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1 Multicast Routing with Minimum Energy Cost in Ad hoc Wireless Networks Xiaohua Jia, Deying Li and Frankie Hung Dept of Computer Science, City Univ of Hong Kong

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2 Energy-efficiency in ad hoc networks Power function: p(u,v) = d α (u,v), 2 ≤ α ≤ 4 Two special features of radio transmission: 1) Broadcast in nature. 2) p(u,w) + p(w,v) < p(u,v), relaying messages by a third node may result in a smaller energy cost.

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3 Multicast Routing with Min Energy Cost Problem: Given a directed network graph G(V,A), and a multicast request (s, D), each node v has a given transmission power p(v), find a multicast tree T rooted at s and covering all nodes in D, such that: where NL(T) is the set of non-leaf nodes of T.

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4 BIP / MIP (Multicast Incremental Power) BIP is based on Prim’s MST algorithm. Starting from s, each time a node with least incremental power is added to the tree, until all nodes are in the tree. MIP tree is obtained by pruning the BIP tree. J.E. Wieselthier, G. D. Nguyen, and A. Ephremides, “On the Construction of Energy-Efficient Broadcast and Multicast Trees in Wireless Networks”, IEEE Infocom’00. s Related Work

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5 Related Wrok (Cont’d) Deying Li, Xiaohua Jia and Hai Liu, “Energy Efficient Broadcast Routing in Ad Hoc Wireless Networks”, IEEE Trans. On Mobile Computing, vol.3, no.2, 2004. An approximation algorithm based on node- weighted Steiner tree method was proposed and the approximation ratio is 2ln(n-1)+1.

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6 Theoretical Results The energy efficient multicast routing problem is NP-Hard and there is NO constant approximation ratio.

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7 Heuristic A. Transforming the problem to the Steiner tree problem The multicast routing problem is transformed to: finding a directed tree T’ in G’ that covers all nodes in D’ and the total weights of T’ (edge weights) is minimized.

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8 Heuristic B. Node-join-Tree method V i : set of outgoing neighbors of node i C = {s};// C: covered set U = D - V s ;// U : uncovered set X = V s ; // X: candidate set for covering nodes While (U ≠ ) do choose v i (X – C) with max(|V i ∩U|/p(i)); C = C {v i }; U = (U – V i ); X = X V i ;

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9 Heuristic C. Tree-join-Tree method The basic idea is: 1. Initially, each node in D is a subtree. 2. Find a node v V that uses the least amount of energy to link the roots of at least two subtrees and merge them into a bigger one (v becomes the root of the new subtree). 3. Repeat (2) until all subtrees form into a single tree rooted at s.

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10 Heuristic C. Tree-join-Tree method (cont’d) Orphan: a subtree whose root is not s. O: the set of orphans. min-quotient-cost(v) = Min S O { TreeCost(v,S)/|S|} for S O. O = {{i}| i D); While (O ≠ ) do choose v V with Min {min-quotient-cost(v)}; choose v V with Min {min-quotient-cost(v)}; link v to the roots of the orphans by shortest paths; link v to the roots of the orphans by shortest paths;

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11 Simulation Results N=50, R=0.2 Energy cost vs. multicast group size

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12 Simulation Results (Cont’d) M=10, R=0.2 Energy cost vs. network size

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13 Simulation Results (Cont’d) M=10, R = Energy cost vs. network size

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14 Simulation Results (Cont’d) M=10, N=50 Energy cost vs. transmission range

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15 Conclusions Investigated energy efficient multicast routing problem. Investigated energy efficient multicast routing problem. Three heuristic methods, Steiner tree based, node-join-tree, and tree-join-tree, were proposed. Three heuristic methods, Steiner tree based, node-join-tree, and tree-join-tree, were proposed. The node-join-tree method can be extended to a distributed version. It uses only the neighbour information to construct the multicast tree. The node-join-tree method can be extended to a distributed version. It uses only the neighbour information to construct the multicast tree. Simulation results suggested the simple node-join-tree method performs the best among the other methods. Simulation results suggested the simple node-join-tree method performs the best among the other methods.

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