WiOpt’03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks

1. 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

2. Presentation Plan • Introduction • Definitions and Problem Formulation • Optimization Algorithms • Generalizations • Numerical Results • Extensions – Issues for Further Study March 3-5, 2003, INRIA Sophia-Antipolis, France

3. 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-maxand Lexicographic node power optimization problem Generalization :Lexicographic optimization under more general cost functions of node powers March 3-5, 2003, INRIA Sophia-Antipolis, France

4. 2. Definitions and Problem Formulation • Wireless Communication Model • Network representation : • Directed graph G (N , L) • Required power for transmission over link l (link cost) cl > 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 , LT) • Node n transmits with power , where if n is a leaf • Example : • T1 : {(A,B) , (B,C) , (B,D)} • T2 : {(A,B) , (A,C) , (B,D)} • Same leaf nodes C , D • Set I : • Set II : March 3-5, 2003, INRIA Sophia-Antipolis, France

5. 2. Definitions and Problem Formulation • 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 ith 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) March 3-5, 2003, INRIA Sophia-Antipolis, France

6. 2. Definitions and Problem Formulation • 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, GR(G,L,p) • A useful transformation of a graph • Eliminate links in L - L with cl ≥p and then set cl = 0 for all l in L • L = {(C,D) , (D,E)} and p = 3 in this example March 3-5, 2003, INRIA Sophia-Antipolis, France

7. 3. Optimization Algorithms Min-maxcriterion : • Finding the spanning tree that minimizes the maximum induced node power is equivalent to finding the tree that minimizes the maximum link power • Bottleneckoptimization 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 March 3-5, 2003, INRIA Sophia-Antipolis, France

8. 3. Optimization Algorithms • 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 March 3-5, 2003, INRIA Sophia-Antipolis, France

9. 3. Optimization Algorithms • Optimal Algorithm for the General Case (cont.) Example : T1* : {(A,B),(A,F),(A,G),(B,C),(B,D),(C,E),(F,H),(G,I)} , path B→C→{F,G}→A T2* : {(A,B),(A,F),(A,G),(B,C),(B,D),(C,E),(F,H),(H,I)} , path B→C→{F,H}→A Node Powers Induced by the Optimal Trees Note:The path A→C is “pruned” from the candidacy tree March 3-5, 2003, INRIA Sophia-Antipolis, France

10. 3. Optimization Algorithms • 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 cl ≤ p have costs “close” to p Complexity :The worst case running time of the proposed heuristic is O(|N|2 |L|) March 3-5, 2003, INRIA Sophia-Antipolis, France

11. 4. Generalizations • Cost functionfn(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 fn(p) = p corresponds to the problem already studied Note II :If we use fn(cl) 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) March 3-5, 2003, INRIA Sophia-Antipolis, France

12. 4. Generalizations • Application I :Node Receive Power Consumption • qn : receive power → + qn : total power consumed by node n ≠ r • → fn(p) = p + qn , if n ≠ r , and fr(p) = p • Application II :Lexicographic Maximization of Remaining Lifetimes • t : duration of transmission , En : battery lifetime , qn = 0 , • : remaining lifetime at node n • fn(p) = pt – En + 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 March 3-5, 2003, INRIA Sophia-Antipolis, France

13. 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 March 3-5, 2003, INRIA Sophia-Antipolis, France

14. 5. Numerical Results 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% March 3-5, 2003, INRIA Sophia-Antipolis, France

15. 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 March 3-5, 2003, INRIA Sophia-Antipolis, France

16. 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 March 3-5, 2003, INRIA Sophia-Antipolis, France