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

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

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

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 :

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)

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

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

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

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* T2*T2* Node Powers Induced by the Optimal Trees Note: The path A→C is “pruned” from the candidacy tree

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|)

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)

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

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

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 ) % %99%98% %93%88% %87%81% %84%77% %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%

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

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