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

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 PapadimitriouCo-Author : Prof. Leonidas Georgiadis

ARISTOTLE UNIVERSITY OF THESSALONIKI, GREECEFACULTY OF ENGINEERING

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERINGDivision of Telecommunications

WiOpt'03 March 3-5, 2003, INRIA Sophia-Antipolis, France 2

Presentation Plan

1. Introduction

2. Definitions and Problem Formulation

3. Optimization Algorithms

4. Generalizations

5. Numerical Results

6. Extensions – Issues for Further Study


1. IntroductionWireless 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


2. Definitions and Problem FormulationA. 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

}{max)(

lnLl

Tn cp

Tout

0}{max l

lc

Example :

T1 : {(A,B) , (B,C) , (B,D)}

T2 : {(A,B) , (A,C) , (B,D)}

• Same leaf nodes C , D

• Set I :

• Set II :

4,2 2121 TB

TB

TA

TA pppp

3,6,4,2 2121 TB

TB

TA

TA pppp


2. Definitions and Problem FormulationB. 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

NnTn

T p )(P

}{max}{max Tn

Nn

Tn

Nnpp

Tlex

T PP *

T

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)


2. Definitions and Problem FormulationB. Optimal Broadcast Trees (cont.)

Example :

T * : {(A,B),(A,C),(C,D),(D,E)} ,

T * : {(A,C),(C,D),(D,E),(E,B)} ,

T )0,3,5,0,5(),,,,( TE

TD

TC

TB

TA

T pppppP

)1,3,5,0,2(),,,,(******

TE

TD

TC

TB

TA

T pppppP

T * satisfies the min-max criterion

T * satisfies the lexicographic criterion

TT

lexT PP

*

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


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

}{max}}{max{max}{max)(

lLl

lnLlNn

Tn

Nnccp

TTout


3. Optimization AlgorithmsA. 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


3. Optimization AlgorithmsA. 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

A B C D E F G H I

T1* 2 5 4 0 0 3 3 0 0

T2* 2 5 4 0 0 3 0 3 0

Node Powers Induced by the Optimal Trees

Note: The path A→C is “pruned”

from the candidacy tree


3. Optimization AlgorithmsB. 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|)


4. Generalizations

)}({max)(

lnnLl

Tn cf

Tout

Cost function fn(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

)0()}({max nlnl

fcf

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

NnTn )(

Tn

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)


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

Tnp

EEEtctpEE nlnLl

Tnn

Tn T

out

}{max)(

}{max nNn

EE



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

2),(),( jiji dc



|N| R–Mean Q(R>0.25 ) Q(R>0.5 ) Q(R>0.75 ) Q(R=1 )

20 0.9925 99% 99% 99% 99%

40 0.9898 100% 99% 98% 98%

60 0.9303 97% 93% 88% 88%

80 0.8901 95% 87% 81% 81%

100 0.8572 93% 84% 77% 77%

120 0.7694 96% 72% 61% 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%


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