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
Presentation Plan
1. Introduction
2. Definitions and Problem Formulation
3. Optimization Algorithms
4. Generalizations
5. Numerical Results
6. Extensions – Issues for Further Study
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
2
1. Introduction
Wireless Networks
Motivation :
• Dissemination of information
• Battery-operated
Broadcasting
Energy Conservation
Assumptions :
• Omnidirectional antennas
• Varying transmission powers
Node-based environment
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'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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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) 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 pnT  max
{cl }, where max {cl }  0 if n is a leaf
T
lLout ( n )
l
Example :
T1 : {(A,B) , (B,C) , (B,D)}
T2 : {(A,B) , (A,C) , (B,D)}
• Same leaf nodes C , D
T
T
T
T
• Set I : p A1  p A2  2, pB1  pB2  4
T
T
T
T
• Set II : p A1  2, p A2  4, pB1  6, pB2  3
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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2. Definitions and Problem Formulation
B. Optimal Broadcast Trees
A spanning tree T induces a vector of node powers PT  ( pnT ) nN
• Objective I : Min-max node power optimization
T
T
Find a tree T : max { pn }  max { pn } for any spanning tree T of G
nN
nN
• Objective II : Lexicographic node power optimization
*
Find a tree T :
T*
P lex PT 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)
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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2. Definitions and Problem Formulation
B. Optimal Broadcast Trees (cont.)
Example :
T * : {(A,B),(A,C),(C,D),(D,E)} , PT  ( pTA , pBT , pCT , pDT , pET )  (5,0,5,3,0)
*
T : {(A,C),(C,D),(D,E),(E,B)} , P
T*
 ( p , p , p , p , p )  (2,0,5,3,1)
T*
A
 T * satisfies the min-max criterion
T*
B
T*
C
T*
D
T*
E
T*
 T * satisfies the lexicographic criterion
P  lex PT
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
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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3. Optimization Algorithms
max { pnT }  max { max
{cl }}  max
{cl }
T
T
Min-max criterion :
nN
nN
lLout ( n )
lL
 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'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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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'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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3. Optimization Algorithms
A. 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
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
Note: The path A→C is “pruned”
from the candidacy tree
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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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 cl ≤ p have costs “close” to p
Complexity : The worst case running time of the proposed heuristic is O(|N|2 |L|)
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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4. Generalizations
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 : nT  max { f n (cl )} , where max { f n (cl )}  f n (0)
l
lLTout ( n )
if n is a leaf node
Objective: Find the tree for which the vector ( T )
is lexicographically minimal
n nN
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 nT ” 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'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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4. Generalizations
Application I : Node Receive Power Consumption
T
 qn : receive power → pn + 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 , E  max {En }
nN
 EnT  En  pnT t   max
{cl t  En  E}  E : remaining lifetime at node n
T
lLout ( 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
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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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 : c(i , j )  d (2i , j )
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'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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5. Numerical Results
Comparison of Heuristic Algorithm vs. Lex-Opt
|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%
 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'03
March 3-5, 2003, INRIA Sophia-Antipolis, France
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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
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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
WiOpt'03
March 3-5, 2003, INRIA Sophia-Antipolis, France