Maximizing the Number of Broadcast

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					208                                               IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,                       VOL. 22, NO. 2,   FEBRUARY 2011




                Maximizing the Number of Broadcast
                 Operations in Random Geometric
                    Ad Hoc Wireless Networks
          Tiziana Calamoneri, Andrea E.F. Clementi, Emanuele G. Fusco, and Riccardo Silvestri

       Abstract—We consider static ad hoc wireless networks whose nodes, equipped with the same initial battery charge, may dynamically
       change their transmission range. When a node v transmits with range rðvÞ, its battery charge is decreased by  rðvÞ2 , where  > 0 is a
       fixed constant. The goal is to provide a range assignment schedule that maximizes the number of broadcast operations from a given
       source (this number is denoted by the length of the schedule). This maximization problem, denoted by MAX LIFETIME, is known to be
       NP-hard and the best algorithm yields worst-case approximation ratio Âðlog nÞ, where n is the number of nodes of the network. We
       consider random geometric instances formed by selecting n points independently and uniformly at random from a square of side length
       pffiffiffi
          n in the euclidean plane. We present an efficient algorithm that constructs a range assignment schedule having length not smaller
       than 1=12 of the optimum with high probability. Then we design an efficient distributed version of the above algorithm, where nodes
       initially know n and their own position only. The resulting schedule guarantees the same approximation ratio achieved by the
       centralized version, thus, obtaining the first distributed algorithm having provably good performance for this problem.

       Index Terms—Energy-aware systems, wireless communication, graph algorithms, network problems.

                                                                                 Ç

1     INTRODUCTION

I  N static ad hoc wireless networks, nodes have the ability
   to vary their transmission ranges (and thus, their energy
consumption) in order to provide good network connectiv-
                                                                                     range values. For this reason, we will assume that nodes
                                                                                     have the ability to choose their transmission range from a
                                                                                     finite set À ¼ f0; r1 ; r2 . . . ; rk g (with 0 < r1 < r2 < Á Á Á < rk )
ity and low energy consumption at the same time. More                                that depends on the particular adopted technology (see [7],
precisely, the transmission ranges determine a (directed)                            [8], [24]). Clearly, the maximal range value rk in À must
communication graph over the set V of nodes. Indeed, a                               be sufficiently large to guarantee that at least one feasible
node v, with range r, can transmit to another node w if and                          solution exists. Further technical constraints on À will be
only if w belongs to the disk of radius r centered in v. The                         given and discussed in Section 2.
transmission range of a node depends, in turn, on the
energy power supplied to the node. In particular, the power                          1.1   Range Assignments in Ad Hoc Wireless
Pv required by a node v to correctly transmit data to another                              Networks
station w must satisfy the inequality (see [24]):                                    A fundamental class of problems, underlying any phase of a
                                                                                     dynamic resource allocation algorithm in ad hoc wireless
                                 Pv
                                           ! ;                                      networks, is the one known as range assignment problems.
                            distðv; wÞ2                                              Given a specific graph connectivity property Å, the
where distðv; wÞ is the euclidean distance between v and w,                          objective of these problems is to find a transmission range
while  is a constant that, without loss of generality, can be                       assignment r : V ! À such that: 1) r induces a communica-
fixed to 1.                                                                          tion graph satisfying Å and 2) its overall cost
   In several previous theoretical works [1], [10], [17], [22], it                                                  X
is assumed that nodes can arbitrarily vary their transmis-                                                costðrÞ ¼    rðvÞ2 ;
sion range over the set fdistðv; wÞ j v; w 2 V g. However, in                                                                 v2V

some network models (like sensor networks), the adopted                              required to deploy the assignment [17], [22], is minimized.
technology allows to have only few possible transmission                                Several research works [1], [10], [17] have been devoted to
                                                                                     the case where Å requires the communication graph to
. T. Calamoneri, E.G. Fusco, and R. Silvestri are with the Dipartimento di           contain a directed spanning tree rooted at a given source
                                    `
  Informatica, Sapienza Universita di Roma, Via Salaria 113, 00198 Roma,             s 2 V (a broadcast tree from s). The relevance of this
  Italy. E-mail {calamo, fusco, silvestri}@di.uniroma1.it.                           problem, denoted by MIN ENERGY BROADCAST, is due to
                                                                    `
. A.E.F. Clementi is with the Dipartimento di Matematica, Universita degli
  studi di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma,
                                                                                     the fact that any communication graph satisfying the above
  Italy. E-mail clementi@mat.uniroma2.it.                                            property allows the source to perform a broadcast operation.
Manuscript received 7 Jan. 2009; revised 21 Apr. 2009; accepted 25 Sept.             Broadcast is a task initiated by the source aiming to send a
2009; published online 2 Apr. 2010.                                                  message to all nodes. This task constitutes a fundamental
Recommended for acceptance by S. Olariu.                                             operation in real-life multihop wireless networks [2], [3],
For information on obtaining reprints of this article, please send e-mail to:
tpds@computer.org, and reference IEEECS Log Number TPDS-2009-01-0005.                [17]. MIN ENERGY BROADCAST is known to be NP-hard
Digital Object Identifier no. 10.1109/TPDS.2010.77.                                  even when jÀj ¼ 3 and r1 is a small positive constant [10].
                                               1045-9219/11/$26.00 ß 2011 IEEE       Published by the IEEE Computer Society
CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS                              209


A series of constant-factor approximation algorithms is              1.3 Our Results
available in the literature (see, e.g., [1], [4], [10], [19]). The   To the best of our knowledge, previous analytical results on
best known approximation factor is close to 4 and it is given        MAX LIFETIME concern worst-case instances only. Some
in [6]. A more general version of MIN ENERGY BROADCAST               experimental studies on MIN ENERGY BROADCAST have
is studied in [5], where a nonuniform node efficiency function       been done on random geometric instances [11], [19]. Such
e : V ! Rþ is considered. Hence, the energy cost required to         input distributions turn out to be very important in the
transmit from node v to w is given by distðv; wÞ2 =eðvÞ. This        study of range assignment problems. On one hand, they
nonsymmetric version of MIN ENERGY BROADCAST seems                   represent the most natural random instance family, where
to be harder: the best known algorithm is given in [5] and           greedy heuristics (such as the MST-based one, see [17]) have
yields approximation ratio Âðlog nÞ.                                 a bad behavior [19]. On the other hand, random geometric
                                                                     distributions provide a good model for well-spread networks
1.2 The MAX LIFETIME Problem
                                                                     located on two-dimensional regions [7], [8], [17], [21].
The MIN ENERGY BROADCAST problem does not consider                      We study MAX LIFETIME in random geometric instances
some important ad hoc wireless network scenarios, where
                                                                     of arbitrary size: the set V is formed by n nodes selected
nodes are equipped with batteries of limited charge: the
                                                                     uniformly and independently at random from the two-
goal here is to maximize the number of broadcast                                                           pffiffiffi
                                                                     dimensional square of side length b nc. Such instances will
operations. Indeed, in a network where each device has
its own battery, minimizing only the overall power                   be simply denoted by random sets. Note that the maximal
consumption can cause early power drains in few key                  euclidean distance between two nodes in random sets is
                                                                     pffiffiffiffiffiffi
nodes, hence, disconnecting the network. This important                2n, soffiffiffiffiffiffi maximal range value rk can be assumed to be at
                                                                             p the
range assignment problem has been first analytically                 most 2n.
studied in [5] and it is the subject of our paper.                      A natural and important open question is to establish
    Time is divided into (time) periods. Period t is devoted to      whether efficiently constructible range assignment sche-
broadcast the tth message from the source s. All nodes are           dules exist for MAX LIFETIME having provably good length
initially equipped with the same battery charge B > 0.               on random sets. Moreover, the design of efficient distributed
    A range assignment schedule S is a sequence of range             implementations of such schedules is of particular rele-
assignment frt : V ! À; t ¼ 1; . . . ; mg.                           vance in ad hoc wireless networks.
    The length m of a range assignment schedule is the                  To this aim, as a first step, we provide an upper bound
number of periods. At every period t, the battery charge of          on the length of an optimal range assignment schedule S for
each node v is reduced by amount  rt ðvÞ2 , where rt ðvÞ            any finite set V in the two-dimensional plane. Note that this
denotes the range assigned to node v during t and  > 0 is a         upper bound holds for any instance, not only for random
fixed constant depending on the adopted technology. So, a            sets. When V is a random set, we present an efficient
range assignment schedule is said to be feasible if, at any          centralized algorithm that, with high probability, returns a
period t, rt yields a broadcast tree from s, and for any v 2 V ,     feasible schedule of length which is not smaller than 1=12 of
it holds that                                                        the optimum. Here and in the sequel, the term with high
                                                                     probability means that the event holds with probability at
                        X
                        m
                                                                     least 1 À 1=nc for some constant c > 0.
                              rt ðvÞ2   B:
                        t¼1
                                                                        We then exploit our algorithm in order to design a fully
                                                                     distributed protocol for MAX LIFETIME. The protocol acts in
In this paper, we assume that  ¼ 1; however, all our results        discrete time slots1 and assumes that every node initially
can be easily extended to any  > 0. The MAX LIFETIME                knows n, its unique label in ½1; . . . ; nŠ (our results remain
problem requires to find a feasible range assignment                 valid even if labels are chosen in ½1; . . . ; NŠ, where
schedule of maximal length.                                          N 2 OðnÞ), and its euclidean position.
   In [5], MAX LIFETIME is shown to be NP-hard. In the                  This assumption is reasonable in static ad hoc wireless
same paper, by means of a rather involved reduction to               networks since the node position can be either stored in the
MIN ENERGY BROADCAST with nonuniform node effi-                      node during the deployment phase or it can be locally
ciency, a polynomial-time algorithm is provided, yielding            computed using a GPS system in a setup phase. This
approximation ratio Âðlog nÞ. This positive result also holds        operation is not too expensive in terms of energy consump-
when the initial node battery charges are not uniform.               tion since it is performed only once during the setup phase.
   A static version of MAX LIFETIME has been studied in              We then show that the resulting schedule is equivalent to
[21]: the broadcast tree is fixed during the entire schedule         the one yielded by the centralized version, and hence, when
and the quality of solutions returned by the MST-based               applied to random sets, it achieves, with high probability,
algorithm is investigated. Such results and techniques are           constant approximation ratio as well. We thus get the first
not useful for solving MAX LIFETIME problem, as the
                                                                     distributed algorithm for MAX LIFETIME having provably
broadcast tree may change at each period.
                                                                     good performance.
   Several other problems concerning network lifetime have
                                                                        We remark that in the analysis of our protocol, we
been studied in the literature [7], [8], [21]. Their definitions
                                                                     consider both the costs due to the construction of the range
vary depending on the particular node technology (i.e.,
                                                                     assignment schedule and that due to its use for the broadcast
fixed or adjustable node power) and on the required
                                                                     operations. Furthermore, our protocol is designed to take
connectivity or covering property. However, both results
and techniques (most of them being experimental) are not                1. Thus, periods are, in turn, divided into discrete time slots: a full
related to ours.                                                     description of the distributed model is given in Section 5.
210                                        IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,          VOL. 22, NO. 2,   FEBRUARY 2011


care about message collisions yielded by the interference              the square grows. In this paper, we adopt the former
problems, thus, no cost is hidden in the analysis.                     approach to simplify calculations; nevertheless, our results
   The MAX LIFETIME problem does not consider the goal                 can be easily translated to suit the latter approach—by
of minimizing the completion time of the broadcast opera-              means of a unit conversion—and thus, handle networks of
tions, i.e., the number of time slots required to get all the          growing sizes where nodes have fixed ranges.
nodes informed (a node is said to be informed if it has                   Assumptions on range set À. We recall that pffiffiffiffiffiffi          À¼
received the source message). However, the completion                  f0; r1 ; r2 . . . ; rk g is such that 0 < r1 < r2 < Á Á Á < rk 2n.
time is clearly an important measure to evaluate the                   In addition, we assume that 1 r1 < CTðnÞ. This condition
efficiency of a solution. We thus provide an analysis of               is motivated by our choice of studying random sets. Indeed,
the amortized completion time of each broadcast operation              define Cs as the connected component containing s in the
yielded by our protocol. If the length of the range assign-            disk graph GðV ; r1 Þ. If r1 ! CTðnÞ, then MAX LIFETIME on
ment schedule generated by the protocol is T , then the
                                                                       random sets admits a trivial schedule which is, with high
amortized completion time is given by the overall number
                                                                       probability, optimal: Since the source must transmit in
of elapsed time slots divided by T . It turns out that our
                                                                       every period with range at least r1 and Cs , with high
protocol has amortized completion time
                                                                       probability, contains all nodes, then an optimal schedule is
                      pffiffiffi          pffiffiffi
                      r2 n n           n                               obtained by assigning range r1 to all other nodes at every
                   O          þ r2 þ       :
                         T       2
                                      r2                               period. This motivates our assumption on r1 .
                                                                          The other values in À can be arbitrarily chosen provided
Assume that r2 2 À is close to the connectivity threshold of           that all of them are not smaller than CTðnÞ and at least one of
randompffiffiffiffiffiffiffiffiffiffi
          geometric graphs [16], [20], [25], [26], i.e.,                                              pffiffiffi pffiffiffiffiffiffiffiffiffiffi
                                                                       them is larger than 2 2c log n, where c >  is a small
r2 ¼ Âð log nÞ. Then, the worst scenario for our protocol is
                                                                       constant that will be defined later in Lemma 4.1. Informally
when the initial battery charge B is very small so that T is
                                                                       speaking, we require that at least one value in À is a bit
small as well. Indeed, if T 2 Oð1Þ, from thepffiffiffiffiffiffiffiffiffiffiffiffiffiffi formula,
                                            previous
                                                                       larger than the connectivity threshold. This is reasonable
           amortized completion time Oðn n log nÞ, which is
we get anpffiffiffiffiffiffiffiffiffiffiffiffiffiffi
                                                                       and relevant in energy problems related to random
a factor n log n larger than the best known distributed
broadcasting time, i.e., OðnÞ [16]. However, this protocol             geometric wireless networks since this value is the minimal
does not take into account node energy costs, and thus, the            one achieving global connectivity with high probability.
lifetime of the network. Our protocol, instead, trades                 Further discussion on such assumptions can be found in
completion time of each broadcast operation with global                Section 6.
network lifetime. This fact clearly arises whenever B is large
                          pffiffiffi
                           nÞ number of broadcast operations:
enough to allow T 2 ð pffiffiffiffiffiffiffiffiffiffi                                     3    THE UPPER BOUND
in this case, we get Oðn log nÞ amortized completion time,
                                                                       In this section, we provide an upper bound on the length of
which is very close to the completion time of the best known
distributed broadcast algorithm.                                       any feasible range assignment schedule for a set V .
                                                                       Lemma 3.1. Given a set V and a source s 2 V , it holds that
                                                                         optðV ; sÞ B=r2 . Furthermore, if the size k1 of Cs is less
2     PRELIMINARIES                                                                    1
                                                                         than n, then
A random set V is formed by n nodes selected uniformly and
independently at random from the square Q of side length                             optðV ; sÞ
pffiffiffi                                                                                        &                        '
   n. The source node s can be any node in V . The length of a                                B BÀ 2     2       2
                                                                                                                   Á
maximum feasible range assignment schedule (in short,                                   min 2 4 k1 r2 þ r1 À k1 r1 :
                                                                                              r1 r2
schedule) for an input ðV ; sÞ is denoted by optðV ; sÞ.
    Given a set V of n nodes in the two-dimensional euclidean          Proof. Since the source must transmit with range at least r1
plane and a positive real r, the disk graph GðV ; rÞ is the              at any period, the first upper bound follows easily.
symmetric graph where edge ðu; vÞ exists iff distðv; wÞ r.                   If k1 < n, then consider any feasible range assignment
When V is a random set, the resulting disk graph distribution            schedule S. Let l1 and l2 be the number of periods, where
is known as geometric random graphs, an important and deeply             the source transmits with range r1 and with range at least
studied model in wireless networks [16], [20], [25], [26]. It is         r2 , respectively. It must hold that
known that for sufficiently large n, a random geometric
graph GðV ; rÞ is connected with high probability if and only if                               l1 r2 þ l2 r2
                                                                                                   1       2   B:
       pffiffiffiffiffiffiffiffiffiffi
                           þ  for
r !  log n, where  ¼ 1 pffiffiffiffiffiffiffiffiffiffi any constant  > 0 [20], [25],
                                                                           Since k1 < n, then, in each of the l1 periods of S, there is
[26]. The value CTðnÞ ¼  log n is known as the connectivity
                                                                           at least one node in Cs À fsg having radius at least r2 .
threshold of random geometric graphs.
                                                                           This yields
    Note that the connectivity threshold grows when the
area of the square grows as a consequence of the                                              l1 r2   ðk1 À 1ÞB:
                                                                                                  2
assumption of having a fixed node density for squares of
any size. A more realistic assumption would be to consider                 The maximum value of l1 þ l2 is achieved when l1 ¼
node ranges as fixed. As a consequence, higher node                        ðk1 À 1ÞB=r2 and l2 ¼ B=r2 À ðk1 À 1ÞBr2 =r4 . As l1 þ l2 is
                                                                                       2              2             1 2
densities would be required in order to maintain the                       the length of the schedule, we obtain the following upper
network connected with high probability while the area of                  bound on the number of periods of S:
CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS                                     211




Fig. 1. Execution example for algorithm BS. The source message is sent through nodes in Cs to the first pivot, using range r1 . When a pivot transmits
with range r2 , all nodes in its cell and in the neighboring cells receive the message.

             l1 þ l2                                                            let Cs be the connected component in GðV ; r1 Þ that
                          &                          '
                            B BÀ 2        2       2
                                                    Á                             contains s;
                       min 2 ; 4 k1 r2 þ r1 À k1 r1 :                           if jCs j r2 then
                           r1 r2                                                              2
                                                                       u
                                                                       t          Ws       Cs ;
                                                                                else
  Note that if V is a random set, then, since r1 < CTðnÞ, it                      Ws        any connected subgraph of Cs s.t. jWs j ¼ r2 and
                                                                                                                                          2
holds with high probability k1 < n.                                                 s 2 Ws ;
                                                                                construct an arbitrary ordering of Ws ;
4    THE ALGORITHM                                                              for any period t ¼ 1; . . . do
                                                                                    if node with index t mod jWs j in Ws has remaining
In this section, we present a simple and efficient algorithm for
                                                                                      battery charge at least r2 then
                                                                                                                   2
MAX LIFETIME and then analyze its performance. For the
                                                                                        it is selected as pivot and range r2 is assigned to it;
       of
sake pffiffiffi simplicity, we restrict ourselves to the case
            pffiffiffiffiffiffiffiffiffiffi                                                             else
r2 ! 2 2c log n. Nevertheless, it is easy to extend all our
                                                                                        the algorithm stops;
results to the more general assumption described in Section 2.                      for any cell Qj do
    The following algorithm partitions Q into square cells
                                                                                      if node with index t mod jVj j in Qj has remaining
and selects, for every period, a set of pivots, i.e., nodes
                                                                                        battery charge at least r2 then
                                                                                                                     2
having assigned range r2 . Each set of pivots is responsible
                                                                                          it is selected as pivot and range r2 is assigned to it;
for spreading the message of its period. This message is
                                                                                      else
delivered to one of the pivots from the source by means of
                                                                                          the algorithm stops;
transmissions with range r1 , thus, exploiting the subgraph
                                                                                    all nodes in Ws not selected yet have radius r1 ;
Cs . Observe that we here assume every node knows the                               all nodes in V n Ws not selected yet have range 0.
positions of all the other nodes and the cell partition: the
algorithm is thus centralized:                                                 In order to analyze the performance of Algorithm BS, we
                                                                            will use the following lemma whose proof is a simple
    See Fig. 1 for an example of execution of Algorithm BS.
                                                                            application of Chernoff’s bound (an alternative proof can
Algorithm 1. BS (Broadcast Schedule)                                        also be obtained from [18, Lemma 1]):
Input: Set V  Q of n nodes; a source s 2 V ; a battery                     Lemma 4.1. There exist two positive constants, c and 
, such that
charge B > 0; the range set À ¼ f0; r1 ; r2 . . . ; rk g.                     the following holds. Given a random set V  Q of n nodes, and
Output: A range-assignment schedule S.                    pffiffiffi                  partition of pffiffiffiinto square cells of side length ‘, where
                                                                              apffiffiffiffiffiffiffiffiffiffi    Q
  Partition Q into square cells of side length r2 =ð2 2Þ;                     c log n ‘         n, every cell contains at least 
‘2 nodes with
  for any cell Qj , let Vj be the set of nodes in Qj ;                        high probability. The constants can be set as c ¼ 12 and
  construct an arbitrary ordering in Vj ;                                     
 ¼ 5=6.
212                                                      IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,         VOL. 22, NO. 2,    FEBRUARY 2011


Proof. Given a fixed cell, let Xi be the random variable that                       During the schedule, every node v in Ws will have range
  gets the value 1 if node i falls into the cell and 0                              either r1 or r2 . Let jWs j ¼ k, then the energy spent by v is
  otherwise. Observe that: 1) the event Xi ¼ 1 has                                  at most
  probability ‘2 =n and 2) the random variables Xi s,                                                              
  1 i n, are independent.                                                                                 T     8T
              P                                                                                             þ         r2 þ T r2 :             ð2Þ
     Let X ¼ n Xi . The expected value of X, i.e., the
                 i¼1
                                                                                                          k 
 r2 22
                                                                                                                              1

  average number of nodes in a cell of side ‘, is E½XŠ ¼ ‘2 .
                                                                                    Indeed, in (2), we have considered that a node in Ws can
  By applying Chernoff’s bound [23], we get
                                                                                    have range r2 because it has been selected as pivot either
                     Pr½X < 
‘2 Š < eÀ‘ ð1À
Þ =2 :
                                                    2        2
                                                                                    of its cell or of Ws .
                         pffiffiffi                                                          Now, two cases may arise:
      There are at most b n=‘c2 n=‘2 cells fully contained in
      Q, and thus, the probability of having one such cell with                        .   If k ! ðr2 Þ2 , since r1 ! 1, from (2), the amount of
                                                                                                   r1
      less than 
‘2 nodes is at most ðn=‘2 ÞeÀ‘ ð1À
Þ =2 by
                                                            2 2
                                                                                           spent energy is at most 2T r2 þ 8=
 T r2 ð2 þ
                                                                                                                               1           1
                                                pffiffiffiffiffiffiffiffiffiffi                                8=
Þ. We require B to exceed the latter value, so T
      applying the Union Bound. As ‘ ! c log n, this prob-
      ability is at most                                                                   can be any value such that

                       2
                           log nð1À
Þ2 =2þlnðn=ðc2 log nÞÞ
                                                                                                                               B
                 eÀc                                         <                                                T                        :             ð3Þ
                                                                                                                       r2 ð2
                                                                                                                        1      þ 8=
 Þ
                                              2
                                 2
                                     log nð1À
Þ =2þlogðn=ðc log nÞÞ
                                                             2
                   < eÀc                     :
                                                                                           Observe that every value T that satisfies (3) also
                                pffiffiffiffiffiffiffiffi                                                                                            B
      By setting 
 ¼ 5=6 and c ! 144 ¼ 12, we obtain                                       satisfies (1). So T can assume value r2 ð2þ8=
Þ , and
                                                                                                                                 1
                                                                                           from Lemma 3.1, we have that
                Pr½X < 
‘2 Š <
                                                                                                                       optðV ; sÞ
                   < eÀ144=72 log nþlogðn=ð144 log nÞÞ                                                            T!              :
                                                                                                                        2 þ 8=

                            eÀ2 log nþlogðn=ð144 log nÞÞ < 1=n:
                                                                            u
                                                                            t          .   If k < ðr2 Þ2 , according to the definition of Ws , we
                                                                                                   r1
                                                                                           have k ¼ k1 . From (2) and some simple calcula-
                                                                                           tions, the energy spent by v 2 Ws is at most
Theorem 4.2. Let V  Q be a random set of n nodes and s 2 V
  be any source node. Then, with high probability, the range                                              r4 þ k1 r2 r2 þ ð8=
Þk1 r2
                                                                                                           2       1 2             2
                                                                                                      T                              ;
  assignment schedule returned by Algorithm BS is feasible and                                                r2 k1 þ r2 À k1 r2
                                                                                                               2        1      1
  it has length at least  optðV ; sÞ, where  ¼ 1=12.
                                                                                           where we used the fact that r2 À k1 r2 0. Observe
                                                                                                                           1      1
Proof. Let us consider any period of the algorithm’s                                       also that, since k1 < ðr2 Þ2 and r1 ! 1, we have
                                                                                                                  r1
  schedule. The component Ws is not empty since it
  contains at least s. Hence, it contains a pivot which is
                                                                                                    k1 r2 r2 þ ð8=
Þk1 r2
                                                                                                        1 2             2
  connected to s by a path using ranges of size r1 only.                                                                                 
  From Lemma 4.1, all cells are nonempty with high                                                           4      8                     8
                                                                                                           r2 1 þ 2                r4 1 þ
                                                                                                                                    2         :
  probability. So, a pivot is selected in every cell with high                                                     
r1                    

  probability. This implies that, with high probability, the
  set of pivots forms a strongly connected subgraph                                          It thus follows that the energy spent by v is at
  whose r2 -disks cover all nodes in V . Moreover, Algo-                                   most
  rithm BS assigns, to every node, an energy power which
                                                                                                                     r4 ð2 þ 8=
Þ
  is never larger than the current battery charge of the                                                      T       2
                                                                                                                                       :
                                                                                                                   2 k þ r2 À k r2
                                                                                                                  r2 1
  node. So the range assignment schedule is feasible, with                                                                  1   1 1

  high probability.                                                                           Hence, T can be any value such that
     We now evaluate the length T of the scheduling
  produced by Algorithm BS. Observe that T equals the
                                                                                                                  r2 k1 þ r2 À k1 r2
                                                                                                                   2        1      1
  index t of the last period performed by Algorithm BS on                                                 T                          B:              ð4Þ
  input ðV ; sÞ.                                                                                                     r4 ð2 þ 8=
Þ
                                                                                                                      2

     Let w be any node in V n Ws ; then, from Lemma 4.1, in
  its cell, there are at least 
 r2 =8 nodes with high                                        Similar to the previous case, every value T that
                                        2
  probability. So, w spends at most energy                                                 satisfies (4) also satisfies (1). Finally, by combin-
                                                                                         ing (4) and Lemma 3.1, we get again
                            8T
                                   r2 :                                                                                 optðV ; sÞ
                            
 r2 2
                               2                                                                              T !                  :
                                                                                                                         2 þ 8=

      Hence, T can be any value such that
                                                                                       So, the theorem is proved for  ¼ 1=ð2 þ 8=
Þ >
                                               
B
                                       T          :                        ð1Þ      1=12.                                            t
                                                                                                                                     u
                                                8
CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS                        213


    We conclude this section by observing that when every             Observe that at the end of the Preprocessing phase, s has
node has full knowledge of the node positions, the time               full knowledge of Ws .
                                               pffiffiffi
complexity of Algorithm BS is Oðr2 þ T Á ð n=r2 ÞÞ. Indeed,
                                       2                             Broadcast operations:
the organization of the nodes according to the cell partition         for t ¼ 0; 1; . . . =à periods à = do
and the construction of an arbitrary order of the nodes in             Execute Procedure BROADCAST(mt )
every cell require linear time. The construction of Ws can be
performed by a standard graph search, so it can be done in           Procedure BROADCAST(mt )
Oðnr2 Þ time. Finally, for each period t, constant time is
     2
required to activate the right pivot in every cell. The activation   Nodes in Ws only:
of those cells having the same r2 -hop distance from the source
can be done in parallel. Since the maximum hop distance from                 s selects the ðt mod minfjWs j; r2 gÞ-th node in Ws as
                            pffiffiffi                                                                                2
the source to any cell is Oð n=r2 Þ, we get the bound.                         pivot (range r2 will be assigned to it);
                                                                               s transmits, with range r1 , hmt ; P i where P is the
5   THE DISTRIBUTED VERSION                                                    path in Tree from s to the pivot.
                                                                            When a node in Ws receives hmt ; P i, it checks
In this section, we present the distributed version of
                                                                               whether its label is the first in P . If this is the case,
Algorithm BS.
                                                                               it transmits, with range r1 , hmt ; P 0 i where P 0 is the
   According to the standard radio communication model
                                                                               residual path to the pivot.
[2], [13], [15], we assume that nodes act in discrete uniform
time slots and are nonspontaneous (but the source, the other                When the selected pivot p of Ws receives
nodes are activated when they get the source message).                       hmt ; P ¼ ðpÞi, it transmits, with range r2 , hmt ; ii
However, we assume a weaker, local synchronous model: if,                    where i is the index of its cell.
at a given time slot t, the range of a message transmission
covers a cell, then, at time slot t þ 1, the nodes of that cell      All nodes: (included the ones in Ws )
are activated, and so, they will agree on the same time. We
assume that every node v knows the number n of points, the                  If ðt ð
=8Þr2 Þ then
                                                                                             2
range À, its own unique label, and its relative coordinates in                  - When a node v receives for the first time hmt ; ii
the square grid Q.                                                                in the period t from the pivot of a neighbor cell
   Our protocol is based on the same partition in cells of                        i, it becomes active.
Algorithm BS. Cells are numbered with consecutive integers.                     - An active node, at every time slot, increments
   The aim of our protocol is to replicate the behavior of                        counter by one and checks whether its label is
Algorithm BS in a distributed fashion. In order to do so,                         equal to the value of its counter. If this is the
nodes need to acquire some knowledge of the network                               case, it becomes the pivot of its cell and
within minimal cost in terms of energy, and as we shall see,                      transmits, with range r2 , hmt ; ii where i is the
time costs.                                                                       index of its cell.
   Let hðWs Þ be the eccentricity of the source s in Ws , i.e.,                 - When an active node in cell i receives hmt ; ii, it
the maximum distance between s and a node in Ws . The tth                         (so the pivot as well) records in P ½tŠ the current
message sent by s is denoted by mt . We assume that mt                            value of counter c, i.e., the label of the pivot,
contains the value of period t. The protocol is described in                      and becomes inactive.
Algorithm 2.                                                                 else (i.e., ðt > ð
=8Þr2 Þ)
                                                                                                       2
                                                                                - When a node v receives for the first time hmt ; ii
Algorithm 2. DBS (Distributed Broadcast Schedule)
                                                                                  in period t from the pivot of a neighbor cell i, it
Preprocessing: construction of Ws  Cs such that
                                                                                  checks if its label is equal to P ½t mod ð
=8Þr2 Š. If
hðWs Þ r2  2
                                                                                                                                 2
                                                                                  this is the case, it becomes the pivot of its cell
 One-to-All
                                                                                  and transmits, with range r2 , hmt ; ji where j is
 Starting from s, use round robin and range transmission r1
                                                                                  the index of its cell.
 to inform all nodes in Cs that are at most within r2 hops
                                                      2
 from s: such nodes will form Ws .                                      Our protocol has the following properties that are a key-
 The one-to-all operation induces a spanning tree Tree of            ingredient in the performance analysis:
 Ws rooted at s.                                                     Fact 5.1. Even though they initially do not know each other, all
 All-to-One                                                            nodes in the same cell are activated (and disactivated) at the
 By a simple bottom-up process on Tree and using round                 same time slot, so their local counters share the same value at
 robin on each level, s collects all node labels and the               every time slot. Furthermore, after the first ð
=8Þr2 broadcast
                                                                                                                             2
 structure of Tree.
                                                                       operations, all nodes in the same cell know the set P of pivots
Initialization:
                                                                       of their cell and the relative order of its elements.
 Every node sets a local counter counter ¼ À1.
 Furthermore, each node has a local array P of length                Proof. A node in a cell is activated when it receives a
 ð
=8Þr2 where it will store the ordered list of the first
         2
                                                                       message from the pivot of a neighboring cell and is set
 ð
=8Þr2 labels belonging to its own cell. This array is
         2
                                                                       inactive when it receives the message sent by the pivot of
 initially empty.                                                      its cell. Each point in a cell is within distance r2 from all
214                                        IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,      VOL. 22, NO. 2,   FEBRUARY 2011


      points in the same cell and in the neighboring cells;              further messages that are sent are the ones required to
      moreover, the pivots transmit with range r2 . So activa-           reach the elected pivot in Ws : the overall number of them
      tions and deactivations of nodes in a cell happen in the           is bounded by r2 .
                                                                                          2                                       t
                                                                                                                                  u
      same time slots. This proves the first part of the claim.       Theorem 5.5. The overall number of time slots required by
         The set of pivots is learned by all nodes in a cell as a       Algorithm DBS to perform T broadcast operations is
      consequence of the transmissions made by the pivots.
                                                                                     À pffiffiffi        À    pffiffiffi ÁÁ
      Since this set has size bounded by ð
=8Þr2 , the proof of
                                                 2                                 O r2 n n þ T Á r2 þ n=r2 :
                                                                                                     2
      the claim is completed.                                   t
                                                                u
                                                                      Proof. For a single broadcast operation performed by
   More practically, the above claim implies that if l0 < l1 <          Algorithm DBS, we define the delay of a cell as the
l2 < Á Á Á < lt are the labels of the nodes in a cell, then, during     number of time slots from its activation time to the
the first ð
=8Þr2 broadcast operations (i.e., periods), the
                    2                                                   selection of its pivot. Observe that the sum of delays
pivot of the cell at period t will be the node having label lt .        introduced by a cell during the first ð
=8Þr2 broadcasts is
                                                                                                                     2
   In order to evaluate the length of the broadcast schedule            at most n À ð
=8Þr2 . Indeed, when a (new) pivot
                                                                                                2
yielded by Algorithm DBS, observe that the distributed                  transmits, a new entry in the pivot-array is determined;
version performs, in parallel, two tasks: 1) it constructs a            while in every time slot there is silence, nodes in the cell
broadcast communication subgraph starting from the                      learn that the node with the corresponding label is not in
source and 2) transmits the source message along this                   the cell. As each cell contains at least ð
=8Þr2 nodes with
                                                                                                                       2
subgraph to all nodes. In our next analysis, all node costs             high probability, the delay introduced by each cell before
due to both the above tasks are taken into account:                     completing the set is with high probability at most
whenever a node transmits any message with range r, its                 n À ð
=8Þr2 . Once the set of pivots is completed (i.e., after
                                                                                    2
battery charge is decreased by r2 .                                     the first ð
=8Þr2 periods), the delay of every cell becomes
                                                                                        2
   The following lemma states the equivalence between the               0 for all the remaining broadcasts. Moreover, a broadcast
performance of Algorithm BS and that of Algorithm DBS:                                            pffiffiffi
                                                                        can cross at most Oð n=r2 Þ cells (as the side length of
                                                                                                                            pffiffiffi
Lemma 5.2. Given a random set V  Q and any source s 2 V , if           each cell is Âðr2 Þ, while the diameter of Q is Âð nÞ). By
  the length of the broadcast schedule yielded by Algorithm BS is       assuming the worst scenario, i.e., a maximal length cell
                                                                                                       pffiffiffi
  T , then the length of the broadcast schedule yielded by              path (this length being Âð n=r2 Þ) together with maximal
  Algorithm DBS is at least T À 2.                                      cell delay can be found in each of the first
Proof. Note that the only difference in terms of power                  minfð
=8Þr2 ; T g broadcasts, we can bound the maximal
                                                                                     2                 pffiffiffi
  consumption between Algorithms BS and DBS lies in the                 overall delay with Oðr2 n nÞ time slots.
  Preprocessing phase required by the latter one. In that                  In the Preprocessing phase, Algorithm DBS uses
  phase, at most two messages with range r1 are sent by a               round-robin to avoid collisions. During the All-to-One
  node to discover Ws . Hence, in the worst case, the                   phase, each node needs to collect all messages from its
  distributed version performs two broadcasts less than                 children before sending a message to its parent in Tree.
  the centralized algorithm. Note that, due to Fact 5.1, the if         Hence, the whole phase is completed in Oðnr2 Þ time  2
  branch of the Broadcast procedure spends time instead of              slots, as the height of Tree is bounded by r2 .2
  power in order to discover the set of Pivots of each cell. t
                                                             u             Finally, the number of time slots required by every
                                                                        broadcast without delays and Preprocessing time is
                                                                                pffiffiffi
   By combining the above lemma and Theorem 4.2, we                     Oðr2 þ n=r2 Þ, since r2 is the upper bound on the height
                                                                            2                     2
easily get the following:                                               of Tree and the length of any path on the broadcast tree
                                                                                           pffiffiffi
                                                                        outside Ws is Oð n=r2 Þ.
Corollary 5.3. Let V  Q be a random set of n nodes and s 2 V              By combining the three contributions, we get the
  be any source node. Then, with high probability, the range            theorem bound without considering collisions among
  assignment schedule yielded by Algorithm DBS is feasible and          pivots of adjacent cells. In order to avoid such collisions,
  it has a length at least  optðV ; sÞ À 2, where  ¼ 1=12.            we further organize Algorithm DBS into iterative phases:
                                                                        in every phase, only cells with not colliding pivot
  We now evaluate message and time complexity of                        transmissions are active. Since the number of cells that
Protocol DBS.                                                           can interfere with a given cell is constant, this further
Lemma 5.4. The overall number of node transmissions (i.e., the          scheduling will increase the overall time of DBS by a
  message complexity) of Algorithm DBS is OðjWs j þ T Á                 constant factor only. This iterative process can be
  ððn=r2 Þ þ r2 ÞÞ, where T is the length of the schedule.
       2      2                                                         efficiently performed in a distributed way since every
Proof. Observe that in the Preprocessing phase, only nodes              node knows n and its position, so it knows its cell.        u
                                                                                                                                    t
  in Cs exchange messages. In particular, all nodes (the
  source as well) in Cs within r2 hops from s send only one              From Theorem 5.5, the amortized completion time of a
                                2
  message; all other nodes, at hop distance from 1 to r2 À 1          single broadcast operation performed by Algorithm DBS is
                                                       2
  from s, send two messages. It follows that the message                                   pffiffiffi         pffiffiffi
                                                                                           r2 n n    2    n
  complexity of the Preprocessing phase is ÂðjWs jÞ. During                             O         þ r2 þ       :
                                                                                              T          r2
  each broadcast, exactly one message per cell is sent (see
  Fact 5.1). As there are Oðn=r2 Þ cells, in each period, an
                                 2                                       Since our protocol returns an almost maximal number T
  Oðn=r2 Þ number of messages are exchanged. The only
        2                                                             of broadcast operations with high probability, unless the
CALAMONERI ET AL.: MAXIMIZING THE NUMBER OF BROADCAST OPERATIONS IN RANDOM GEOMETRIC AD HOC WIRELESS NETWORKS                                    215


available battery charge of nodes is small, the analysis we                 [10] A. Clementi, P. Crescenzi, P. Penna, G. Rossi, and P. Vocca, “On
                                                                                 the Complexity of Computing Minimum Energy Consumption
made at the end of Section 1.3 on the amortized completion                       Broadcast Subgraphs,” Proc. 18th Ann. Symp. Theoretical Aspects of
time of Algorithm DBS is likely to fall in a scenario in which                   Computer Science (STACS), pp. 121-131, www.dia.unisa.it/
T is large enough to significantly shrink the gap between                        ~penna, Feb. 2001.
                                                                            [11] A. Clementi, G. Huiban, P. Penna, G. Rossi, and Y.C. Verhoeven,
our distributed algorithms and the best known distributed                        “On the Approximation Ratio of the MST-Based Heuristic for the
broadcasting time [16].                                                          Energy-Efficient Broadcast Problem in Static Ad-Hoc Radio
                                                                                 Networks,” Proc. Int’l Parallel and Distributed Processing Symp.
                                                                                 (IPDPS ’03), vol. 222, 2003.
6     OPEN PROBLEMS                                                         [12] A.E.F. Clementi, A. Monti, and R. Silvestri, “Distributed Broadcast
                                                                                 in Radio Networks of Unknown Topology,” Theoretical Computer
In this paper, we provided efficient solutions for the MAX                       Science, vol. 302, nos. 1-3, pp. 337-364, 2003.
LIFETIME problem on random sets. Further interesting                        [13] M. Chrobak, L. Gasieniec, and W. Rytter, “Fast Broadcasting and
future studies should address other basic operations such as                     Gossiping in Radio Networks,” J. Algorithms, vol. 43, no. 2,
                                                                                 pp. 177-189, 2002.
the gossiping operation which is known to be NP-hard as                     [14] W. Chu, C.J. Colbourn, and V.R. Syrotiuk, “The Effects of
well [5]. A more technical problem, left open by our work, is                    Synchronization on Topology Transparent Scheduling,” Wireless
the study of MAX LIFETIME when À contains more than one                          Networks, vol. 12, pp. 681-690, 2006.
                                                                            [15] A. Czumaj and W. Rytter, “Broadcasting Algorithms in Radio
positive value smaller than the connectivity threshold CTðnÞ                     Networks with Unknown Topology,” J. Algorithms, vol. 60, no. 2,
of random geometric graphs. This case seems to be very                           pp. 115-143, 2006.
hard since it concerns the size and the structure of the                    [16] A. Dessmark and A. Pelc, “Broadcasting in Geometric Radio
                                                                                 Networks,” J. Discrete Algorithms, vol. 5, no. 1, pp. 187-201, 2007.
connected components of such random graphs under the                        [17] A. Ephremides, G.D. Nguyen, and J.E. Wieselthier, “On the
connectivity threshold [20], [25].                                               Construction of Energy-Efficient Broadcast and Multicast Trees in
   Finally, we emphasize that after the presentation of the                      Wireless Networks,” Proc. IEEE INFOCOM, pp. 585-594, 2000.
                                                                            [18] A.D. Flaxman, A.M. Frieze, and J.C. Vera, “On the Average Case
conference version of this work, a new protocol for MIN
                                                                                 Performance of Some Greedy Approximation Algorithms for the
ENERGY BROADCAST has been given in [9]. This new                                 Uncapacitated Facility Location Problem,” Proc. ACM Symp.
protocol is inspired by ours and achieves provably good                          Theory of Computing (STOC ’05), pp. 441-449, 2005.
performances on random-grid instances yielded by nonuni-                    [19] M. Flammini, A. Navarra, and S. Perennes, “The Real Approx-
                                                                                 imation Factor of the MST Heuristic for the Minimum Energy
form node distributions.                                                         Broadcast,” Proc. Int’l Workshop Experimental and Efficient Algo-
                                                                                 rithms (WEA ’05), pp. 22-31, 2005.
                                                                            [20] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic
ACKNOWLEDGMENTS                                                                  Connectivity in Wireless Networks,” Stochastic Analysis, Control,
                                                                                 Optimization and Applications, pp. 547-566, Birkhauser, 1999.
An extended abstract of this work has been presented at                     [21] I. Kang and R. Poovendran, “Maximizing Network Lifetime of
OPODIS 2007. This research is partially supported by the                         Wireless Broadcast Ad Hoc Networks,” J. ACM Mobile Networks
European Union under the Project IP-FP6-015964 AEOLUS.                           and Applications, vol. 10, no. 6, pp. 879-896, 2005.
                                                                            [22] L.M. Kirousis, E. Kranakis, D. Krizanc, and A. Pelc, “Power
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                                                                                 Science, vol. 243, pp. 289-305, 2000.
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[3]   R. Bar-Yehuda, A. Israeli, and A. Itai, “Multiple Communication                              Tiziana Calamoneri received the graduate
      in Multi-Hop Radio Networks,” SIAM J. Computing (SICOMP),                                    degree in mathematics in 1992 and the PhD
      vol. 22, no. 4, pp. 875-887, 1993.                                                           degree in computer science in 1997 from the
[4]   G. Calinescu, X.Y. Li, O. Frieder, and P.J. Wan, “Minimum-Energy                             University of Rome “La Sapienza,” Italy. Since
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      IEEE INFOCOM, pp. 1162-1171, Apr. 2001.                                                      Department of Computer Science, University of
[5]   G. Calinescu, S. Kapoor, A. Olshevsky, and A. Zelikovsky,                                    Rome “La Sapienza,” where she was also an
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216                                            IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,              VOL. 22, NO. 2,    FEBRUARY 2011

                        Andrea E.F. Clementi received the laurea                                      Riccardo Silvestri received the degree in
                        degree in mathematics in 1990 and the PhD                                     mathematics (magna cum laude) from the
                        degree in computer science in 1994 from the                                   University of Rome in 1987 and the PhD degree
                        University “La Sapienza” of Rome. From 1996 to                                in computer science in 1992. He is currently an
                        1998, he was an assistant professor at “La                                    associate professor in the Department of Com-
                        Sapienza.” From 1998 to 2002, he was an                                       puter Science. In 1994, he was first appointed at
                        associate professor at the University “Tor Ver-                               the University of Rome “La Sapienza.” Before
                        gata” of Rome. Since 2002, he is a full professor                             joining “La Sapienza” at Rome, he worked from
                        at the same university. His main research activity                            1987 to 1988 in two Italian software companies.
                        focuses on algorithms and complexity theory                                   In 1991, he was a guest at the University of
with special interest in randomized and/or distributed models. He            Rochester, New York. His main research interests include algorithms
published more than 50 papers in the most important international            and computational complexity, especially randomized algorithms and
journals and conferences of the area. He has visited several universities,   distributed models and algorithms. He is an author of more than
such as UCSD of San Diego, University of Geneva (where he was an             60 scientific papers in the main international journals and conferences.
assistant professor for two years), and the French Research Institute
INRIA (Sophia Antipolis). He has been the local coordinator of several
national and international research projects. He has been a member of
                                                                             . For more information on this or any other computing topic,
the Program and/or Organizing Committees of several international
                                                                             please visit our Digital Library at www.computer.org/publications/dlib.
conferences such as RANDOM/APPROX, ARACNE, ACM DIALM,
WMAN. ALGOSENSORS, the IEEE/ACM Distributed Computing in
Sensor Systems, ACM-IEEE MSWIM, and SIROCCO.

                        Emanuele G. Fusco received the graduate
                        degree in computer science in 2005 and the
                        PhD degree in computer science in 2009 from
                        Sapienza, University of Rome. He is currently a
                        research fellow in the Computer Science De-
                        partment at Sapienza, University of Rome. His
                        research interests include distributed algorithms
                        and graph theory.

				
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