending on their battery power, e.g., agb cgb,d ggf,e,d,a

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. 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] d

7. Our results • General graph formulation • Similarity to Steiner tree problem • t-restricted decompositions • Improved approximation results • 1+ln2 +   1.69+  • 15/8 for a practical greedy algorithm • Efficient exact algorithm for Min-Power Symmetric Unicast • Experimental study

8. 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

9. 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)

10. n points    1 1 1 1+  1+  1+  p(Q) = 2c(T) = n (1+ ) p(T) = n/2 (1+ 2) Size-restricted Tree Decompositions • A t-restricted decomposition Q of tree T is a partition into edge-disjoint sub-trees with at most t vertices • Power-cost of Q = sum of power costs of sub-trees • t = supT min {p(Q):Q t-restricted decomposition of T} / p(T) • E.g., 2 = 2

11. Size-restricted Tree Decompositions Theorem:For every T and t, there exists a 2t-restricted decomposition Q of T such that p(Q)  (1+1/t) p(T) • t  1 + 1 / log k • t  1 when t   • Theorem:For every T, there exists a 3-restricted decomposition Q of T such that p(Q)  7/4 p(T) • 3  7/4

12. 8 8 d d 2 2 8 8 f f 10 13(+3) 2 2 10 10 c c 2 2 2(-10) g g 12 10 10 13 13 b b 10 10 13 13 2 2 12 12 a a h h 10 13 (+3) 2 12 2 13 (+1) e e Fork {ac,ab} decreases the power-cost by gain = 10-3-1-3=3 Gain of a Sub-tree • t-restricted decompositions are the analogue of t-restricted Steiner trees • Fork = sub-tree of size 2 = pair of edges sharing an endpoint • The gain of fork F w.r.t. a given tree T = decrease in power costobtained by • adding edges in fork F to T • deleting two longest edges in two cycles of T+F

13. Approximation Algorithms • For a sub-tree H of G=(V,E) the gain w.r.t. spanning tree T is defined by gain(H) = 2 c(T) – 2 c(T/H) – p(H) where G/H = G with H contracted to a single vertex • [Camerini, Galbiati & Maffioli 92 / Promel & Steger 00] • 3 +  7/4 + approximation • t-restricted relative greedy algorithm [Zelikovsky 96] • 1+ln2 +   1.69 +  approximation • Greedy triple (=fork) contraction algorithm [Zelikovsky 93] • (2 + 3) / 2  15/8 approximation

14. Greedy Fork Contraction Algorithm 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

15. 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

16. 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

17. Percent Improvement Over MST

18. Runtime (CPU seconds)

19. Percent Improvement Over MST

20. Summary and Ongoing Research • Graph-based algorithms handle practical constraints • Obstacles, power level upper-bounds • Improved approximation algorithms based on similarity to Steiner tree problem in graphs • Ideas extend to Min-Power Symmetric Multicast • Ongoing research -- Every tree has 3-decomposition with at most 5/3 times larger power-cost • 5/3+ approximation using [Camerini et al. 92 / Promel & Steger 00] • 11/6 approximation factor for greedy fork-contraction algorithm

21. Symmetric Connectivity With Minimum Power Consumptionin Radio Networks G. Calinescu (IL-IT) I.I. Mandoiu (UCSD) A. Zelikovsky (GSU)