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

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

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 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 b d g f e c 1 1 1

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 b d g f e c Asymmetric Connectivity Increase range of “b” by 1 and decrease “g” by 2 a b d g f e c Symmetric Connectivity

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 Power levels for k=2 Distances

Previous Results d 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 Similarity to Steiner tree problem –t-restricted decompositions Improved approximation results –1+ln2 +    – 15/8 for a practical greedy algorithm Efficient exact algorithm for Min-Power Symmetric Unicast Experimental study

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 Power costs of nodes are yellow Total power cost of the tree is 68

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

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 = sup T min {p(Q):Q t-restricted decomposition of T} / p(T) E.g.,  2 = 2 1+  1   11  p(Q) = 2c(T) = n (1+  ) p(T) = n/2 (1+ 2  ) n points

Size-restricted Tree Decompositions Theorem: For every T and t, there exists a 2 t -restricted decomposition Q of T such that p(Q)  (1+1/t) p(T)   t  /  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

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 cost obtained by –adding edges in fork F to T –deleting two longest edges in two cycles of T+F Fork {ac,ab} decreases the power-cost by Fork {ac,ab} decreases the power-cost by gain = =3 a b d g f e c 12 2 h a b d g f e c 12 2 h (+3) (+1) 13 (+3) 2(-10)

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 +    approximation Greedy triple (=fork) contraction algorithm [Zelikovsky 93]  (  2 +  3 ) / 2  15/8 approximation

Greedy Fork Contraction Algorithm 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

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

Edge Swapping Heuristic a b d g f e c 12 2 h 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 For each edge do Delete an edge Connect with min increase in power-cost Undo previous steps if no gain a b d g f e c 2 h

Percent Improvement Over MST

Runtime (CPU seconds)

Percent Improvement Over MST

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

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