1. © Yamacraw, Fall 2002 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) ES0036

2. © Yamacraw, Fall 2002 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 d k  Transmission range radius = kth root of node power

3. © Yamacraw, Fall 2002 Asymmetric Connectivity Strongly connected Nodes transmit messages within a range depending on their battery power, e.g., 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 Range radii Message from “a” to “b” has multi-hop acknowledgement route a 2 3 1 1 b d g f e c 1 1 1

4. © Yamacraw, Fall 2002 Symmetric Connectivity Per link acknowledgements  symmetric connectivity Two nodes are symmetrically connected iff they are within transmission range of each other Node “a” cannot get acknowledgement directly from “b” a 2 3 1 1 b d g f e c 1 1 1 Asymmetric Connectivity Increase range of “b” by 1 and decrease “g” by 2 a 2 1 1 1 b d g f e c 1 1 2 Symmetric Connectivity

5. © Yamacraw, Fall 2002 Min-power Symmetric Connectivity Problem Given: set S of 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 Power assigned to a node = largest power requirement of incident edges k=2  total power p(T)=257 a b d g f e c 4 2 h 2 4 2 1 10 100 16 4 4 1 Power levels for k=2 Distances

6. © Yamacraw, Fall 2002Results d 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

7. © Yamacraw, Fall 2002 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 a b g f e c 12 2 h 2 4 2 13 10 13 12 13 12 4 2 2 Power costs of nodes are yellow Total power cost of the tree is 68

8. © Yamacraw, Fall 2002 MST Algorithm Theorem: The power cost of the MST is at most 2 OPT Proof (1)power cost of any tree is at most twice its cost p(T) =  u max v~u c(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) 1+  1   11  Power cost of MST is n Power cost of OPT is n/2 (1+  ) + n/2   n/2 n points

9. © Yamacraw, Fall 2002 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 T  MST(G) H   Repeat forever Find fork F with maximum gain If gain(F) is non-positive, exit loop H  H U F T  T/F Output T  H

10. © Yamacraw, Fall 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

11. © Yamacraw, Fall 2002 Integer Linear Program Formulation y uv = range variable, =1 if for uv is maximum weight edge from u in tree T x uv = tree variable, =1 if uv is in tree T - choose a single power range - power range connects endpoints - connectivity requirement

12. © Yamacraw, Fall 2002 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. © Yamacraw, Fall 2002 Percent Improvement Over MST

14. © Yamacraw, Fall 2002 Runtime (CPU seconds)

15. © Yamacraw, Fall 2002 Percent Improvement Over MST