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Efficient Algorithms for Periodic Scheduling


									                                           Computer Networks 45 (2004) 155–173

                   Efficient algorithms for periodic scheduling
                 Amotz Bar-Noy a, Vladimir Dreizin b, Boaz Patt-Shamir                                           b,*

                                AT&T Research Labs, 180 Park Avenue, Florham Park, NJ 07932, USA
                             Department of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
               Received 4 February 2003; received in revised form 14 November 2003; accepted 19 December 2003
                                                     Responsible Editor: N. Shroff


    In a perfectly periodic schedule, time is divided into time slots, and each client gets a slot precisely every predefined
number of time slots. The input to a schedule design algorithm is a frequency request for each client, and its task is to
construct a perfectly periodic schedule that matches the requests as ‘‘closely’’ as possible. The quality of the schedule is
measured by the ratios between the requested frequency and the allocated frequency for each client (either by the
weighted average or by the maximum of these ratios over all clients). Perfectly Periodic schedules enjoy maximal
fairness, and are very useful in many contexts of asymmetric communication, e.g., push systems and Bluetooth net-
works. However, finding an optimal perfectly periodic schedule is NP-hard.
    Tree scheduling is a methodology for developing perfectly periodic schedules based on hierarchical round-robin,
where the hierarchy is represented by trees. In this paper, we study algorithms for constructing scheduling trees. First,
we give optimal (exponential time) algorithms for both the average and the maximum measures. Second, we present a
few efficient heuristic algorithms for generating schedule trees, based on the structure and the analysis of the optimal
algorithms. Simulation results indicate that some of these heuristics produce excellent schedules in practice, sometimes
even beating the best known non-perfectly periodic scheduling algorithms.
Ó 2004 Elsevier B.V. All rights reserved.

Keywords: Periodic scheduling; Fair scheduling; Broadcast disks; Hierarchical round robin

1. Introduction                                                         difficulty is to use scheduling strategies that allow
                                                                        mobile devices to keep their radios turned off for
   One of the major problems of mobile commu-                           most of the time. For example, BluetoothÕs Park
nication devices is power supply, partly due to the                     Mode and Sniff Mode allow a client to sleep except
fact that radio communication is a relatively high                      for some pre-defined periodic interval [10]. An-
power consumer. A common way to mitigate this                           other example is Broadcast Disks [1], where a
                                                                        server broadcasts ‘‘pages’’ to clients. The goal is to
                                                                        minimize the waiting time and, in particular, the
    Corresponding author. Tel.: +972-3-640-7036; fax: +972-3-           ‘‘busy waiting’’ time of a random client that wishes
   E-mail addresses: (A. Bar-Noy),
                                                                        to access one of the pages [16]. (V. Dreizin), (B. Patt-               One class of particularly attractive schedules
Shamir).                                                                (from the clientÕs point of view) is the class of

1389-1286/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
156                             A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

perfectly periodic schedules, where each client i gets
one time slot exactly every bi time slots, for some                            A                <ABACAD>
bi called the period of i. Under a perfectly periodic
schedule, a client needs to record only the period
length and the offset relative to a globally known             Fig. 1. An example of a tree and its corresponding schedule (see
time zero to get a full description of its own                Section 2.3 for explanation).
schedule. In Broadcast Disks, other non-perfectly
periodic schedules that guarantee low waiting time
may require the client to actively listen until its           hard [6], so one must resort to near-optimal
turn arrives (busy waiting), while perfectly periodic         algorithms. One of the most effective ways to
schedules allow the client to actually shut down its          construct perfectly periodic schedules is tree
receiver. Note that allocating many consecutive               scheduling [7]. A tree schedule is essentially a
slots to the a client with large bandwidth demand             hierarchical composition of round-robin sched-
will in fact increase the average waiting time of             ules, where the hierarchy is represented by a tree
that client. Thus, the appeal for mobile devices              (see example in Fig. 1; a detailed explanation is
from the power consumption point of view is                   given later).
therefore obvious. In addition, observe that per-                 In this paper, we study trees that represent
fectly periodic schedules have, in some sense, the            ‘‘good’’ perfectly periodic schedules. Our strategy
best fairness among all schedules (cf., for example,          is to analyze the (exponential time) algorithm that
the ‘‘chairperson assignment problem’’ [18]).                 constructs the optimal tree scheduling, and using it
    The main question related to perfectly periodic           as the starting point, we develop ‘‘good’’ heuristic
scheduling is how to find good schedules. More                 algorithms that run in polynomial-time. The
specifically, the model is roughly as follows. We              goodness of the heuristics is tested by experimen-
assume that we are given a set of share requests              tally comparing them with the best known non-
fa1 ; . . . ; an g where each ai represents the fraction      periodic algorithm.
Pn the bandwidth requested by client i, i.e.,
   i¼1 ai ¼ 1. Given this input, an algorithm com-            1.1. Related work
putes a perfectly periodic schedule that matches
the clients requests as ‘‘closely’’ as possible. The             Motivated by the goal of minimizing the wait-
schedule implies a period bi and a share bi ¼ 1=bi            ing time, much research has focused on scheduling
for each client i. Measuring the goodness of a                which is not perfectly periodic, using the average
schedule is done based on the ratio of the re-                measure as the target function. For example, Bar-
quested shares to the granted shares ai =bi : it makes        Noy et al. [6] presents an algorithm that produces
sense, depending on the target application, to be             a schedule whose average measure is at most 9     8
concerned either by the weighted average of these             times the best possible. Their algorithm uses the
ratios, or by the maximum of these ratios. (We will           golden ratio schedule [13], and hence gaps between
show that weighted maximum and unweighted                     consecutive occurrences of a client can have any of
average do not make sense.) Formally, for each i,             three distinct values (whereas in perfect schedules
let qi ¼ ai =bi . Using the qi Õs we define the perfor-        there is exactly one possible value). In [15], Ken-
mance measures:                                               yon et al. describe a polynomial approximation
                                                              scheme for the average measure; their solution is
Maximum: MAX ¼ maxfqi j1 6 i 6 ng;                            not perfectly periodic either. Hameed and Vaidya
                                                              [19,20] propose, using Weighted Fair Queuing, to
Weighted average: AVE ¼             ai qi :                   schedule broadcasts (which results in non-perfectly
                              i¼1                             periodic schedules).
                                                                 Ammar and Wong [2,3], consider the problem
Unfortunately, finding an optimal schedule (under              of minimizing the average response time in Tele-
either the maximum or average measure) is NP-                 text Systems, which is equivalent to Broadcast
                               A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                             157

Disks. They show that the optimal schedule is                  Zipf distribution [22] under both the MAX and
cyclic, and give nearly optimal algorithms for this            AVE measures. The Zipf distribution is believed
problem. Khanna and Zhou [16] show how to use                  to be the best distribution to approximate ‘‘real
indexing with perfectly periodic scheduling to                 life’’ frequencies (see e.g., [11]).
minimize busy waiting, and they also show how to             • Finally, we present experimental results. In
get a 3 approximation w.r.t. the average measure.
                                                               extensive tests that we ran, we have found that
Bar-Noy et al. prove in [6] that it is NP-hard to              our best algorithm manages to beat even the best
find the optimal perfectly periodic schedule.                   known non-periodic algorithm for both distri-
    Baruah et al. [9] and Anderson et al. [4] give             butions. This is interesting, since perfect period-
algorithms for fair periodic scheduling. Their                 icity is not always possible (consider, for
algorithms build schedules where at every time t a             example, the requests a1 ¼ 1=2 and a2 ¼ 1=3).
client with share demand ai , must have been
scheduled either dai Á te or bai Á tc times. These              We note that to the best of our knowledge, this
schedules are not perfectly periodic: gaps between           work is the first to introduce the MAX criterion for
consecutive occurrences of a client can be as high           measuring performance of schedules in this or
as twice the requested period. Additional papers             similar settings.
with analysis of the average measure (motivated by
Broadcast Disks and related problems) are                    1.3. Paper organization
[1,8,17]. The machine maintenance problem [5,21]
and the chairperson assignment problem [18] are                 The remainder of the paper is organized as
also closely related to periodic scheduling.                 follows. In Section 2, we formally define periodic
    The general notion of perfect periodicity and            schedules and the performance measures. In Sec-
the tree methodology were introduced in [7]. This            tion 3, we present the optimal tree scheduling
paper presents algorithms for constructing sched-            algorithms. Our heuristic algorithms are described
ules that are guaranteed to be close to optimum              in Section 4. Experimental results are presented in
w.r.t. the average measure, depending on the value           Section 5. We conclude in Section 6.
of the maximal requested share.

1.2. Our contribution                                        2. Definitions and preliminaries

   In this paper, we study the problem of effi-                   In this section, we define basic terms and give
ciently constructing good tree schedules. Given a            several preliminary results (see Fig. 2 for a sum-
set of frequencies requests, our goal is to find a tree       mary). In Section 2.1, we formally define sched-
schedule that grants these frequencies exactly or            ules. Quality measures are defined and discussed in
approximately. The quality of a schedule is mea-             Section 2.2. Tree scheduling is described in Section
sured by either the weighted average or the maxi-            2.3.
mum of the ratios between the requested frequency
and the allocated frequency for each client.                 2.1. Schedules

• We first give optimal (exponential time) algo-                  We are given a set of n clients 1; 2; . . . ; n. Each
  rithms for both measures. Their construction               client i requests a share 0 < ai < 1 (of a common
  is based on a well structured bottom–up ap-                resource). We refer to ai as a requested share or a
  proach and is not straightforward.                         frequency demand of client i. A frequency demand
• Next, based on the structure of the optimal                vector is a vector A ¼ ha1 ; a2 ; . . . ; an i, where
  algorithms, we develop a few efficient (good                    i¼1 ai ¼ 1. Sometimes we state client demands in
  polynomial time) heuristic algorithms. These               terms of requested periods ai ¼ 1=ai .
  heuristics perform well on frequencies that are                A schedule is an infinite sequence S ¼ s0 ; s1 ; . . .,
  produced by the uniform distribution and the               where Sj 2 f1; 2; . . . ; ng for all j. A schedule is
158                                   A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

                                                   Fig. 2. Glossary of notations.

cyclic if it is an infinite concatenation of a finite                      Intuitively, the ‘‘best’’ (or ‘‘fairest’’) perfect
sequence C ¼ hs0 ; s1 ; . . . ; sjCjÀ1 i. C is a cycle of S. A       schedule should be the one that provides each
schedule is perfectly periodic (or just perfect for                  client exactly with its demand, i.e., bi ¼ ai for all i.
short) if slots allocated to each client are equally                 In this case we get that MAX ¼ AVE ¼ 1. The
spaced, i.e. for each client i there exist non-nega-                 following lemmas show that this is indeed the best
tive integers bi ; oi 2 Zþ called the period and offset               possible for the MAX and the AVE measures.
of i, respectively, such that i is scheduled in slot j if
and only if j  oi (mod bi ). Note that perfectly                    Lemma 2.1. For all frequency vectors A,B, we
periodic schedules are cyclic. The frequency bi of a                 have that MAXA;B P 1.
client i in a perfect schedule is the reciprocal of its
                                                                                  P              P
period, i.e., bi ¼ 1=bi . We refer to bi as the granted              Proof. Since 1 6 i 6 n ai ¼ 1 6 i 6 n bi ¼ 1, it follows
share and to bi as the granted period for client i.                  from the pigeon hole principle that it cannot be the
                                                                     case that bi > ai for all 1 6 i 6 n. Thus, there exists
2.2. Measures                                                        an index j such that aj P bj . As a result we get that
                                                                     MAXA;B P 1. h
    Given a frequency demand vector A and a
granted frequency vector B we consider for each                      Lemma 2.2. For all frequency vectors A,B, we
client the ratio of the requested shares ai , to the                 have that AVEA;B P 1.
granted shares bi : qi ¼ ai =bi . Using qi Õs we define
the following measures:                                              Proof. Let A ¼ ha1 ; a2 ; . . . ; an i be the frequency
MAXA;B ¼ maxfqi j1 6 i 6 ng                                          vector and let B ¼ hb1 ; b2 ; . . . ; bn i be a vector such
                                                                    that       P i 6 n bi ¼ 1.
                                                                                 16                    We         show       that
       ¼ max      j1 6 i 6 n                                         AVEA;B ¼ 1 6 i 6 n a2 =bi gets the minimum value
               bi                                                    when bi ¼ ai for all 1 6 i 6 n.
               bi                                                       For simplicity, we show the proof for n ¼ 2, the
       ¼ max      j1 6 i 6 n ;                                       general case follows similar arguments. For the
                                                                     n ¼ 2 case A ¼ ha; 1 À ai and B ¼ hb; 1 À bi. We
              X                 X a2      X b                        view     AVE       as      a       function        on     b:
                                      i             i
AVEA;B ¼             qi ai ¼            ¼             :                                        2
                                                                     aveðbÞ ¼ a2 =b þ ð1 À aÞ =ð1 À bÞ. Differentiating,
             16i6n             16i6n
                                     bi 1 6 i 6 n a2
                                                   i                                                                  2        2
                                                                     we get that dðaveÞ=db ¼ Àa2 =b2 þ ð1 À aÞ =ð1 À bÞ .
We omit subscripts when they are clear by the                        Solving dðaveÞ=db ¼ 0, we find that the minimum
context.                                                             is obtained when a ¼ b. h
                                       A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                                         159

Table 1
  Schedule                     ðb1 ; q1 Þ                ðb2 ; q2 Þ                ðb3 ; q3 Þ                 MAX            AVE
  RR                           (3,3/2)                   (3,1)                     (3,1/2)                    3/2            7/6
  C                            (2,1)                     (4,4/3)                   (4,2/3)                    4/3            19/18

  For both MAX and AVE, the closer to 1 the                                                         1 X ai
                                                                            U AVEA;B ¼                             :
measure is, the more satisfied clientsÕ share de-                                                    n 1 6 i 6 n bi
mands are. Therefore, our goal is to find schedules
minimizing performance measures.                                               We say that a measure ‘‘makes sense’’ if
                                                                            granting the clients with their requested shares
Example 1. Let a1 ¼ 1=2, a2 ¼ 1=3, and a3 ¼ 1=6.                            would yield the best performance. This is because
We      compare     the     round-robin     schedule                        in most applications the requested shares were
RR ¼ h1; 2; 3i with the only other perfect schedule                         calculated to yield the best performance or to
with three clients C ¼ h1; 2; 1; 3i. Table 1 summa-                         satisfy fairness issues. It may be surprising, but
rizes the performance of the two schedules for the                          unweighted average and weighted max do not
two measures. Table 1 shows that C outperforms                              make sense in this respect. The point with un-
RR in both measures.                                                        weighted average is that if many low-weight re-
                                                                            quests get larger shares, they bias the measure
Example 2. Let a1 ¼ 1=3, a2 ¼ 1=3, a3 ¼ 1=4, and                            merely by having many of them, even if their total
a4 ¼ 1=12. We compare the following (and only)                              weight is small. The flip side of this anomaly is
four perfect schedules with four clients:                                   weighted average: in this case, the measure sup-
RR ¼ C1 ¼ h1; 2; 3; 4i; C2 ¼ h1; 2; 3; 1; 2; 4i;                            presses the effect of granting a larger share to the
                                                                            heaviest client at the expense of the small clients.
C3 ¼ h1; 2; 1; 3; 1; 4i; C4 ¼ h1; 2; 1; 3; 1; 2; 1; 4i:                     In what follows, we construct instances for which
                                                                            granting the requested shares is not optimal for the
   Table 2 summarizes the performance of the four
                                                                            above two measures.
schedules. The table shows that the round-robin
                                                                               The perfect schedule that gives each client its
schedule C1 is the best for the MAX measure, while
                                                                            exact shares yields W MAX ¼ maxfai j1 6 i 6 ng
C2 is the best schedule for the AVE measure.
                                                                            and U AVE ¼ 1. Consider the following two fre-
                                                                            quency vectors:
2.2.1. Other performance measures: weighted max
and unweighted average                                                                 1Àa         1Àa
   In this paper, we work with unweighted MAX                               A1 ¼ a;          ;...;         ;
                                                                                       nÀ1         nÀ1
and weighted AVE measures. We show why we do                                                                   
                                                                                     1     1              1
not consider the other two possible measures. The                           A2 ¼       ;         ;...;            :
                                                                                     2 2ðn À 1Þ        2ðn À 1Þ
first is the weighted MAX:
                     2                                                        Assume that a ¼ 1=a for an integer a such that
W MAXA;B ¼ max           j1 6 i 6 n :                                       a À 1 divides n À 1. For both vectors there exist
                                                                            corresponding schedules for which each client gets
   The second is the unweighted AVE:                                        its exact share:

Table 2
  Schedule                ðb1 ; q1 Þ           ðb2 ; q2 Þ             ðb3 ; q3 Þ                ðb4 ; q4 Þ             MAX     AVE
  C1                      (4,4/3)              (4,4/3)                (4,1)                     (4,1/3)                4/3     7/6
  C2                      (3,1)                (3,1)                  (6,3/2)                   (6,1/2)                3/2     13/12
  C3                      (2,2/3)              (6,2)                  (6,3/2)                   (6,1/2)                2       47/36
  C4                      (2,2/3)              (4,4/3)                (8,2)                     (8,2/3)                2       11/9
160                                     A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

C1 ¼ h1; 2; . . . ; a; 1; a þ 1; . . . ; 2a À 1; . . . ; 1;           child of u, and u is the parent of v. The degree of a
                                                                      node in a rooted tree is the number of its children.
       n À a þ 1; . . . ; n À 1i;                                     A leaf is a node with degree 0.
                                                                          Tree scheduling is a methodology for con-
C2 ¼ h1; 2; 1; 3; . . . ; 1; n À 1i:
                                                                      structing perfect schedules that can be represented
   To show the unnatural effect of W MAX, con-                         by rooted trees [7]. The basic idea is that each
sider A1 as the frequency vector, and let C1 and C2                   leaf corresponds to a distinct client, and the period
denote the frequency vectors corresponding to C1                      of each client is the product of the degree of all
and C2 , respectively. We have that W MAXA1 ;C1 ¼                     the nodes on the path leading from the root to
maxfa; ð1 À aÞ=ðn À 1Þg       and     W MAXA1 ;C2 ¼                   the corresponding leaf. In the example of Fig. 1, the
maxf2a2 ; 2ð1 À aÞ2 =ðn À 1Þg. If n > 1 þ ða À 1Þ2                    period of A is 2 because the root degree is 2, and the
then 2a2 > 2ð1 À aÞ2 =ðn À 1Þ and a > ð1 À aÞ=                        periods of B,C and D are 6, because the root degree
ðn À 1Þ. For such large values of n we get                            is 2 and the degree of their parent is 3. We refer to a
W MAXA1 ;C1 ¼ a and W MAXA1 ;C2 ¼ 2a2 . Under                         tree that represents a schedule as a scheduling tree.
the W MAX measure, for a < 1=2, C2 is better than                         Given a scheduling tree, one can build a corre-
C1 that meets exactly the demands of all clients!                     sponding schedule in several ways. One possible
Moreover, the ratio of the performance of the                         way to construct a schedule is to explicitly compute
schedules is a=ð2a2 Þ ¼ a=2, which can be arbi-                       offsets of clients from the tree [7]. Another way is to
trarily large (by selecting n large enough such that                  use a tree directly. One can build a full listing of
a À 1 divides n À 1). Note that in order to achieve                   schedule cycle from the tree as follows. We con-
a better performance for W MAX, the schedule                          struct a schedule cycle recursively: Each leaf of the
might prefer the first client as is the case with C2 .                 scheduling tree corresponds to a schedule cycle of
   To show the unnatural effect of U AVE, con-                         length 1 consisting of its client. To construct a
sider A2 as the frequency vector. We have that                        schedule cycle of an internal node, schedules of its
U AVEA2 ;C2 ¼ 1                                                       children are brought to the same size by replica-
                                                                      tion, and then the resulting schedules are inter-
                                                                    leaved in the round-robin manner. Finally, the
                    1        1   nÀ1                                  schedule cycle associated with the root is the output
U AVEA2 ;C1 ¼                  þ         :
                    n       2a 2ð1 À aÞ                               of the algorithm. In the example of Fig. 1, the
Plugging in the value of a ¼ 1=a, we get                              schedule associated with the parent of B, C, and D
                                                                      is hBCDi. The final schedule hABACADi is obtained
                    an þ a2 À 2a
U AVEA2 ;C1 ¼                    :                                    by interleaving the two schedules hAAAi and
                     2ða À 1Þn                                        hBCDi. More details on usage of scheduling trees
We choose a large value for a and then a larger                       can be found in [12]. We summarize basic proper-
value for n to get that U AVEA2 ;C1 ¼ 1 þ Oð1Þ.                       ties of scheduling trees in the following lemma:
Thus, for the U AVE measure, C1 performs almost
twice better than the natural schedule C2 that                        Theorem 2.3 (Bar-Noy et al. [7, Theorem 3.1]). Let
meets exactly the demands of all clients. Note that                   T be a scheduling tree with n leaves labeled 1; . . . ; n,
in order to achieve a better performance for                          where leaf i corresponds to client i. Then there exists
U AVE, the schedule might give the first client less                   a perfect schedule for clients 1; . . . ; n; the period of
slots as is the case with C1 .                                        each client i in the schedule is the product of the
                                                                      degrees of all ancestors of i in T .
2.3. Scheduling trees

   A tree is a connected acyclic graph. A rooted                      3. Optimal tree scheduling
tree is a tree with one node designated as the root.
We assume that all edges are directed away from                          In this section, we describe optimal (exponential
the root. If ðu; vÞ is a directed edge, then v is the                 time) algorithms that construct scheduling trees,
                                 A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                          161

for both the MAX and the AVE measures. Since the               ai 6 aj but bi > bj . Let S 0 be the schedule S in
algorithms for MAX and AVE are nearly identical,               which the clients i and j are switched: S 0 grants
we describe MAX in detail and only point out the               client i period bj and grants client j period bi . Let
differences for AVE.                                            T 0 be the scheduling tree that is associated with the
    Our optimal algorithms use the bottom–up ap-               schedule S 0 . Note that T 0 is the tree T in which
proach, in the sense that they combine a set of                clients i and j switch leaves. By Lemma 3.1, the
leaves into a single node and then continue                    cost of S 0 is at most the cost of S for both mea-
recursively. This approach is similar to the con-              sures. Therefore, T 0 is also an optimal scheduling
struction of Huffman codes [14].                                tree. We proceed with these switches to get an
    The optimal algorithms are based on the                    optimal scheduling tree T 00 for which bi 6 bj for all
observation that for both MAX and AVE measures,                pairs of clients 1 6 i; j 6 n such that ai 6 aj . h
it is always better to give more time slots in the
schedule for clients whose requested shares are                   The above corollary implies that there exists an
larger. This means that there exists an optimal tree           optimal scheduling tree that preserves the non-
where the smallest share in the schedule belongs to            increasing order of the requested shares. We use
the client with the smallest share demand. To prove            this to prove the following key theorem that is
this idea, we need the following algebraic lemma.              valid for both MAX and AVE measures. This the-
                                                               orem serves as the first step in constructing the
Lemma 3.1. For a1 6 a2 and b1 6 b2 ,                           bottom–up optimal algorithms.

1. max fb2 =a1 ; b1 =a2 g P max fb1 =a1 ; b2 =a2 g,            Theorem 3.3. For each frequency demand vector A,
2. b2 =a2 þ b1 =a2 P b1 =a2 þ b2 =a2 .                         there exists an optimal scheduling tree T , an integer
        1        2         1       2
                                                               2 6 k 6 n, and a node q with k children such that the
Proof                                                          children of q in T are the clients with the smallest k
1. a1 6 a2 implies that b2 =a1 P b2 =a2 and b1 6 b2            requested shares.
   implies that b2 =a1 P b1 =a1 . Hence, b2 =a1 P
   max fb1 =a1 ; b2 =a2 g, and therefore, max fb2 =a1 ;        Proof. First, note that by definition siblings leaves
   b1 =a2 g P max fb1 =a1 ; b2 =a2 g.                          have the same granted share. Second, note that if a
2. a1 6 a2 and b1 6 b2 imply that ðb2 À b1 Þða2 À              leaf has an internal node as a sibling than all of the
   a2 Þ P 0. Hence, b2 a2 þ b1 a2 P b1 a2 þ b2 a2 and          leaves in the subtree rooted at this node are granted
    1                      2      1      2       1
   therefore b2 =a2 þ b1 =a2 P b1 =a2 þ b2 =a2 . h             a smaller share. Therefore, in any tree there exists
                   1         2        1      2
                                                               at least one node q that has k P 2 children all of
   We say that a scheduling tree is optimal for a              them leaves whose granted shares are the smallest
given frequency request vector if its corresponding            in the tree. By Corollary 3.2, there exists an optimal
schedule achieves the best performance achievable              tree in which these k leaves are associated with the
by schedules that can be represented as trees. The             smallest k requested shares. h
following corollary states a property of some opti-
mal trees.                                                        Our optimal tree algorithms rely on Theorem
                                                               3.3. The idea is to coalesce k clients with the
Corollary 3.2. Let fa1 6 Á Á Á 6 an g be the requested         smallest share demands into a new client, and then
periods. Then, for both the MAX and the AVE                    solve the new problem recursively.
measures, there exists an optimal scheduling tree
whose corresponding granted periods preserve the               3.1. The optMax algorithm
non-decreasing order, i.e., b1 6 Á Á Á 6 bn .
                                                                  We first explain the optimal algorithm for the
Proof. Let S be any optimal tree schedule associ-              MAX measure. The algorithm loops through all
ated with the optimal scheduling tree T . Assume               values of k, and for each value, it coalesces the k
that in S there exist two clients 1 6 i; j 6 n such that       smallest clients, and solves the new set recursively.
162                                   A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

      Fig. 3. An optimal algorithm for the MAX measure. For the AVE measure, replace the line marked by H with Eq. (2).

                                                                                         (              )
The best k is chosen. Pseudo code is presented in                                      aq akþ1       an
                                                                           0   0
Fig. 3.                                                             valðA ;T Þ ¼ max 0 ; 0 ;...; 0
                                                                                       bq bkþ1       bn
   The crux of the algorithm is the weight a0k that is                               (                                    )
assigned to a new client that replaces k old clients                                   k Á maxfa1 ;...;ak g akþ1       an
with the smallest share demands                                                ¼ max                         ; 0 ;...; 0
                                                                                                b0q           bkþ1     bn
a0k ¼ k Á maxfaja 2 Mk g:                                    ð1Þ                     (                                )
                                                                                        a1          ak akþ1        an
   To explain this choice, we first make the fol-                               ¼ max 0 ;...; 0 ; 0 ;...; 0
                                                                                       bq =k     bq =k bkþ1        bn
lowing definition. Let A ¼ ha1 ; a2 ; . . . ; an i be a                                                          
vector of frequencies. We do not require that                                          a1      ak akþ1        an
Pn                                                                             ¼ max      ;...; ;       ;...;
  i¼1 ai ¼ 1. Let T be a tree with n leaves and                                        b1      bk bkþ1        bn
granted frequency vector BðT Þ ¼ hb1 ; . . . ; bn i.                               ¼ valðA;T Þ:       Ã
Then we define
                                                                      We say that an algorithm finds an optimal tree
                    a1       an
valðA; T Þ ¼ max       ;...;      :                                 T Ã for n clients with a frequency request vector A
                    b1       bn
                                                                    if    for     any    tree    T     with    n   leaves
   We now prove that when optMax coalesces                          valðA; T Ã Þ 6 valðA; T Þ. The next lemma justifies
several clients, val is preserved.                                  the recursive step of the optMax algorithm. We use
                                                                    the     following     notation.    For     a  vector
Lemma 3.4. Let T be a tree with leaves 1; . . . ; n;                A ¼ ha1 6 a2 6 Á Á Á 6 an i of frequencies whose sum
where clients 1; . . . ; k are siblings with parent q. Let          is not necessarily 1, we denote
A ¼ ha1 ; . . . ; an i be the frequency requests of the
                                                                    A0 ðkÞ ¼ ha0k ; akþ1 ; . . . ; an i for 1 < k 6 n;
clients of T . Let T 0 be the tree generated from T by
replacing clients 1; . . . ; k with q, and let A0 be the            where a0k ¼ k Á maxfa1 ; . . . ; ak g:
frequency request vector resulting from A by
replacing a1 ; . . . ; ak with aq ¼ k Á maxfa1 ; . . . ; ak g.      Lemma 3.5. Let A be a frequency demand vector,
Then valðA; T Þ ¼ valðA0 ; T 0 Þ.                                   and let T 0 ðkÞ be an optimal tree for A0 ðkÞ. Let T ðkÞ
                                                                    be a tree generated from T 0 ðkÞ by adding k clients
Proof. Let B ¼ hb1 ; . . . ; bn i and B0 ¼ hb0q ; b0kþ1 ; . . . ;   with frequency demands a1 ; . . . ; ak as children of the
b0n i be granted frequency vectors implied by T and                 node with frequency demand a0k . Let k à be such that
T 0 , respectively. By definition, bi ¼ b0i for k þ 1 6              valðA0 ðkÞ; T 0 ðkÞÞ is minimized. Then T ðk Ã Þ is an
i 6 n and b1 ¼ b2 ¼ Á Á Á ¼ bk ¼ b0q =k. Hence,                     optimal tree for A w.r.t. the MAX measure.
                               A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                                           163

Proof. Let T à denote T ðk à Þ. Assume that T à is not       computation of the new client share demand. In
optimal for A and let R be an optimal tree for A,            optAve, coalescing k clients with share demands
i.e., valðA; T Ã Þ > valðA; RÞ. By Theorem 3.3, with-        ða1 ; . . . ; ak Þ produces the new client with the share
out loss of generality, there exists 2 6 m 6 n such          demand
that m clients with the smallest frequency demands                   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
are siblings in R. Let R0 be the tree generated from R       a0k ¼ kða2 þ Á Á Á þ a2 Þ:                            ð2Þ
                                                                               1                  k
by coalescing these m clients. Note that R0 corre-
sponds to the frequency demand vector A0 ðmÞ.                   (In the pseudo code of Fig. 3, the equation
Since coalescing clients does not change the value           above replaces the line marked by H.)
of the val function by Lemma 3.4, we get                        For      the    AVE       measure,     we   define
valðA0 ðmÞ; R0 Þ ¼ valðA; RÞ < valðA; T Ã Þ                  valðA; T Þ ¼ a2 =b1 þ Á Á Á þ a2 =bn . The following
                                                                             1              n
                   ¼ valðA0 ðk à Þ; T 0 ðk à ÞÞ              lemma shows that when optAve coalesces several
                   6 valðA0 ðmÞ; T 0 ðmÞÞ:                   clients, val is preserved.

Hence, valðA0 ðmÞ; R0 Þ < valðA0 ðmÞ; T 0 ðmÞÞ, con-
tradicting the assumption that T 0 ðmÞ is optimal            Lemma 3.7. Let T be a tree with leaves 1; . . . ; n,
tree for A0 ðmÞ. h                                           where clients 1; . . . ; k are siblings with parent q. Let
                                                             A ¼ ha1 ; . . . ; an i be the frequency requests of the
  Theorem 3.6 summarizes the correctness and                 clients of T . Let T 0 be the tree generated from T by
optimality of Algorithm optMax.                              replacing clients 1; . . . ; k with q, and let A0 be the
                                                             frequency request vector resulting from A by
Theorem 3.6. Algorithm optMax finds the optimal              replacing a1 ; . . . ; ak with aq ¼ kða2 þ Á Á Á þ a2 Þ:
                                                                                                            1                    k
tree w.r.t. the MAX measure in time Oð2n Þ.                  Then valðA; T Þ ¼ valðA0 ; T 0 Þ.

Proof. We first prove optimality by induction on n.           Proof.          Let        B ¼ hb1 ; . . . ; bn i and B0 ¼
For n ¼ 1 the claim is trivial. For the inductive              0    0              0
                                                             hbc ; bkþ1 ; . . . ; bn i be granted frequency vectors im-
step, we have that by Lemma 3.5, optMax finds the             plied by T and T 0 , respectively. Obviously, bi ¼ b0i
tree T that minimizes valðA; T Þ. Since                      for k þ 1 6 i 6 n and b1 ¼ b2 ¼ Á Á Á ¼ bk ¼ b0c =k.
valðA; T Þ ¼ MAXA;BðT Þ , we conclude that optMax            Hence,
finds the optimal tree. As for the running time, let
T ðnÞ denote the running time of optMax for n cli-                              a2 a2             a2
                                                             valðA0 ; T 0 Þ ¼    c
                                                                                   þ kþ1 þ Á Á Á þ n
ents. Clearly, we have that the time is given by the                            b0c b0kþ1         b0n
recursive relation T ð1Þ ¼ Oð1Þ and                                             kða2 þ Á Á Á þ a2 Þ a2            a2
        nÀ1                                                                 ¼      1            k
                                                                                                   þ kþ1 þ Á Á Á þ n
T ðnÞ ¼     T ðiÞ þ Oðn2 Þ;                                                            b0c          b0kþ1         b0n
        i¼1                                                                      a2             a2   a2            a2
whose solution is T ðnÞ ¼ Oð2n Þ. h                                         ¼      1
                                                                                 0 =k
                                                                                      þ Á Á Á þ 0 k þ kþ1 þ Á Á Á þ n
                                                                                bc             bc =k bkþ1          b0n
   We remark that there exists a more efficient                                   a2          a2 a2             a2
                                                                            ¼    1
                                                                                   þ Á Á Á þ k þ kþ1 þ Á Á Á þ n
implementation of the optMax algorithm that                                     b1          bk bkþ1           bn
performs at each recursion step OðnÞ operations                             ¼ valðA; T Þ:             Ã
rather than Oðn2 Þ, but this solution improves the
asymptotic time complexity only by a constant
factor. Details are omitted.
                                                             Theorem 3.8. Algorithm optAve finds the optimal
3.2. The optAve algorithm                                    tree w.r.t. the AVE measure in Oð2n Þ time.

   Algorithm optAve for the AVE measure is                      The proof is identical to the proof of Theorem
identical to the optMax algorithm, except for the            3.6, and it is therefore omitted.
164                                  A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

4. Heuristics                                                      4.1. The bin algorithms

   In this section, we describe various ways to re-                    The binMax algorithms find the best binary tree
duce the running time of the optimal algorithms by                 for the given frequencies request vector by coa-
restricting their exhaustive search. At each recur-                lescing at each stage two leaves with minimal re-
sive step, our algorithms coalesce k clients with the              quests. Pseudo code appears in Fig. 5. As is the
smallest share demands and continue recursively.                   case with the optimal algorithms, the binAve
The optimal algorithms try each 2 6 k 6 n, result-                 algorithm is identical to binMax except for the
ing in exponential running time. To save time,                     computation of the share demand of the new
we examine only some restricted subsets of values                  client.
for k.                                                                 The following lemma analyzes the running time
   We now list all of our heuristics. We use the                   of both binMax and binAve.
following convention in naming algorithms: the
suffixes Ave and Max denote the measure targeted                     Lemma 4.1. The running time of the bin algorithms
by the algorithm.                                                  for n clients is Oðn log nÞ.
   bin: The bin algorithm performs recursive calls
for k ¼ 2 only, i.e., it generates the best possible               Proof. We use a heap for storing frequencies. The
binary trees.                                                      procedure of building a heap from the original
   rr: It turns out that the bin algorithms perform                frequencies array costs OðnÞ. At each step we
badly especially when all remaining clients have                   perform two delete-min operations and one insert
similar demands, and this gave rise to the follow-                 operation, i.e., Oðlog nÞ operations. Therefore, the
ing algorithm: at each step, the algorithm tries                   running time T ðnÞ is defined by T ð1Þ ¼ Oð1Þ and
both the recursive call, and the round-robin option                T ðnÞ ¼ T ðn À 1Þ þ Oðlog nÞ. The solution of this
(which will finalize the tree). We add the rr-prefix                 recursive equation is T ðnÞ ¼ Oðn log nÞ. h
to names of these algorithms.
   binMixed: In the recursive step where ‘ clients                    It is not difficult to show that the binMax
are considered, binMixed recursively calls the bin                 algorithm guarantees an approximation ratio of at
algorithm if ‘ > logðn log nÞ, and otherwise it calls              most 2; we omit the details (see [12]). We remark
the optimal algorithm.                                             that it is known [7] that some binary tree schedules
   pseudoOpt: The idea in pseudoOpt is to check,                   have approximation ratio of at most 4 þ 2 maxfai g
                                                                                                         3   3
at each step, what is the best k according to some                 for the AVE measure. Since binAve produces the
heuristic. Specifically, pseudoOpt tests the k                      best possible binary tree schedule, it follows that
smallest elements for all k, by coalescing them and                its approximation ratio is never worse than
applying rrBinMixed. The best k is chosen, the                     3
                                                                     þ 2 maxfai g. Simulations show that practically,
corresponding clients are coalesced, and the pro-                  the performance of the algorithm is much better.
cess continues.                                                    Note that our other heuristics for both measures
   The hierarchy among the algorithms is de-                       outperform bin, and hence their approximation
scribed in Fig. 4. In the following subsections, we                ratio is also at most 2 for the MAX measure and at
study these algorithms in more detail.                             most 4 þ 4 maxfai g for the AVE measure.
                                                                          3   3

                                                                   4.2. The rrBin algorithms

                                                                      The rrBin algorithms choose at each step the
                                                                   best solution among the recursive solution and the
                                                                   round-robin solution. Clearly, the rrBin heuristics
Fig. 4. The hierarchy among heuristic algorithms. An algo-
                                                                   outperform the bin heuristics. Pseudo code for
rithm to the left of another looks at more trees and runs longer   rrBinMax is presented in Fig. 6; the code for
time.                                                              rrBinAve is nearly identical.
A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173   165

               Fig. 5. binMax algorithm.

              Fig. 6. rrBinMax algorithm.
166                            A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

                                           Fig. 7. binMixedMax algorithm.

   A na€ implementation of these algorithms                  ine also the best binary tree as a possible solution,
calculates the performance of the round-robin                and therefore, outperform the bin algorithms. We
schedule at each step by looking at all elements             present the pseudo code for the binMixedMax
of A, thus performing OðnÞ computations at                   procedure in Fig. 7. The binMixedAve algorithm is
each step. We employ a more efficient imple-                   identical to binMixedMax except for a computation
mentation, that calculates the performance of                of a share demand of a new client. The analysis of
the round-robin schedule in the current step                 both binMixed algorithms appears in the next
using its value in the previous step in constant             lemma.
time. Since the overhead of the round-robin
extension is only constant at each step, the total           Lemma 4.3. The running time of the binMixed
running time remains asymptotically the same as              algorithms is Oðn log nÞ.
in the bin algorithms, namely Oðn log nÞ. Thus,
                                                             Proof. TbinMixed ðnÞ 6 OðTbin ðnÞ þ Topt ðlogðn log nÞÞÞ ¼
we have
                                                             Oðn log n þ 2logðn log nÞ Þ ¼ Oðn log nÞ. h

Lemma 4.2. The running time of the rrBin algo-               4.4. The rrBinMixed algorithms
rithms for n clients is Oðn log nÞ.
                                                                The rrBinMixed algorithms are the round-robin
4.3. The binMixed algorithms                                 extension of the binMixed algorithms. At each step
                                                             they try both the round-robin solution and the
   The main idea of the binMixed algorithms is as            recursive call, and then choose the better solution.
follows. In the recursive step where ‘ clients are           We omit the pseudo codes of rrBinMixed for both
considered, recursively call the bin algorithms if           measures, since they result from a straightforward
‘ > logðn log nÞ, otherwise call the optimal algo-           merging of the pseudo codes of rrBin and
rithms. Note that, the binMixed algorithms exam-             binMixed.
                              A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                      167

                                         Fig. 8. pseudoOptMax algorithm.

   We note that the result achieved by the                  main idea of the pseudoOpt algorithms is to sim-
rrBinMixed algorithms on an input fai g is the              ulate recursive calls in the optimal algorithms by
minimum of the results of rrBin and binMixed. It            calls to rrBinMixed, thus saving on the running
follows from the following observation:                     time. Clearly, the pseudoOpt algorithms outper-
                                                            form the rrBinMixed algorithms, since among
• if the rrBinMixed algorithms choose the round-            others they examine all the trees examined by
  robin solution at the first stage of the algorithm         the rrBinMixed algorithms. Pseudo code of
  (a recursive call of the binary algorithm), their         pseudoOptMax is given in Fig. 8; the pseudo code
  performance is equal to the performance of                of the pseudoOptAve algorithm is very similar to
  the rrBin algorithm.                                      pseudoOptMax. The running time of pseudoOpt is
• if the rrBinMixed algorithms never choose the             Oðn3 log nÞ for both measures, since at each step
  round-robin solution at the first stage of the             the algorithm call up to n times the rrBinMixed
  algorithm, their performance is equal to the per-         algorithms, which run in Oðn log nÞ time.
  formance of the binMixed algorithms.
                                                            Lemma 4.5. The running time of the pseudoOpt
   The running time of the rrBinMixed algorithms is         algorithms for n clients is Oðn3 log nÞ.
the same as the running time of the binMixed algo-
rithms following the same reasoning that showed
that the running time of the rrBin algorithms is the        5. Simulations
same as the running time of the bin algorithms.
                                                               We study the performance of our heuristics for
Lemma 4.4. The running time of the rrBinMixed               two distributions of share demands: the Zipf dis-
algorithms for n clients is Oðn log nÞ.                     tribution [11,22] and the uniform distribution,
                                                            under both the MAX and AVE measures.
4.5. The pseudoOpt algorithms
                                                            • The Zipf distribution. This distribution depends
    The optimal algorithms choose the number of               on a skew coefficient h. The value of the share of
clients to be coalesced by a recursive call. The              the ith client is proportional to iÀh . That is,
168                                                              A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

                 ð1=iÞ                                                                                             the AVE measures. Quantitatively, the algorithms
      ai ðhÞ ¼ Pn         h
                            :                                                                                      get better results for AVE. For small numbers of
                j¼1 ð1=jÞ
                                                                                                                   clients, we compared our algorithms to the optimal
  We use h ¼ 0:8 in our simulations [11].                                                                          algorithm (Figs. 9 and 10). For large numbers of
• The uniform distribution. Each client randomly                                                                   clients, we present in Figs. 11 and 12 the perfor-
  chooses an ‘‘absolute’’ share demand a0i , which                                                                 mance achieved by our algorithms, noting that for
  is later normalized. Formally, a0i $ U ð0; 1Þ and
            Pn                                                                                                     both measures, the best possible performance of
  ai ¼ a0i = j¼1 a0j . We repeat each experiment                                                                   any perfect schedule is 1 by Lemmas 2.1 and 2.2.
  for the uniform distribution 10 times, and then                                                                      Our empirical results follow the hierarchy of the
  average results.                                                                                                 heuristic algorithms (Fig. 4) for both distributions
                                                                                                                   considered. The weakest results are due to the bin
   In general, we found that the relative quality of                                                               algorithms (about 1.45 for MAX and 1.04 for the
the algorithms is the same for both the uniform                                                                    AVE). While the rrBin algorithms perform only
and the Zipf distributions, for both the MAX and                                                                   slightly better than bin, the binMixed algorithms

                                          Algorithms for the MAX measure (Zipf distribution)                             Algorithms for the MAX measure (Uniform distribution)
                                    1.6                              binMax/OptMax                                                                      binMax/OptMax
                                                                     binMixedMax/OptMax                                                                 binMixedMax/OptMax
                                    1.5                              rrBinMax/OptMax                               1.5                                  rrBinMax/OptMax
                                                                     rrBinMixedMax/OptMax                                                               rrBinMixedMax/OptMax
                                                                     pseudoOptMax/OptMax                                                                pseudoOptMax/OptMax
                                    1.4                                                                            1.4
                      Alg. result

                                                                                                     Alg. result

                                    1.3                                                                            1.3


                                      0         5        10    15         20        25         30                    0               5        10          15        20         25
                                                          Number of clients                                                                  Number of clients

                                                Fig. 9. The performance of algorithms for small number of clients for the MAX measure.

                                          Algorithms for the AVE measure (Zipf distribution)                                 Algorithms for the AVE measure (Uniform distribution)

                          1.1                                        binAve/OptAve                                                                        binAve/OptAve
                                                                     binMixedAve/OptAve                                                                   binMixedAve/OptAve
                                                                     rrBinAve/OptAve                              1.08                                    rrBinAve/OptAve
                      1.08                                           rrBinMixedAve/OptAve                                                                 rrBinMixedAve/OptAve
                                                                     pseudoOptAve/OptAve                                                                  pseudoOptAve/OptAve
        Alg. result

                                                                                                    Alg. result

                                     0         5        10    15         20         25         30                        0               5     10          15         20         25
                                                         Number of clients                                                                    Number of clients

                                               Fig. 10. The performance of algorithms for small number of clients for the AVE measure.
                                                 A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                                                            169

                         Algorithms for the MAX measure (Zipf distribution)                              Algorithms for the MAX measure (Uniform distribution)

                                                            binMax                                                                           binMax
                   1.7                                      binMixedMax                            1.7                                       binMixedMax
                                                            rrBinMax                                                                         rrBinMax
                                                            rrBinMixedMax                                                                    rrBinMixedMax
                   1.6                                      pseudoOptMax                           1.6                                       pseudoOptMax
     Alg. result

                                                                                     Alg. result
                   1.5                                                                             1.5

                   1.4                                                                             1.4

                   1.3                                                                             1.3

                   1.2                                                                             1.2

                   1.1                                                                             1.1
                              50           100             150                200                              50           100             150               200
                                         Number of clients                                                                Number of clients

                               Fig. 11. The performance of algorithms for large number of clients for the MAX measure.

                         Algorithms for the AVE measure (Zipf distribution)                              Algorithms for the AVE measure (Uniform distribution)

                   1.1                                       binAve                                1.1                                        binAve
                                                             binMixedAve                                                                      binMixedAve
                                                             rrBinAve                                                                         rrBinAve
                  1.08                                       rrBinMixedAve                        1.08                                        rrBinMixedAve
                                                             pseudoOptAve                                                                     pseudoOptAve
    Alg. result

                                                                                    Alg. result

                  1.06                                                                            1.06

                  1.04                                                                            1.04

                  1.02                                                                            1.02

                     1                                                                               1
                              50           100             150                200                               50           100             150              200
                                         Number of clients                                                                 Number of clients

                                Fig. 12. The performance of algorithms for large number of clients for the AVE measure.

provide a significant improvement. The pseudoOpt                                                       Recall that in Section 4.4 we argued that the
algorithms improve on the results of the                                                           rrBinMixed algorithms achieve the minimum be-
rrBinMixed algorithms as expected. To conclude,                                                    tween the rrBin and the binMixed algorithms. In this
for both distributions, our best heuristic scheme                                                  section, the binMixed algorithms demonstrate in our
produces schedules which are about 0.5% over the                                                   experiments much better performance than the
optimum for the AVE measure and about 15%                                                          rrBin algorithms. This observation explains why the
over the optimum for the MAX measure.                                                              graphs of rrBinMixed and binMixed almost coincide.
   All algorithms behave similarly for both the                                                    It can be seen that for the uniform distribution,
Zipf and the uniform distributions of share de-                                                    when very small number of clients is considered, the
mands. Changing the h parameter of the Zipf                                                        graphs of rrBinMixed and binMixed are very close to
distribution does not change the performance                                                       each other, but do not coincide completely (see
significantly, as can be seen in Fig. 13.                                                           Figs. 9 and 10). Each point on the graph for the
170                                                 A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

                     Algorithms for the MAX measure (Zipf distribution, 100 clients)                   Algorithms for the AVE measure (Zipf distribution, 100 clients)
                                                               binMax                                                                            binAve
                                                               rrBinMixedMax                         1.16                                        rrBinMixedAve
                                                               pseudoOptMax                                                                      pseudoOptAve
      Alg. result

                                                                                       Alg. result

                    1.3                                                                              1.06

                    1.2                                                                              1.04

                          0   0.2     0.4     0.6       0.8    1       1.2     1.4                          0   0.2     0.4     0.6       0.8    1      1.2      1.4
                                                    Theta                                                                             Theta

                                    Fig. 13. The performance of algorithms for different values if h in Zipf distribution.

uniform distribution is a result of 10 independent                                                   clients in order to choose one to schedule, i.e., XðnÞ
experiments. If for some n, rrBin is better than                                                     operations. In addition, non-trivial computations
binMixed in at least one experiment, and, hence, for                                                 need to be carried out. Therefore, in many cases
this particular experiment rrBinMixed achieves                                                       one computes the schedule up to some point, and
better result than binMixed, the overall average                                                     repeats it [8]. We use the same approach in order
result of 10 experiments achieved by rrBinMixed is                                                   to be able to measure the performance of the
better as well. Thus, for such values of n, rrBinMixed                                               greedy heuristics. We found that the best perfor-
is slightly better than binMixed.                                                                    mance is achieved by greedy when the schedule
                                                                                                     length is about 100n. Following empirical obser-
5.1. Performance for the broadcast disks problem                                                     vation of [8] that the performance of greedy as a
                                                                                                     function of the schedule length performs oscilla-
    The Broadcast Disks problem [1] can be viewed                                                    tions with period n, we chose the schedule length
as a relaxation of the AVE measure, in the sense that                                                to be the best point in the interval [100n, 101n]. We
the schedule needs not be perfectly periodic. In this                                                assume that the demand probability of pages fol-
case, a bi value represents only the average fre-                                                    lows the Zipf distribution. It is known that the
quency for client i. The best known results for the                                                  optimal schedule for the Broadcast Disks problem,
Broadcast Disks problem, from simulations point                                                      if exists, is a perfect schedule in which page i is
                                                                                                                                  pffiffiffiffi P pffiffiffiffi
of view, are based on some variations of a greedy                                                    granted the share qi ¼ pi = n       i¼1 pi . Hence, we
heuristic (e.g., [6,19]). The idea in the greedy algo-                                               build a perfect broadcast schedule by calling our
rithms is to compute, every time slot, for each client                                               algorithms with the frequency request vector
i, what would be the ‘‘cost’’ of the schedule in case i                                              A ¼ hq1 ; . . . ; qn i. Then we compare a ratio be-
is now scheduled, where cost is computed according                                                   tween the average waiting time achieved by perfect
the AVE measure. The client that minimizes the cost                                                  schedules to the one achieved by the greedy algo-
increase is scheduled. Note that the greedy algo-                                                    rithm. It can be seen in Fig. 14, that our algorithms
rithm requires knowledge of the demand vector, or                                                    perform very well compared to greedy and even
else the cost cannot be computed. In the next par-                                                   give better results for some database sizes.
agraph we explain how we compare the output of                                                           We note that for the Zipf distribution, many
the greedy heuristics with our heuristics.                                                           studies demonstrated that greedy is very close to
    The greedy algorithms, in general, produce non-                                                  optimal. In most of the cases, greedy does not
cyclic schedules. In each time slot, they scan all the                                               produce a perfect schedule. Our results show that
                                                     A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173                                     171

                                                             Perfectly periodic schedules vs. greedy schedule
                                                    1.008                                pseudoOptAve/greedy

                                ratio: alg/greedy





                                                         0       50      100       150       200       250      300
                                                                               Database size

Fig. 14. The performance of our algorithms vs. greedy heuristic algorithm for Broadcast Disks problem. The heuristics are better than
greedy whenever their graph is below 1.

we can approach optimality with perfectly periodic
schedules as well.

5.2. Performance for perfectly periodic schedules
without a tree representation

    In this paper we considered perfectly periodic                                   Fig. 15. The performance of pseudoOptAve and pseudoOptMax
schedules that have a tree representation. In [7] it                                 on instances that have a perfect schedule but do not have a tree
was demonstrated that there are perfect schedules                                    representation.
that have no tree representation. In this section, we
evaluate the performance of our algorithms with
such instances.
                                                                                     can be scheduled optimally (there is no known
    Consider the following parametric instances.
                                                                                     algorithm for finding such schedules in general). In
Let p, q, r be three distinct prime numbers. The
                                                                                     Fig. 15, we present the results for four such in-
requested shares are either 1=pq, 1=pr, 1=qr, or
                                                                                     stances. For each instance, the number of clients
1=pqr. It is straightforward to construct demand
                                                                                     and the performance of pseudoOptAve and
vectors that can be satisfied optimally, i.e., each
                                                                                     pseudoOptMax are given. By construction, there
client gets precisely its requested share. However,
                                                                                     are perfect schedules for these instances with value
there cannot be any tree representation for clients
                                                                                     of 1 for both the MAX and the AVE measures. The
with periods pq, pr, and qr: this follows from the
                                                                                     table demonstrates that the performance penalty
observation that the degree of the root of the tree
                                                                                     for tree schedules is not very large.
must divide all periods, and the greatest common
divisor of these periods is 1.
    More concretely, let p ¼ 2, q ¼ 3, and r ¼ 5.
We have constructed a few examples where some                                        6. Discussion and open problems
clients have requested periods 6, 10 and 15, and
remaining ones have period of 30. The requests                                          In this paper, we studied a scheduling trees
were manually constructed so that each instance                                      design problem. We considered the problem of
172                           A. Bar-Noy et al. / Computer Networks 45 (2004) 155–173

constructing good scheduling trees and described a              to maintain than perfect schedule, but on the
bottom–up structure exponential time algorithms                 other hand they might yield better performance.
to find the optimal scheduling trees for both the
MAX and the AVE measures. Then we proposed
several efficient polynomial-time heuristic algo-
rithms based on the structure of the optimal
                                                               We thank the anonymous referees for their
algorithms. We tested our solutions by simulation.
                                                            useful suggestions.
We concluded that for the Zipf and the Uniform
frequency distributions, our approximation algo-
rithms perform very well sometimes even beating             References
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     Effort, Addison-Wesley, Reading, MA, 1949.                                            Weizmann Institute in 1989, and Ph.D.
                                                                                          in Computer Science from MIT in
                                                                                          1995. He was an assistant professor in
Amotz Bar-Noy received the B.Sc. degree in 1981 in Mathe-                                 Northeastern University between 1994
matics and Computer Science and the Ph.D. degree in 1987 in                               and 1997, and since 1997 he is with the
Computer Science, both from the Hebrew University, Israel.                                Department of Electrical Engineering
From October 1987 to September 1989 he was a post-doc fellow                              in Tel Aviv University, where he di-
in Stanford University, California. From October 1989 to Au-                              rects the Computer Communication
gust 1996 he was a Research Staff Member with IBM T.J.                                     and Multimedia Laboratory. In 2002–
Watson Research Center, New York. From February 1995 to                                   2004 he is visiting HP Labs in Cam-
September 2001 he was an associate Professor with the Elec-                               bridge, Mass.

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