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WiOpt’03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks March 3-5, 2003, INRIA Sophia-Antipolis, France Session : Energy Efficiency Paper : Energy-aware Broadcasting in Wireless Networks Ioannis Papadimitriou Co-Author : Prof. Leonidas Georgiadis ARISTOTLE UNIVERSITY OF THESSALONIKI, GREECE FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Division of Telecommunications
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France2 Presentation Plan 1.Introduction 2.Definitions and Problem Formulation 3.Optimization Algorithms 4.Generalizations 5.Numerical Results 6.Extensions – Issues for Further Study
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France3 1. Introduction Wireless Networks Motivation : Dissemination of information Broadcasting Battery-operated Energy Conservation Assumptions : Omnidirectional antennas Node-based environment Varying transmission powers Directed graph model Common approach : Min-sum (of node powers consumption) criterion Our setup : Min-max and Lexicographic node power optimization problem Generalization : Lexicographic optimization under more general cost functions of node powers
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France4 2. Definitions and Problem Formulation A.Wireless Communication Model Network representation : Directed graph G (N, L) Required power for transmission over link l (link cost) c l > 0 If node i transmits with power p, it can reach any node j for which c (i, j) ≤ p Determining broadcast transmissions : Define an r-rooted spanning tree T = (N, L T ) Node n transmits with power, where if n is a leaf Example : T 1 : {(A,B), (B,C), (B,D)} T 2 : {(A,B), (A,C), (B,D)} Same leaf nodes C, D Set I : Set II :
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France5 2. Definitions and Problem Formulation B.Optimal Broadcast Trees A spanning tree T induces a vector of node powers Objective I : Min-max node power optimization Find a tree : for any spanning tree T of G Objective II : Lexicographic node power optimization Find a tree T * : for any spanning tree T of G Stronger optimization criterion Provided that we minimize the i th maximum consumed node power, we also seek to minimize the (i+1) th maximum No node in the network consumes excessive power For example, vector (3,4,8) is lexicographically smaller than (5,8,2)
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France6 2. Definitions and Problem Formulation B.Optimal Broadcast Trees (cont.) Example : T * : {(A,B),(A,C),(C,D),(D,E)}, T * : {(A,C),(C,D),(D,E),(E,B)}, T * satisfies the min-max criterion T * satisfies the lexicographic criterion Definition: “Reduction” of G, G R (G,L,p) A useful transformation of a graph Eliminate links in L - L with c l ≥ p and then set c l = 0 for all l in L L = {(C,D), (D,E)} and p = 3 in this example
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France7 3. Optimization Algorithms Min-max criterion : Finding the spanning tree that minimizes the maximum induced node power is equivalent to finding the tree that minimizes the maximum link power Bottleneck optimization problem – polynomial time algorithms exist Lexicographic criterion :NP-complete in general Equivalent to finding an optimal MPR set, when all link costs in G are equal Optimal algorithm with O(|N| 2 log|N| + |N||L|) complexity, under the condition that the powers of links outgoing from different nodes are different Main idea : Solve min-max problem → identify the unique node that has to transmit with the given power → form the corresponding reduced graph → solve min-max problem on that graph → reiterate, until the value of the solution is zero
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France8 3. Optimization Algorithms A.Optimal Algorithm for the General Case Min-max solution still minimizes the maximum consumed node power However, in general there may be many nodes in the network that can reach others with a given power An optimal set of nodes has to be determined Candidacy tree : A useful structure with levels and nodes Each level corresponds to a “distinct” value of the optimal node power vector Each node is associated with a set of nodes of G, candidate to be optimal Upon completion, the candidacy tree provides all lexicographically optimal (with respect to node powers) spanning trees
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France9 3. Optimization Algorithms A.Optimal Algorithm for the General Case (cont.) Example : T 1 * : {(A,B),(A,F),(A,G),(B,C),(B,D),(C,E),(F,H),(G,I)}, path B→C→{F,G}→A T 2 * : {(A,B),(A,F),(A,G),(B,C),(B,D),(C,E),(F,H),(H,I)}, path B→C→{F,H}→A ABCDEFGHI T1*T1* 254003300 T2*T2* 254003030 Node Powers Induced by the Optimal Trees Note: The path A→C is “pruned” from the candidacy tree
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France10 3. Optimization Algorithms B.Heuristic Algorithm Motivation : The general optimal algorithm runs in reasonable time for moderate size random networks, but requires exponential number of computations in |N| in the worst case However, its steps are useful for the development of an efficient heuristic Approach : The heuristic algorithm avoids the most computing intensive operations by Selecting efficiently appropriate sets of nodes to transmit with a given power, approximating the optimal ones Eliminating the branchings in the candidacy tree (only one node at each level and, therefore, a single path at each step of the iteration) Main idea : If some node has to transmit with power p, it is preferable to select one whose outgoing links such that c l ≤ p have costs “close” to p Complexity : The worst case running time of the proposed heuristic is O(|N| 2 |L|)
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France11 4. Generalizations Cost function f n (p) : Strictly increasing in p and nonnegative Expresses the cost incurred at node n if it transmits with power p Given a spanning tree T :, where if n is a leaf node Objective: Find the tree for which the vector is lexicographically minimal Note I : The case f n (p) = p corresponds to the problem already studied Note II : If we use f n (c l ) as link cost functions, then the main difference is that the “power ” of a leaf node n may be non zero in the general case It is proved that the same algorithms can be used in this case as well, by appropriately modifying G (N, L) to a new network G (N, L)
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France12 4. Generalizations Application I : Node Receive Power Consumption q n : receive power → + q n : total power consumed by node n ≠ r → f n (p) = p + q n, if n ≠ r, and f r (p) = p Application II : Lexicographic Maximization of Remaining Lifetimes t : duration of transmission, E n : battery lifetime, q n = 0, : remaining lifetime at node n f n (p) = pt – E n + E : nonnegative by definition of E Application III : Node Importance Different cost functions for different nodes, according to their importance The previously developed methods can also solve this generalized problem
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France13 5. Numerical Results Algorithms compared : 1) “Min-Max” 2) “Lex-Opt” 3) “Heuristic” Networks created : (20,40,…,120) nodes in a rectangular grid of 100×100 points, 100 randomly generated networks for a given |N|, link costs : Main observations : Lex-Opt algorithm gives optimal (lexicographically smallest) node power vector Heuristic algorithm provides satisfactory performance relative to the optimal one Min-Max algorithm’s performance rapidly deteriorates as the network size increases, since it ensures only the minimization of the maximum node power Min-Max algorithm has the shortest running times Heuristic algorithm has satisfactory running times for all network sizes Lex-Opt algorithm’s running time is reasonable for no more than 80 nodes
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France14 5. Numerical Results |N|R–MeanQ(R>0.25 )Q(R>0.5 )Q(R>0.75 )Q(R=1 ) 200.992599% 400.9898100%99%98% 600.930397%93%88% 800.890195%87%81% 1000.857293%84%77% 1200.769496%72%61% Comparison of Heuristic Algorithm vs. Lex-Opt R, 0 < R ≤ 1 : a measure of how close the Heuristic algorithm comes to providing the optimal (lexicographically smallest) vector of node powers For 40-node networks for example, the Heuristic algorithm provides the optimal solution, Q(R=1), in 98% of the performed experiments For 120-node networks, the percentage of the experiments for which at least the first 30 (0.25×120) maximal node powers are optimal, Q(R>0.25), is 96%
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France15 6. Extensions – Issues for Further Study Distributed Implementation : If each node has knowledge of its one, two, …, k-hop neighbors, then the proposed algorithms can be applied locally in a manner similar to MPR algorithm In general, they can be directly applied in network environments where at least partial information of network topology is proactively maintained at each node, as in OLSR and ZRP Min-max node power optimization problem can be solved distributively by replacing the sum operation with the maximum operation in an existing distributed implementation of Edmond’s algorithm for finding a minimum-sum spanning tree Multicast Extensions : The optimal algorithms solve the lexicographic optimization problem, based on algorithms solving the bottleneck multicast tree problem New heuristics must be developed, since in general not all nodes are destinations
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WiOpt'03March 3-5, 2003, INRIA Sophia-Antipolis, France End of Presentation Thank you for your attention Paper : Energy-aware Broadcasting in Wireless Networks Ioannis Papadimitriou Co-Author : Prof. Leonidas Georgiadis ARISTOTLE UNIVERSITY OF THESSALONIKI, GREECE FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Division of Telecommunications
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