Energy efficient communication in ad hoc networks from user's and designer's perspective

Abstract

We consider a game that models the creation of a wireless ad hoc network, where nodes are owned by selfish agents. We study a novel cost sharing model in which agents may pay for the transmission power of the other nodes. Each agent has to satisfy some connectivity requirement in the final network and the goal is to minimize its payment with no regard to the overall system performance. We analyze two fundamental connectivity games, namely broadcast and convergecast. We study pure Nash equilibria and quantify the degradation in the network performance called the price of anarchy resulting from selfish behavior. We derive tight bounds on the price of anarchy for these games. We also study centralized network design. One of the most important problems in wireless ad hoc networks is the minimum-energy broadcast. Recently, there appeared many new applications such as real-time multimedia, battlefield communications and rescue operations that impose stringent end-to-end latency bounds on the broadcasting time. However, the existing algorithms that minimize the broadcasting energy tend to produce solutions with high latency. In this paper we consider the problem of bounded-hop broadcast. We present approximation and heuristic algorithms for this problem.

Full text

Energy Efficient Communication in Ad Hoc Networks
from User’s and Designer’s Perspective∗
Alex Kesselman†
akessel@mpi-sb.mpg.de
Dariusz Kowalski‡
darek@mpi-sb.mpg.de
Michael Segal§
segal@cse.bgu.ac.il
June 17, 2004
Abstract
Wireless ad hoc networks have gained a lot of attention in recent years. We consider
a game that models the creation of such a network, where nodes are owned by selfish
agents. We study a novel cost sharing model in which agents may pay for the transmission
power of the other nodes. Each agent has to satisfy some connectivity requirement in the
final network and the goal is to minimize its payment with no regard to the overall sys-
tem performance. We analyze two fundamental connectivity games, namely broadcast and
convergecast. We study pure Nash equilibria and quantify the degradation in the network
performance called the price of anarchy resulting from selfish behavior. We derive tight
†Max Planck Institut fur Informatik, Saarbrucken, Germany.
‡Max Planck Institut fur Informatik, Saarbrucken, Germany and Instytut Informatyki, Uniwersytet Warszawski,
Banacha 2, 02-097 Warszawa, Poland.
§Communication Systems Engineering Dept., Ben Gurion University of the Negev, Beer-Sheva, Israel.
∗The work of second author is supported in part by the KBN Grant 4T11C04425.
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bounds on the price of anarchy for these games.
We also study centralized network design. One of the most important problems in wire-
less ad hoc networks is the minimum-energy broadcast. Recently, there appeared many
new applications such as real-time multimedia, battlefield communications and rescue op-
erations that impose stringent end-to-end latency requirement on the broadcasting time.
However, the existing algorithms that minimize the broadcasting energy tend to produce
solutions with high latency. In this paper we consider the problem of bounded-hop broad-
cast. We present approximation and heuristic algorithms for this problem.
1 Introduction
The next generation communication networks are likely to be a combination of wireline and
ad hoc networks, which are expected to fulfill a critical role where wired backbone networks
are not available or not economical to build [10, 34]. A communication session in a wireless
network is achieved either through a single-hop transmission if the communication parties are
close enough, or through relaying by intermediate nodes otherwise. Depending on its power
level and on the nature of environmental interference, a node can reach all nodes in a certain
range. Typically, the signal power falls as 1/dα, where dis the distance from the transmitter
antenna and αis a constant between 2and 4depending on the environment [35]. All receivers
have the same power threshold for signal detection, which is typically normalized to one. Under
the above assumptions, the power required to establish a link between two nodes at distance d
is dα. In this paper we assume that nodes are located in a Euclidean space and consider the
2

symmetric energy model. In ad hoc networks devices are usually equipped with battery that
has a limited power. Thus, among the most crucial issues is that of developing energy-efficient
topology control algorithms, which maximize the network lifetime [20].
Most of the existing works on wireless networks are based on the assumption that there is a
centralized control, or the nodes run the same distributed algorithm. While this assumption may
hold for some networks, such as military or government networks, it is not valid in general for
Internet-like networks. In such a network, nodes are owned by different commercial entities,
which are strongly driven by their economic interests. This naturally gives rise to many game-
theoretic issues. The stable outcomes of the interaction between non-cooperative selfish agents
correspond to Nash equilibria. A Nash equilibrium [32] can be viewed as a solution that selfish
agents can agree upon, i.e., the agents have no incentive to deviate. Since Nash equilibria in
network games can be much more expensive than the best centralized design, it is important
to consider the implications of selfish behavior on the network performance. Recently, game-
theoretical analysis of ad hoc networks has received a great deal of attention [14, 4, 30, 19, 38,
13].
Traditionally, in Computer Science research has been focused on finding a global optimum.
With the emerging interest in computational issues related to game theory, Koutsoupias and
Papadimitriou [28] introduced so called price of anarchy. The pessimistic price of anarchy is
the ratio between the cost of the worst possible Nash equilibrium (the one with the maximum
social cost) and the cost of the social optimum. Roughgarden and Tardos [36] derive the price of
anarchy of selfish routing in networks. Ansheleich et al. [2] propose to consider the optimistic
price of anarchy, which is the ratio between the cost of the best possible Nash equilibrium (the
3

one with the minimum social cost) and that of the social optimum.
In this paper we consider the following game modeling the creation of a wireless ad hoc
network. Each agent (node) has a specific connectivity requirement, i.e., each agent has to build
a network in which this requirement is satisfied. We study a novel cost sharing model in which
the transmission power of each node can be paid by different agents. In the model without cost
sharing, each agent pays only for the transmission power of its own node. The goal of each
agent is to pay as little as possible. Note that the price of anarchy measures the cost of lack
of coordination between the agents. Thus, we assume that agents have complete information.
We consider only pure strategies since mixed (probabilistic) Nash equilibria do not seem to
be suitable for network design [17]. We study broadcast and convergecast problems, which
are fundamental communication tasks in ad hoc networks. In the Broadcast Game, each agent
has to establish a directed path between a designated root node and its own node while in the
Convergecast Game, each agent has to establish a directed path between its own node and a
designated root node.
The rapidly increasing capabilities and low costs of computing and communication devices
have made it possible to use wireless networks in a wide range of applications such as real-time
multimedia, battlefield communications, and rescue operations. The above applications impose
stringent end-to-end latency requirement on the broadcasting time, which is the time taken
by the message to reach all the nodes in the network. The latency of a broadcast scheduling
algorithms is at least Ω(D), since the message needs to reach the furthest node from the source,
where Dis the network diameter. Gaber and Mansour [22] show that for any graph with D=
Ω(log5n), there is a centralized randomized schedule with latency O(D), where nis the number
4

of nodes.
Unfortunately, in the existing minimum-energy broadcast algorithms the latency of the
broadcasting tends to be quite large since the minimum energy is attained when the number
of the relay nodes is maximized, which results in a large network diameter. We consider the
problem of bounded-hop broadcast, introduced in [12]. In this problem, we aim to construct a
minimum-energy communication graph of bounded diameter rooted at the source node in which
all nodes are covered. Note that there is a trade-off between reaching more nodes in a single
hop using higher power and thus decreasing the network diameter and reaching fewer nodes
using lower power and thus increasing the network diameter. The unrestricted version of this
problem is NP-hard [6]. For a line topology, this problem can be solved optimally using the
dynamic programming algorithm of Clementi et al. [11].
Our results. First we consider the network creation games. For the Broadcast Game with
cost sharing, we show that in contrast to the single-source connection game of [2], a pure Nash
equilibrium may not exist and the optimistic price of anarchy is bounded away from one. We
also establish that the pessimistic price of anarchy is Θ(n). We note that the Broadcast Game is
impossible without cost sharing. For the Convergecast Game without cost sharing, we demon-
strate that the optimistic price of anarchy is 1while the pessimistic price of anarchy is Θ(nα−1).
Interestingly,the Convergecast Game has a lower pessimistic price of anarchy compared to that
of the strong connectivity game in [15], which is Ω(nα). We show that if we allow cost sharing,
the pessimistic price of anarchy is improved significantly to Θ(n)while the optimistic price of
anarchy remains the same.
Then we consider algorithms for centralized network design. For the bounded-hop broadcast
5

problem, we present a simple algorithm that has an approximation factor of min(Dα−1,12(n/D)log β),
where Dis the bound on the network diameter and βis a constant that depends on α. In partic-
ular, for α= 2 we have β≤2, and the approximation factor is at most O(√n)for any Dand
at most O(log n)for D=O(log n)and D= Ω(n/ log n). Thereafter, we derive the bounded-
diameter minimum spanning tree (BDMST) algorithm that achieves an approximation factor
of O(f(n)·log n), where f(n)is the worst-case ratio between the energy cost of an optimal
BDMST and that of an optimal solution for the bounded-hop broadcast problem. However, the
running time of this algorithm is rather high because of the complexity of the BDMST problem.
In addition, we propose the decremental distance heuristic, which is easy to implement from the
practical point of view and has a polynomial running time (without large hidden coefficients).
Finally, we show how to solve optimally in polynomial time the full-duplex (bi-directional) con-
nection problem and the generalized multicast and the web-conferencing problems for a fixed
number of nodes.
Paper organization. Our model is presented in Section 3. Section 2 describes the related
work. We analyze the networks creation games in Section 4. Algorithms for the bounded-
hop broadcast problem are presented in Section 5. Optimal algorithms for some connectivity
problems are given in Section 6. We conclude with Section 7.
2 Related Work
Our paper is closely related to the work of Eidenbenz et al. [15], which considers topology
control problems in ad hoc networks, where nodes choose their power levels in order to ensure
6

the desired connectivity properties. Eidenbenz et al. give asymptotically tight bounds on the
price of anarchy for the strong connectivity game and demonstrate that for the connectivity
game it is NP-complete to decide whether a pure Nash equilibrium exists. In contrast to [15],
we allow cost sharing in our games. Ansheleich et al. [2] consider a network design game for
wired networks, where each agent has to connect a set of terminals. Ansheleich et al. show that
determining whether a pure Nash equilibrium exists is NP-complete and present a polynomial-
time algorithm that computes a 3-approximate Nash equilibrium. We define a cost sharing
model similar to that of [2] for ad hoc networks. Fabricant et al. [16] study a wireline network
creation game, where the cost of a node depends on the distances to the other nodes.
The problem of bounded-hop connectivity has been studied extensively. Optimal algorithms
based on dynamic programming for the problem of bounded-hop strong connectivity and the
problem of bounded-hop broadcast on a line are presented by Kirousis et al. [26] and Clementi
at al. [11], respectively. Beier et al. [3] and Funke et al. [21] give efficient algorithms for finding
a minimum-energy route between two given nodes that uses a bounded number of hops. The
problem of bounded-hop broadcast is studied in a recent paper by Ambuhl et al. [1]. Ambuhl
et al. give a polynomial-time algorithm based on dynamic programming for finding an optimal
solution for 2-hop broadcast with running time O(n7)and derive a PTAS for the case in which
the bound on the number of hops is fixed. Contrary to [1], in this paper we consider the whole
range of possible values of D.
The problem of minimum-energy broadcasting in which the transmission power of each
node has to be determined so that the total power is minimized has received recently a great deal
of attention. Cagalj et al. [6] give a proof of NP-hardness of the minimum-energy broadcast
7

problem in Euclidean space. Wieselthier et al. [40] propose three greedy heuristics, namely the
minimum spanning tree (MST), the shortest path tree (SPT) and the broadcasting incremental
power (BIP), and evaluate them through simulations. Wan et al. [39] present the first analytical
results for this problem. In particular, they prove that the approximation ratio of MST is between
6and 12 while the approximation ratio of SPT is at least n/2. Several approximation methods
with analytical bounds for the asymmetric model are proposed by Liang [29]. Some of these
results have been improved by Caragiannis at al. [8] for the symmetric model. Calinescu et
al. [7] consider the problem of maximizing network lifetime as well as different energy efficient
network connectivity problems. Cartigny et al. [9] develop localized algorithms for minimum-
energy broadcasting.
3 Model and Notation
We assume that there are nnodes, which are located on a Euclidean space. We denote by P
the placement of nodes. A wireless network is represented by a directed communication graph
G= (V, E ), where Vis the set of nodes and Eis the set of edges. The neighbors of a node
uare determined by its transmission power Pu. Namely, node ucan reach all nodes within its
transmission range Ru=P1/α
u, where 2≤α≤4is the the distance-power gradient.∗Thus,
edge (u, v)belongs to Eif the distance between uand v,d(u, v), is at most Ru. We consider
the symmetric model in which the energy cost of an edge (u, v)is the same as that of (v, u).
In a minimum-energy network design problem, the goal is to assign transmission powers to
∗For simplicity, we assume that the maximum transmission range of a node is unbounded.
8

the nodes in order to establish the desired connectivity requirement while minimizing the total
energy of the nodes. Note that it is always sufficient to consider at most ndifferent transmission
ranges for each node, which are the distances to the other nodes and zero transmission range.
We denote by OP T an optimal solution. We say that an algorithm Ahas the approximation
factor of c, if the total power of the solution produced by A,P(A), is at most ctimes that of
OP T ,P(OP T ), for any instance of the problem.
3.1 Broadcast and Convergecast Games
We assume that each node is an agent. A strategy of node uis a payment function Pu
v, that
is how much energy uis willing to contribute to the transmission power of node v. The actual
transmission power of node uis defined as Pu=Pv∈VPv
u. The goal of node uis to minimize its
total payment, that is Pv∈VPu
v. We assume that nodes have complete information. The game is
said to be in a Nash equilibrium if the connectivity requirement of each agent is satisfied and no
agent can find a better (with lower cost)alternative strategywith respect to the current strategies
of the other agents (i.e., assuming that the strategies of the other agents are fixed).
The social cost of the outcome of a game is the total power of the nodes. For a placement of
nodes P, we denote the cost of a Nash equilibrium by CP=Pu∈VPuand the cost of OP T by
C∗
P=Pu∈VP∗
u, where P∗
uis the transmission power of uunder OP T . The pessimistic price
of anarchy is maxPCP/C∗
Pwhile the optimistic price of anarchy is minPCP/C∗
P.
We study the Broadcast Game: there is a designated root node rthat has a message that must
be delivered to all nodes, i.e., each node uhas to establish a directed path r→u. In this game,
the transmission power of each node can be paid by different nodes. This model is inspired by
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the model of [2] for wired networks. We assume that each node, once it is connected, forwards
data packets of the other nodes. Observe that the Broadcast Game requires cost sharing, since a
node cannot establish a path from the root by controlling only its own power.
We also consider the Convergecast Game: each node has a message that must be delivered
to a designated root node r, i.e., each node uhas to establish a directed path u→r. In this
game, each agent ucontributes only to the transmission power of its own node, i.e., Pu
v= 0
for v6=u. This model has been introduced in [15]. We assume that multiple messages can be
aggregated into a single packet and thus we count the energy cost of each edge only once. We
will also extend the analysis of the Convergecast Game to the cost sharing model.
3.2 Bounded-Hop Broadcast
For each node u, we consider a shortest path in Gbetween a designated root node rand u.
The diameter of the communication graph is the maximal length of such a path. The bounded-
hop broadcast problem is to find the energy assignment for broadcast that minimizes the total
energy, that is PuPu, subject to the constraint that the diameter of the resulting communication
graph is bounded by D.
Note that if D=n, the problem is just the classical minimum-energy broadcast problem,
which is NP-hard [6]. On the other hand, if D= 1, there exists only a trivial solution in which
rhas the transmission range equal to the distance to the furthest node. We denote this distance
by L, i.e., L= maxud(r, u).
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4 Network Creation Games
In this section we consider the network creation games.
4.1 Broadcast Game
We show that in contrast to the single-source connection game of [2], a pure Nash equilibrium
for the Broadcast Game may not exist and the optimistic price of anarchy is bounded away from
one. We also demonstrate that the pessimistic price of anarchy is Θ(n).
Definition 4.1 We say that node ucovers node vif there exists a directed simple path in the
underlying communication graph Gfrom rto vsuch that uprecedes von this path.
The following observation characterizes the cost sharing in a Nash equilibrium.
Observation 1 Consider a Nash equilibrium. We have that for each node vcovered by usuch
that d(u, v)< Ru,Pv
u= 0, i.e., vdoes not contribute anything to the transmission power of u.
If it is not the case, vwould always have incentive to decrease Pv
ubecause it would still
remain covered by u.
First we give an example of the Broadcast Game for which a pure Nash equilibrium does
not exist.
Theorem 4.1 The Broadcast Game may possess no pure Nash equilibrium.
Proof: Consider the following scenario (see Figure 1). Node uis at distance dfrom the root in
one direction and two other nodes vand ware at distance d/2and d+ǫin the opposite direction,
11

respectively. Obviously, in a Nash equilibrium the transmission range Rrof ris either d/2,d,
or d+ǫ. We will show that all these situations are unstable.
dd/2 d/2+eps
root wu v
Figure 1: An example of the Broadcast Game without a pure Nash equilibrium.
If Rr=d/2, then uwould always increase its payment to rto assure Rr=dbecause it is
the cheapest way for uto get covered.
If Rr=d, then by Observation 1, vdoes not pay anything to r. We argue that the payment
of wto ris also zero because it would only to pay to vfor a transmission range of d/2 + ǫ.
Therefore, ufully pays to the root for a transmission range of d. In this case whas incentive
to increase its payment to the root from 0to (d+ǫ)α−(d)αto make its transmission range be
d+ǫand decrease its payment to vto zero.
If Rr=d+ǫ, Observation 1 implies that wfully pays for the energy of the root Pr= (d+ǫ)α.
However, wcan benefit from decreasing its payment to the root to (d/2)αand increasing its
payment to vto (d/2 + ǫ)α.
The next theorem shows that the optimistic price of anarchy is greater than 1.
Theorem 4.2 The optimistic price of anarchy for the Broadcast Game is bounded away from 1.
Proof: Consider the following scenario (see Figure 2). There are three rays emanating from
the root and the the angle between two adjacent rays is 120 degrees. On the first two rays there
12

are two nodes at distance d−xand d+xfrom the root, respectively. On the third ray there is
a node at distance dfrom the root.
root
d
d−x
2x
Figure 2: An example of the Broadcast Game in which the optimistic price of anarchy is
bounded away from 1.
Consider a situation in which the node on the last ray pays to the root for a transmission
range of dand the furthest node on each of the first two rays pays to the first node on its ray for
a transmission range of 2x. It is a unique Nash equilibrium provided that no furthest node can
benefit from increasing the transmission range of the root, i.e.,
(d+x)α−dα>(2x)α.(1)
We have that the price of anarchy is bounded away from 1if
dα+ 2(2x)α>(d+x)α.(2)
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Note that OP T just assigns a transmission range of d+xto the root. In case α= 2, the
inequalities (1, 2) hold for 2
7d < x < 2
3d.
Now we proceed to study the pessimistic price of anarchy. The following theorem estab-
lishes a simple upper bound of non the price of anarchy, which turns out to be asymptotically
tight.
Theorem 4.3 The pessimistic price of anarchy for the Broadcast Game is at most n.
The theorem holds due to the fact that no agent spends more energy than OP T does, other-
wise it can connect to the root by paying at most C∗.
The following theorem derives a lower bound of Ω(n)on the pessimistic price of anarchy.
Theorem 4.4 The pessimistic price of anarchy for the Broadcast Game is at least Ω(n).
Proof: Consider the following scenario (see Figure 3). There is a circle c1with radius d
whose center is the root and a circle c2with a smaller radius. A set S1of nodes is located on the
external part of c2w.r.t. c1such that the first and the last nodes are located at the intersection
of c1and c2. Another set S2of nodes is located on the radius connecting the last point in S1to
the root and the distance between the last point of S1and the first point of S2is x. We assume
that |S1|=|S2|=n/2and the distance between any two adjacent nodes in S1and in S2is
δ= 2(d−x)/n.
Suppose that each node in S1pays to the root dα/|S1|and all but the first and the last nodes
pay to the succeeding node in S1for the energy worth δα. Nodes in S2do not pay anything.
The only alternative strategy for a node in S1is to decrease its payment to the root to zero and
14

S2
d
x
root
S1
c1 c2
Figure 3: An example of the Broadcast Game in which the price of anarchy is Ω(n).
pay xαto the first node in S2to connect the last node in S1. Thus, the system is in a Nash
equilibrium if xα≥dα/|S1|.We have that C=dα+n−4
2δα.
Note that OP T sets the transmission ranges of all but the first nodes in S1and S2to δ
and assigns a transmission range of 0and xto the first node in S1and to the first node in S2,
respectively. Thus, C∗=xα+ (n−2)δα. Therefore, the pessimistic price of anarchy is at least
n
2·dα+n−4
2δα
dα+n(n−2)
2δα= Ω(n),
when ntends to infinity.
15

4.2 Convergecast Game
We show that for the Convergecast Game, the optimistic price of anarchy is 1while the pes-
simistic price of anarchy is Θ(nα−1). Then we extend our results to the cost sharing model. We
demonstrate that the pessimistic price of anarchy becomes Θ(n)while the optimistic price of
anarchy remains 1. We note that the MST algorithm for strong connectivity by Kirousis et al.
[26] implicitly finds an optimal solution for the minimum-energy convergecast in polynomial
time.
The following theorem shows that the optimistic price of anarchy is one.
Theorem 4.5 The optimistic price of anarchy for the Convergecast Game is 1.
Clearly, OP T is a Nash equilibrium since no node can decrease its transmission range and
still stay connected to the root.
Corollary 4.6 There always exists a pure Nash equilibrium for the Convergecast Game.
Next we derive an upper bound on the pessimistic price of anarchy.
Theorem 4.7 The pessimistic price of anarchy for the Convergecast Game is at most O(nα−1).
Proof: Consider a Nash equilibrium. Enumerate the nodes in order of non-increasing trans-
mission range: R1≥R2·· · ≥ Rn. Let Zbe the sum of the transmission ranges of the nodes
under OP T . We claim that Ri≤Z/i. Consider a set of nodes Sicontaining the first inodes
and the root. We claim that the distance between any two nodes in Siat least Ri. If it is not the
case, at least one node can decrease its transmission power without losing connectivity to the
16

root, which contradicts the stability of a Nash equilibrium. We obtain that Z≥iRisince OP T
must connect all these nodes to the root. Hence, the cost of the Nash equilibrium is at most
C≤
n
X
i=1
(Z/i)α=Zα
n
X
i=1
1/iα≤Zα·const(α),
where const(α)is a positive constant which depends on α > 1. For example, for α= 2 we
have const(α)≤π2/6. On the other hand, C∗≥n(Z/n)α, since the transmission power is
minimized when all nodes are evenly spaced. Dividing the bound for Cby the bound for C∗we
conclude the proof of the theorem.
In the next theorem we establish a lower bound of Ω(nα−1)on the pessimistic price of
anarchy, which matches our upper bound.
Theorem 4.8 The pessimistic price of anarchy for the Convergecast Game is at least Ω(nα−1).
Proof: Let ǫbe a small positive constant. Consider the following scenario (see Figure 4). All
nodes are located on the line and the root is the first node. The distance between the root and
the first regular (other than root) node is 1 +ǫand the distance between any two adjacent regular
nodes is 1. The transmission range of the regular nodes but the last one is 1and the transmission
range of the last node is n−1 + ǫ.
Clearly, it is a Nash equilibrium since all nodes are connected to the root and no node can
decrease its transmission power. The cost of this Nash equilibrium is
C=n−2 + (n−1 + ǫ)α,
17

root 1+eps 1
Figure 4: A bad example for the Convergecast Game.
while
C∗=n−2 + (1 + ǫ)α,
since OP T sets the transmission range of all regular nodes but the first one to 1and the trans-
mission range of the first node to 1 + ǫ.
Note that if we allow cost sharing, the optimistic price of anarchy remains the same. That is
due to the fact that OP T is still a Nash equilibrium if each node pays for its own transmission
energy. However, the pessimistic price of anarchy drops to Θ(n), which shows the benefit of
cost sharing. The proof of the upper bound is trivial. The proof of the lower bound is almost
identical to that of Theorem 4.8 and is omitted from this abstract.
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5 Bounded-hop Broadcast
In this section we present approximation algorithms for the bounded-hop broadcast problem.
5.1 Simple Algorithm
In this section we give a simple min(Dα−1,12(n/D)log β)-approximation algorithm, where βis
a constant that depends on α. First we derive two obvious lower bounds on P(OP T ). The next
lower bound will be useful for small values of D. Recall that L= maxud(r, u).
Lemma 5.1 For a given D,P(OP T )≥D(L/D)α.
Proof: The energy of OP T is at least D(L/D)αsince it must establish a path between rand
the furthest node and the transmission power is minimized when nodes are evenly spaced and
the number of relay nodes is maximized.
The following lower bound will be useful for large values of D.
Theorem 5.2 ([39]) For any value of D,P(O P T )≥P(MS T )/12.
Now we present two basic algorithms. The first algorithm called root cover (RC) is a trivial
approach in which the root covers all the nodes. Note that this algorithm has a good performance
for small values of D. A similar claim has been made in [1] without a proof.
Theorem 5.3 The approximation factor of the RC algorithm is at most Dα−1.
Proof: Note that the total energy of the root cover algorithm is Lα. On the other hand, Lemma
5.1 implies that the total energy of OP T is at least D(L/D)α. We obtain that P(RC)/P (OP T )≤
Dα−1.
19

In the second algorithm, we start from a shortest path tree Tcorresponding to the communi-
cation graph produced by the MST algorithm. The l-th layer of Tcontains all nodes at distance
lfrom the root. We will contract the odd layers of Tuntil the diameter drops below D. The
layer contraction algorithm is presented in Figure 5. Observe that this algorithm would perform
better for large values of D.
1. Apply the MST algorithm [39] and set the transmission range of each node accord-
ingly.
2. If the diameter of the communication graph is smaller than D, return the current solu-
tion.
3. Construct a shortest path tree Tof Grooted at r.
4. Contract all odd layers of Tin the following way:
(a) For each layer ls.t. lis even and each node uin layer lset the transmission range
of uto maxv,w:(u,v)∈T,(v,w)∈T(d(u, v) + d(v, w)). (Note that vis a child of uat
layer l+ 1 and wis a child of vat layer l+ 2 and unow will be able to reach w
directly.)
(b) For each layer ls.t. lis odd and each node uin layer lset the transmission range
of uto zero.
5. Goto Step 2.
Figure 5: The layer contraction (LC) algorithm.
Theorem 5.4 The approximation factor of the LC algorithm is at most 12(n/D)log β, where β
is a constant that depends on α.
Proof: From Theorem 5.2 it follows that P(O P T )≥P(MS T )/12. If Step 4is never exe-
cuted, we are done. Otherwise, during each iteration we decrease the diameter (the number of
layers) by half. We argue that Step 4incurs at most a constant blowup in the energy. Let βbe
20

the smallest constant s.t. for any positive x,ythe following holds
xα+yα≥((x+y)α−yα)/β. (3)
Note that for α= 2, we have β≤2. Consider a node uat an odd layer of Twhose power is
increased by Step 4and let vand wbe the nodes for which the maximum of d(u, v) + d(v, w)is
achieved. The transmission range of uis increased to d(u, v) + d(v, w)while the transmission
range of vis decreased to zero. Thus, the overall increase in energy is bounded by
d(u, w)α−d(v, w)α
d(u, v)α+d(v, w)α≤β(by inequality (3)).
Summing over all nodes, we get the total energy is increased by at most a factor of β. Observe
that we count the decrease in the energy of even-layer nodes exactly once since each node has
a unique parent in T.
Obviously, after at most log(n/D)iterations the diameter is at most D. Therefore, P(LC)≤
βlog(n/D)·P(M ST ). The theorem follows.
It follows from the proof that β≤2for α= 2. Finally, we consider the combined algorithm
that selects the best solution among the root cover and the layer contraction algorithms.
Corollary 5.5 The approximation factor of the combined algorithm is at most
min(Dα−1,12(n/D)log β).
In particular, for α= 2 we have β≤2, and the approximation factor is at most O(√n)for
any Dand at most O(log n)for D=O(log n)and D= Ω(n/ log n).
21

5.2 Bounded-Diameter MST Algorithm
In this section we describe an algorithm, which reduces our problem to the BDMST problem. In
the BDMST problem, we are given a connected, undirected graph Gr= (Vr, Er)on nr=|Vr|
vertices and an integer bound K. The goal is to find a spanning tree Tof Grwhose diameter
does not exceed Kso as to minimize the weight of T. The BDMST problem is NP-hard for 4≤
K < nr[23]. Kortsarz and Peleg [27] show a lower bound of Ω(log nr)on the approximation
ratio and describe a O(Klog nr)approximation algorithm that combines a greedy heuristic and
exhaustive search. Naor and Schieber [31] give a polynomial time O(log nr)-approximation
algorithm for directed graphs that uses linear programming, where the path lengths from the
root to the rest of the vertices are at most twice the given bound.
We construct a complete weighted graph in which the weight of an edge is the energy re-
quired to establish it. Then we apply the algorithm of [31] to this graph and obtain an arbores-
cence, which is used to specify the energy assignment. The BDMST algorithm is described in
Figure 6.
1. Construct a complete weighted graph G′with the set of nodes V, where the weight of
an edge (u, v)is d(u, v)α(the energy required to establish this edge).
2. Compute a BDMST of G′,T, using the algorithm of [31] with K=D.
3. Set the transmission power of each node to be equal to the weight of the heaviest
incident edge in Tdirected outward the root.
4. If the diameter of Gis greater than D, apply Steps 3,4of the layer contraction algo-
rithm. (We may need to decrease the diameter of Gby half since the diameter of Tis
bounded by 2D.)
Figure 6: The BDMST algorithm.
22

Note that Step 4of the BDMST algorithm incurs only a constant blowup in the total energy
(see the proof of Theorem 5.4). Suppose that the energy cost of an optimal BDMST with
diameter D(we count only the heaviest incident edge to each node) is at most f(n)times larger
than the energy cost of an optimal solution for the bounded-hop broadcast problem. We obtain
that the BDMST algorithm achieves an approximation factor of O(f(n)·log n). We note that
in [39] the complete cost (all edges are counted) of a minimum spanning tree with unbounded
diameter is shown to be a constant factor larger than the energy cost of an optimal broadcast
with unbounded latency. That is not true in our model since for a star topology and D= 1, the
complete cost of an optimal BDMST is n−1times larger than the cost of an optimal solution
for the bounded-hop broadcast problem. However, in this case the energy cost is exactly the
same for both solutions and it may still turn out that f(n)is a constant.
5.3 Decremental Distance Heuristic for Bounded-Hop Broadcast
In this section we describe the decremental distance heuristic. In a nutshell, we begin with
the solution of the MST algorithm. Then we greedily find either the maximal absolute power
decrease or the minimal average power increase that decreases the distance of some nodes
that are more than Dhops apart from the root. This process terminates when the diameter
constraint is satisfied. The decremental distance heuristic is presented in Figure 7. Intuitively,
this algorithm will tend to increase the power of the nodes that are closer to the root, which
allows us to simultaneously decrease the distance from the root for many nodes.
The following theorem considers the running time of the decremental distance algorithm.
Theorem 5.6 The decremental distance algorithm terminates after at most n2−nD iterations
23

1. Apply the MST algorithm [39] and set the transmission power of each node accord-
ingly.
2. If the diameter of the communication graph is smaller than D, return the current solu-
tion.
3. For each node uand for each possible level of the transmission power P > PuDo:
(a) Let mbe the number of nodes at distance more than Dfrom the root for which
this distance is decreased if the transmission power of uis increased to P.
(b) Assume that the transmission power of ubecomes P. Go over all nodes w6=u
and find the the minimal transmission power of w,P′′ < Pw(if any) so that
(i) the distance from the root to any node does not increase beyond Dand (ii)
the coverage is maintained. Let P′be the total power decrease over all consid-
ered w. (When the transmission range of uis increased, the transmission ranges
of the other nodes may be decreased without affecting feasibility of the current
solution.)
4. Select a pair u, P with m > 0with the maximum negative value of (P−Pu−P′)
(the absolute energy decrease); otherwise select a pair u, P with the minimum positive
value of (P−Pu−P′)/m (the average energy increase).
5. Set the transmission power of uto be P.
6. Decrease the transmission power of all nodes that contribute to P′on Step 3(b).
7. Goto Step 2.
Figure 7: The decremental distance (DD) heuristic.
and has running time of at most O(n8).
Proof: Clearly, during each iteration we decrease the distance from the root to at least one
node that is more than Dhops apart and condition (i) guarantees that we do not increase the
distance from the root to any node beyond D. Therefore, the algorithm terminates after at most
n2−nD iterations.
We argue that Step 3, which dominates the running time of each iteration, takes at most
O(n6)time. Note that we have to consider at most npossible transmission for each node
24

u, w. Thus, there can be at most n4possible quadruples u, P, w, P ′′. Clearly, we can check the
conditions (i) and (ii) in O(n2)time. The theorem follows.
6 Communication Problems
In this section we consider different minimum-energy communication problems in ad hoc net-
works and reduce them to connectivity problems in a directed weighted graph. A similar reduc-
tion has been used in [29, 8].†
Given the placement of nodes P, we construct a directed graph G′= (V′, E ′)as follows.
For each node vi∈V, we create n−1new nodes vj
i:j= 1,...,n,j 6=i. Then for each pair
of nodes (vi, vj
i), we add a directed edge between viand vj
iof weight d(vi, vj)α. We also add a
directed edge of weight zero between vj
iand vmif d(vi, vj)≥d(vi, vm). Note that G′contains
n2vertices and O(n3)edges.
Full-Duplex Connection. We are a given two endpoints s, d ∈V. We wish to establish a
full-duplex connection between sand d. We use an algorithm proposed by Natu and Fang [33]
in order to solve the following problem. Given a directed weighted graph Grand a pair of nodes
s, d, our goal is to find a minimum-weight subgraph Hof Grthat contains paths from sto d
and from dto s. Natu and Fang propose an algorithm with running time O(mrnr+n2
rlog nr),
where nris the number of nodes and mris the number of edges in Gr. We apply the algorithm
of [33] with G′as the input graph to obtain an optimal solution in O(n5)time.
Generalized Multicast. We are a given a subset P={v1,...,vq} ⊂ Vof senders and a
†We note that our reduction works also for the asymmetric energy model, where the energy cost of an edge
(u, v)may differ from that of (v, u).
25

subset P′={u1,...,uq′} ⊂ Vof receivers that have fixed cardinality q, q′. We wish the trans-
mission from each sender to reach all the receivers, i.e., to establish directed paths from every
node of Pto every node of P′. We assume that different transmissions can be aggregated if
they pass through the same edge. We can use the algorithm provided by Feldman and Ruhl [18]
for the p-Directed Steiner Network problem. In this problem, we are given a directed weighted
graph Grand ppairs of nodes {(s1, d1),...,(sp, dp)}and our goal is to find a minimum-weight
subgraph Hof Grthat contains all paths si→di. Feldman and Ruhl propose an algorithm
with running time O(mrn4p−2
r+n4p−1
rlog nr), where n(resp. m) is the number of nodes (resp.
edges) in Gr. We apply the algorithm of [18] to G′with qq′input pairs of nodes {(vi, uj)}. In
this way, we get an optimal solution in O(n8qq′−1)time.
Web-Conferencing. We are a given a subset P={v1,...,vq} ⊂ Vof fixed cardinality p
that contains nodes participating in a conference. We wish that all nodes in Pwould be able
to communicate with each other. We can use another algorithm provided by Feldman and Ruhl
[18], where they solve the p-Strongly Connected Steiner Subgraph problem. In this problem,
we are given a directed weighted graph Grand pvertices w1,...,wpand we need to find a
minimum-weight strongly connected subgraph Hof Gr, that contains w1, . . . , wp. Feldman
and Ruhl give an algorithm with running time O(mrn2p−3
r+n2p−2
rlog nr), where nr(resp. mr)
is the number of nodes (resp. edges) in Gr. We can apply this algorithm to G′obtaining an
optimal solution in O(n4q−3)time.
26

7 Conclusion and Open Problems
We have presented tight analysis of the broadcast and the convergecast games in ad-hoc net-
works under a novel model of energy cost sharing. We show that cost sharing significantly
improves the price of anarchy. Unfortunately, the price of anarchy still remains quite high,
which suggests a need for some sort of coordination mechanism. An interesting future research
direction is to study the case of mobile nodes.
We have also proposed approximation and heuristic algorithms for the bounded-hop broad-
cast problem. There is an open question of whether the BDMST algorithm achieves an approx-
imation factor of O(log n).
References
[1] C. Ambuhl, A. E. F. Clementi, M. Di Ianni, N. Lev-Tov, A. Monti, D. Peleg, G. Rossi and
Riccardo Silvestri, “Efficient algorithms for low-energy bounded-hop broadcast in ad-hoc
wireless networks,” Proc. of the 21st Annual Symposium on Theoretical Aspects of Computer
Science (STACS’04), pp. 418-427, 2004.
[2] E. Ansheleich, A. Dasgupta, E. Tardos and T. Wexler, “Near-optimal network design with
selfish agents,” Proc. of the 35th Symp. on Theory of Computing (STOC’03), pp. 511-
520, 2003.
[3] R. Beier, P. Sanders and N. Sivadasan, “Energy optimal routing in radio networks using geo-
metric data structures,” Proc. of the 29th International Colloquium on Automata, Languages
27

and Programming (ICALP’02), pp. 366-376, 2002.
[4] L. Buttyan and J. P. Hubaux, “Stimulating cooperation in self-organizing mobile ad hoc
networks,” Mobile Networks and Applications, Vol. 8, pp. 579-592, 2003.
[5] J. Goodman and J. O’Rourke, Eds, “Handbook of Discrete and Computational Geometry,”
CRC Press, 1997.
[6] M. Cagalj, J. P. Hubaux and C. Enz, “Minimum-energy broadcast in all-wireless networks:
NP-completeness and distribution issues,” Proc. of the 8th Annual International Conference
on Mobile Computing and Networking (MobiCom’02), pp. 172-182, 2002.
[7] G. Calinescu, S. Kapoor, A. Olshevsky and A. Zelikovsky, “Networks lifetime and power
assignment in ad hoc wireless networks,” Proc. of the 11th European Symposium on Algo-
rithms (ESA’03), pp. 114-126, 2003.
[8] I. Caragiannis, C. Kaklamanis and P. Kanellopoulos, “Energy efficient wireless network
design,” Proc. of 14th Int. Symp. on Algorithms and Computation (ISAAC’03), pp. 585-
594, 2003.
[9] J. Cartigny, D. Simplot and I. Stojmenovic, “Localized minimum-energy broadcasting in
ad-hoc networks,” Proc. of the 22nd Annual Joint Conference of the IEEE Computer and
Communications Societies (INFOCOM’03), 2003.
[10] I. Chlamtac and A. Farago, “A new approach to the design and analysis of peer-to-peer
mobile networks,” Wireless Networks, Vol. 5, pp. 149-156, 1999.
28

[11] A. E. F. Clementi, A. Ferreira, P. Penna, S. Perennes and R. Silvestri, “The minimum
broadcast range assignment problem on linear multi-hop wireless networks,” Theor. Comput.
Sci., Vol. 1-3(299), pp. 751-761, 2003.
[12] A. E. F. Clementi, G. Huiban, P. Penna, G. Rossi and P. Vocca, “Some recent theoretical
advances and open questions on energy consumption in static ad hoc wireless networks,”
Proc. of the 3rd International Workshop on Approximation and Randomized Algorithms in
Communication Networks (ARACNE’02), pp. 23-38, 2002.
[13] M. Conti, E. Gregori and G. Maselli, “Cooperation issues in mobile ad hoc networks,”
Proc. of the International Workshop on Wireless Ad Hoc Networking (WWAN’04), 2004, to
appear.
[14] J. Crowcroft, R. Gibbens, F. Kelly and S. Ostring, “Modelling Incentives for Collaboration
in Mobile Ad Hoc Networks,” Proc. of the Modeling and Optimization in Mobile, Ad Hoc
and Wireless Networks (WiOpt’03), 2003.
[15] S. Eidenbenz, V. S. Anil Kumar and S. Zust, “Equilibria in topology control games for ad
hoc networks,” Proc. of the Joint Workshop on Foundations of Mobile Computing (DIALM-
POMC’03), pp. 2-11, 2003.
[16] A. Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, and S. Shenker, “On a network
creation game,” Proc. of the 22nd ACM Symposium on Principles of Distributed Computing
(PODC’03), pp. 347-351, 2003.
29

[17] A. Fabrikant, C. H. Papadimitriou, and K. Talwar, “On the complexity of pure equilibria,”
Manuscript.
[18] J. Feldman and M. Ruhl, “The Directed Steiner Network problem is tractable for a con-
stant number of terminals”, Proc. of the 40th IEEE Symposium on Foundations of Computer
Science (FOCS’99), pp. 299-308, 1999.
[19] M. Felegyhazi, L. Buttyan, and J. P. Hubaux, “Equilibrium analysis of packet forwarding
strategies in wireless ad hoc networks – the static case,” Proc. of the 8th International Con-
ference on Personal Wireless Communications (PWC’03), Venice, Italy, 23-25 September,
2003.
[20] M. Frodigh, P. Johansson and P. Larsson, “Wireless ad hoc networking - The art of net-
working without a network,” Ericsson Review, No. 4, pp. 248-263, 2000.
[21] S. Funke, D. Matijevic, and P. Sanders, “Approximating energy efficient paths in wireless
multi hop networks,” Proc. of the 11th European Symposium on Algorithms (ESA’03), pp.
230-241, 2003.
[22] I. Gaber and Y. Mansour, “Centralized broadcast in multihop radio networks,” Journal of
Algorithms, Vol. 46(1), pp. 1-20, 2003.
[23] M. R. Garey and D. S. Johnson, “Computers and Intractability: A Guide to the Theory of
NP Completeness,” W. H. Freeman, San Francisco, 1979.
30

[24] R. Gandhi, S. Parthasarathy, and A. Mishra, “Minimizing broadcast latency and redun-
dancy in ad hoc networks,” Proc. of the 4th ACM International Symposium on Mobile Ad
Hoc Networking and Computing (MobiHoc’03), pp. 222-232, 2003.
[25] S. Guha and S. Khuller, “Improved methods for approximating node weighted steiner trees
and connected dominating sets,” Inf. Comput., 150(1), pp. 57-74, 1999.
[26] L. M. Kirousis, E. Kranakis, D. Krizanc, and A. Pelc, “Power consumption in packet radio
networks,” Theoretical Computer Science, Vol. 243, pp. 289-305, 2000.
[27] G. Kortsarz and D. Peleg, “Approximating the weight of shallow Steiner trees,” Discrete
Applied Mathematics, Vol. 93, pp. 265-285, 1999.
[28] E. Koutsoupias, C. H. Papadimitriou, “Worst-case equilibria,” Proc. of the 16th Annual
Symposium on Theoretical Aspects of Computer Science (STACS’99), pp. 404-413, 1999.
[29] W. Liang, “Constructing minimum-energy broadcast trees in wireless ad hoc networks,”
Proc. of the 3rd ACM International Symposium on Mobile Ad Hoc Networking and Comput-
ing (MobiHoc’02), pp. 112-122, 2002.
[30] P. Michiardi and R. Molva, “A game theoretical approach to evaluate cooperation enforce-
ment mechanisms in mobile ad hoc networks,” Proc. of the Modeling and Optimization in
Mobile, Ad Hoc and Wireless Networks (WiOpt’03), 2003.
[31] J. Naor and B. Schieber, “Improved approximations for shallow-light spanning trees,”
Proc. of the IEEE Conf. on Foundations of Computer Science (FOCS’97), pp. 536-541, 1997.
[32] J. F. Nash, “Non-cooperative games,” Annals of Mathematics, vol. 54, pp. 286-295, 1951.
31

[33] M. Natu and S. Fang, “On the point-to-point connection problem,” Information Processing
Letters, Vol. 53(6), pp. 333-336, 1995.
[34] C. Perkins, “Ad hoc networking: an introduction,” In C. Perkins, ed. Ad Hoc Networking,
pp. 314-318, 2001.
[35] T. S. Rappaport, “Wireless Communications: Principles and Practices,” Prentice Hall,
1996.
[36] T. Roughgarden and E. Tardos, “How bad is selfish routing?” Journal of the ACM, Vol.
49(2), pp. 236-259, 2002.
[37] S. Singh, C. Raghavendra, and J. Stepanek, “Power-aware broadcasting in mobile ad hoc
networks,” Proc. of the 10th IEEE International Symposium on Personal, Indoor and Mobile
Radio Communications (PIMRC’99), pp. 181-190, Osaka, Japan, September 1999.
[38] V. Srinivasan, P. Nuggehalli, C. F. Chiasserini and R. R. Rao, “Cooperation in wireless
ad hoc networks,” Proc. of the 22nd Annual Joint Conference of the IEEE Computer and
Communications Societies (INFOCOM’03), pp. 808-817, 2003.
[39] P. J. Wan, G, Calinescu, X. Y. Li and O. Frieder, “Minimum-energy broadcast routing in
static ad hoc wireless networks,” Wireless Networks, 2002.
[40] J. E. Wieselthier, G. D. Nguyen and A. Ephremides, ”Energy-Efficient Broadcast and
Multicast Trees in Wireless Networks,” Mobile Networks and Applications (MONET), Vol.
7(6), pp. 481-492, 2002.
32