Wireless Networks 9, 117–129, 2003 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Efficient Channel Allocation Tree Generation for
Data Broadcasting in a Mobile Computing Environment
WEN-CHIH PENG and MING-SYAN CHEN∗
Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, ROC
Abstract. The research issue of broadcasting has attracted a considerable amount of attention in a mobile computing system. By utilizing broadcast channels, a server continuously and repeatedly broadcasts data to mobile users. These broadcast channels are also known as “broadcast disks” from which mobile users can retrieve data. Using broadcasting, mobile users can obtain the data of interest efficiently and only need to wait for the required data to present on the broadcast channel. The issue of designing proper data allocation in the broadcast disks is to reduce the average expected delay of all data items. We explore in this paper the problem of generating hierarchical broadcast programs with the data access frequencies and the number of broadcast disks in a broadcast disk array given. Specifically, we first transform the problem of generating hierarchical broadcast programs into the one of constructing a channel allocation tree with variant-fanout. By exploiting the feature of tree generation with variant-fanout, we develop a heuristic algorithm VFK to minimize the expected delay of data items in the broadcast program. In order to evaluate the solution quality obtained by algorithm VFK and compare its resulting broadcast program with the optimal one, we devise an algorithm OPT based on a guided search to obtain the optimal solution. Performance of these algorithms is comparatively analyzed. Sensitivity analysis on several parameters, including the number of data items and the number of broadcast disks, is conducted. It is shown by our simulation results that by exploiting the feature of variant-fanout in constructing the channel allocation tree, the solution obtained by algorithm VFKis of very high quality and is in fact very close to the optimal one resulted by algorithm OPT. Moreover, algorithm VFK is of very good scalability which is important for algorithm VFK to be of practical use to generate hierarchical broadcast programs dynamically in a mobile computing environment.
Keywords: broadcast disks, mobile computing, broadcast programs, multiple broadcast channels
1. Introduction
In a mobile computing environment, a mobile user with a power-limited mobile computer can access various informa-tion via wireless communicainforma-tion. Applicainforma-tions such as stock activities, traffic reports and weather forecast have become in-creasingly popular in recent years [26,27]. It is noted that mo-bile computers use small batteries for their operations without directly connecting to any power source, and the bandwidth of wireless communication is in general limited. As a result, an important design issue in a mobile system is to conserve the energy and communication bandwidth of a mobile unit while allowing mobile users of the ability to access informa-tion from anywhere at anytime [5,9,15,28].
In order to conserve the energy and communication band-width of a mobile computing system, a data delivery architec-ture in which a server continuously and repeatedly broadcasts data to a client community through a broadcast channel was proposed in [1,13,14,17,25]. In a push-based information sys-tem, a server generates a broadcast program to broadcast data to the mobile users. This broadcast channel is referred to as a “broadcast disk” from which mobile clients can retrieve data. The mobile users need to wait for the data of interest to appear on the broadcast channel, and the corresponding waiting time is called the expected delay of that data item. One objective of designing proper data allocation in the broadcast disks is to reduce the average expected delay of data items. Broadcast-∗Corresponding author.
ing schemes in this context have been previously addressed by other researchers [3,20,22,24]. Also, a significant amount of research effort has been elaborated on developing the index mechanism in multiple broadcast channels [18,21]. A sys-tem of multiple broadcast channels can be viewed as a broad-cast disk array. The broadbroad-cast disks in a broadbroad-cast disk array can be categorized according to the speed of broadcast disks, where the speed of a broadcast disk corresponds to the ex-pected delay for the data items in that broadcast disk. The data items in each broadcast disk are sent out in a round robin man-ner. Clearly, as the number of data items in a broadcast disk increases, the expected delay of those data items increases. As a result, the data items that are more frequently requested by mobile users should be put in fast broadcast disks, whereas cold data items can be pushed to slow broadcast disks to mini-mize the average expected delay of data items in the broadcast disk array. Organizing data in a broadcast disk array raises a number of new research problems. The most important issue is to develop algorithms to allocate data items to the broad-cast disk array according to their access frequencies so as to minimize the average expected delay of data items. This is the very problem that we shall address in this paper.
The problem we study can be best understood by the illus-trative example in figure 1 where two broadcast programs are presented. Assume that the data items Ri, 1 i 6, are of
the same size and figure 1(a) shows a flat broadcast program, where data items are evenly allocated to the two broadcast disks and the speeds of two broadcast disks in the broadcast
Table 1
Expected delays and access frequencies of data items under two broadcast programs. (a)
Expected delay of data items
dR1 dR2 dR3 dR4 dR5 dR6
In figure 1(a) 1 1 1 1 1 1
In figure 1(b) 0.5 0.5 1.5 1.5 1.5 1.5
(b)
Access frequency Average expected delay Pr(R1) Pr(R2) Pr(R3) Pr(R4) Pr(R5) Pr(R6) in figure 1(a) in figure 1(b)
Case 1 0.167 0.167 0.167 0.167 0.167 0.167 1 1.16
Case 2 0.25 0.25 0.125 0.125 0.125 0.125 1 1
Case 3 0.3 0.3 0.1 0.1 0.1 0.1 1 0.9
Case 4 0.4 0.4 0.05 0.05 0.05 0.05 1 0.7
Figure 1. Two broadcast programs for the broadcast disk array of two broad-cast disks.
disk array are the same, meaning that the expected delays for all data items are equal from one to another. In contrast, fig-ure 1(b) shows another broadcast program with its channel al-location tree (or abbreviatedly as alal-location tree). As will be described in section 2 later, the depth of the allocation trees corresponds to the number of broadcast disks, and those leaf nodes in the same level of the allocation tree correspond to those data items to be put in the same broadcast disk. In fig-ure 1(b), the upper channel is allocated with two data items and the lower channel is allocated with four data items. No-tice that in figure 1(b) the data items in the fast disk (i.e., the upper broadcast channel) spin twice as fast as those data items in the slow disk (i.e., the lower broadcast channel). The allocation tree in figure 1(b) is called an allocation tree with variant-fanout in this paper. Denote the expected delay and the access frequency of data item Ri, respectively, as dRi and
Pr(Ri). Table 1(a) shows the expected delay for data items
under the two different allocations of broadcast arrays, and four sets of access frequencies of data items are given in ta-ble 1(b) for illustrative purposes. The average expected delay in table 1(b) is obtained by multiplying the access frequency of each data item by the expected delay of that data item and summing up the results, i.e.,6i=1dRi· Pr(Ri). Same as in
[1,23], the expected delay for each data item in the broad-cast disk i is formulated asNi
x=1(Ni− x)/Ni where i is the
number of data items allocated in the broadcast disk i. For ex-ample, for the allocation in figure 1(a), N1 = 3 and N2= 3, and for the allocation in figure 1(b), N1= 2 and N2= 4. For ease of exposition, the details of formulating the average ex-pected delay, which are in essence the same as those used in [1,23], are given in the appendix for interested readers. It can be verified that the expected delays of data items R1and R2in figure 1(a) are both equal to dR1 = dR2 = (2 + 1 + 0)/3 = 1.
On the other hand, the expected delays of data items R1 and R3 in figure 1(b) are dR1 = (1 + 0)/2 = 0.5 and
dR3 = (3 + 2 + 1 + 0)/4 = 1.5, respectively.
1
It can be seen from table 1(b) that when the access frequen-cies of all data items are the same, the flat broadcast program in figure 1(a) is the one to use, and we do not want to allocate different numbers of data items to different broadcast disks as in figure 1(b). Explicitly, the average expected delay of data items for case 1 is6i=1dRi· Pr(Ri)= 1 · 0.167 + 1 · 0.167+
1·0.167+1·0.167+1·0.167+1·0.167 = 1, whereas that in figure 1(b) is6i=1dRi· Pr(Ri)= 0.5 · 0.167 + 0.5 · 0.167 +
1.5· 0.167 + 1.5 · 0.167 + 1.5 · 0.167 + 1.5 · 0.167 = 1.16. It can be verified that the average expected delay resulting from the flat broadcast program will remain the same as the access frequencies of data items vary. However, when the access fre-quencies become increasingly skewed, the average expected delay of allocation with variant-fanout is reduced from 1.16 in case 1 to 0.7 in case 4. With the access frequencies being skewed, the average expected delay of the hierarchical broad-cast program based on variant-fanout becomes much smaller than that of the flat broadcast program, showing the very ad-vantage of exploiting the feature of hierarchy in designing broadcast programs for a broadcast disk array.
Consequently, we explore in this paper the issue of gen-erating hierarchical broadcast programs with the data access 1Note that as described in the appendix, same as in prior work [1,23], the expected delay dRi we consider in this paper does not include the
resid-ual waiting time which is half of the size of a data item. This provision will facilitate our presentation, but will not affect the quality of solutions derived.
frequencies and the number of broadcast disks in a broadcast disk array given. Specifically, we first formulate the prob-lem of generating hierarchical broadcast programs and trans-form this problem into the one of constructing a channel al-location tree with variant-fanout. A hierarchical broadcast program for the broadcast disk array of K broadcast disks can then be represented by a channel allocation tree with a height of K. By exploiting the feature of tree generation with variant-fanout, we develop a heuristic algorithm VFK (stand-ing for variant-fanout with the constraint K), which is basi-cally a family of algorithms with different values of K, to minimize the expected delay of the corresponding broadcast program. In order to evaluate the solution quality obtained by algorithm VFK and compare its resulting broadcast program with the optimal one, we devise an algorithm OPT (stand-ing for OPTimal) to obtain the optimal solution. Algorithm OPT is mainly a guided search and is similar to the well-known A∗search by using a cost function to guide the search and to ensure the optimality of the goal node reached [19]. With the proper design of the guide function, algorithm OPT can obtain the optimal solution very efficiently. Performance of these algorithms is comparatively analyzed and sensitivity analysis on several parameters, including the number of data items and the number of broadcast disks, is conducted. It is shown by our simulation results that by exploiting the fea-ture of variant-fanout in constructing the channel allocation tree, the solution obtained by algorithm VFK is of very high
quality and is in fact very close to the optimal one resulted by algorithm OPT. With its polynomial time complexity, al-gorithm VFK incurs a much shorter execution time than
al-gorithm OPT. As the number of broadcast disks and the num-ber of data items increase, the advantage of algorithm VFK over algorithm OPT becomes more prominent. It is worth mentioning that even when the number of broadcast chan-nels K changes dynamically, algorithm VFK can reach the new configuration very efficiently with the number of data items required to be moved around broadcast channels min-imized, showing another advantage of algorithm VFK. It is also shown by our experimental results that algorithm VFK is not only able to produce the solutions of very high quality but also of good scalability which is important for VFKto be
of practical use to generate hierarchical broadcast programs dynamically in a mobile computing environment.
A significant amount of research effort has been elaborated upon issues of data broadcast [1,3,4,12,24]. We mention in passing that the authors in [1] explored a push-based data de-livery architecture using the broadcast disk to meet the need of mobile applications. The research on striking a balance between push and pull data delivery methods was conducted in [2]. The authors in [11] proposed a mechanism to capture data access patterns which can be utilized in a data delivery model composed of push-based and pull-based delivery meth-ods. Without fully exploiting the advantage of multiple broad-cast channels, the attention of those studies in [1,6,8,11,18] was mainly paid to the design of data delivery model and in-dex methods for a single broadcast channel, but not for ple broadcast channels. The design of index methods in
multi-ple broadcast channels was addressed in [18,21], where, how-ever, the problem of generating hierarchical broadcast pro-grams was not considered.
The contributions of this paper are twofold. First, we pro-pose an innovative concept to broadcast data in a hierarchi-cal manner through multiple broadcast channels, and then, in light of this concept, we develop algorithm VFK which uti-lizes the feature of variant-fanout tree generation to minimize the average expected delay of data items. In addition, in or-der to evaluate the quality of the solution obtained by algo-rithm VFK, we develop algorithm OPT which exploits the feature of A∗search to obtain the optimal solution for perfor-mance comparison with algorithm VFK. To the best of our knowledge, priori works neither fully explored the impact of using a broadcast disk array, nor utilized the feature of hier-archy to allocate data items for broadcasting, let alone devel-oping efficient algorithms to generate hierarchical broadcast programs and conducting their performance studies. These features distinguish this paper from others. The fast increase in mobile applications which use broadcasting for informa-tion disseminainforma-tion justifies the timeliness and importance of this study.
This rest of this paper is organized as follows. Prelimi-naries are given in section 2. In section 3, we develop algo-rithm VFK for allocating data items to a broadcast disk
ar-ray. We devise algorithm OPT in section 4 to obtain the opti-mal solution. Performance studies are conducted in section 5. This paper concludes with section 6.
2. Preliminaries
Note that as the number of data items in a broadcast disk increases, the expected delay of those data items within the broadcast disk increases. Theoretically, generating such a broadcast program can be viewed as a partition problem for data items. Given the number of broadcast disks in a disk array and the access frequencies of all data items, we shall determine the proper set of data items that should be allo-cated to each broadcast disk in a broadcast disk array with the purpose of minimizing the average expected delay of all data items. Table 2 shows the description of symbols used in mod-elling the program. Figure 2 shows the program formulation of generating a hierarchical broadcast program for a broadcast disk array of K broadcast disks. Denote the total number of
Table 2 Description of symbols.
Description Symbol
Number of broadcast disks in a broadcast disk array K Number of data items within broadcast disk i Ni
The expected delay of data items within broadcast disk i di
The j th data item Rj
The access frequency of data item Rj Pr(Rj)
The expected delay of data item Rj dRj
The number of leaf nodes in level i of the allocation tree Li
The weight of leaf nodes in level i of the allocation tree wi
Figure 2. Generating hierarchical broadcast programs for a broadcast disk array of K broadcast disks, where Niis the number of data items allocated
to disk i and ti=ij=1Nj.
data items as n, and a data item as Ri, 1 i n. Recall that Pr(Ri)is the access frequency of Ri and
n
i=1Pr(Ri)= 1.
Assume that all data items in figure 2 have been sorted ac-cording to the descending order of the access frequencies of all data items. As can be seen in figure 2, a broadcast disk ar-ray of K broadcast disks corresponds to the partition of n data items into K groups, where Ni is the number of data items to
be allocated to broadcast disk i. We then haveKi=1Ni = n.
According to the characteristics of broadcast disks, we have the following properties.
Property 1. The expected delay of the data items within the broadcast disk i, denoted by di, is di =
Ni
j=1(Ni− j)/Ni,
where Ni is the number of data in that broadcast disk.
Property 2. Let ti =
i
h=1Nh and t0 = 0. Then, Ni data
items, denoted by Rj, ti−1+ 1 j ti, are allocated to
broadcast disk i, and di = dRj for j∈ [ti−1+ 1, ti].
For example, in figure 1(b), d1 = dR1 = dR2 = 0.5
since R1 and R2 are in broadcast disk 1, and d2 = dR3 =
dR4 = dR5 = dR6 since R3, R4, R5 and R6 are in
broad-cast disk 2. The problem that we study in this paper can be formally defined as follows:
Problem of generating a hierarchical broadcast program. Given the number of broadcast disks in a disk array and the access frequencies of all data items, we shall determine the proper set of data items that should be allocated to each broad-cast disk in a broadbroad-cast disk array with the purpose of mini-mizing the average expected delay of all data items. The aver-age expected delay of all data items in a broadcast disk array of K broadcast disks can be formulated as follows:
n j=1 dRj · Pr(Rj) = K i=1 di ti j=ti−1+1 Pr(Rj) = K i=1 N i q=1 Ni− q Ni t i j=ti−1+1 Pr(Rj), (1) where ti = i h=1Nhand t0= 0.
Problem transformation. As mentioned above, generating such a broadcast program can be viewed as a partition prob-lem. We now would like to transform this partition problem
Figure 3. Grouping a set of nodes and moveing them to a lower level to reduce the weights of leaf nodes.
on data items into the one of generating a channel allocation tree. The initial allocation tree is mainly a tree of all data items attached to the root node (as shown in figure 3(a)). Note that the leaf nodes in the same level of the allocation tree cor-respond to a set of data items to be put in the same broadcast disk. Thus, the allocation tree in figure 3(a) corresponds to the case that all data items are put in one broadcast disk. We next expand the allocation tree by moving a set of nodes to one level lower as shown in figure 3(b) (the criterion to determine such a set of nodes will be explained in section 3 later). Fig-ure 3(b) corresponds to the case that two broadcast disks are used to store the data items. This procedure continues by per-forming more partitions on data items until all broadcast disks in the broadcast disk array are used. As such, a hierarchical broadcast program for a broadcast disk array of K broadcast disks can be represented as a channel allocation tree with a height of K.
It can be seen that similar to the formulation of the ex-pected delay for those data items residing in a broadcast disk, the weight of leaf nodes in level i, denoted by wi, can be
for-mulated as wi =
Li
x=1(Li− x)/Li, where Li is the number
of leaf nodes in the level i. The weight wiin fact corresponds
to the expected delay of each data item in level i. Explic-itly, when the construction of an allocation tree is completed, Li equals Ni and wi equals di. Let PLi be the summation
of access frequencies of data items associated with leaf nodes in level i. For example, it can be seen from figure 1(b) that the depth of the allocation tree is 2 and those leaf nodes in the same level are allocated in the same broadcast disk. For that allocation tree with access frequencies given in case 3 of table 1(b), we have w1 = 0.5, w2 = 1.5, PL1 = 0.6, and
PL2 = 0.4. According to the formulation of average expected
delay for broadcast disk array, we have the cost of the tree below: K i=1 wiPLi = K i=1 L i q=1 Li− q Li · PLi, (2) where PLi= si j=si−1+1 Pr(Rj),
si= i
h=1
Lh and s0= 0.
As such, the problem of generating a hierarchical broad-cast program can be transformed into the one of building an allocation tree with the minimal cost.
3. Algorithm VFK: using variant-fanout for allocation
tree generation
We devise in section 3.1 a heuristic algorithm VFK to gener-ate the channel allocation tree which explores the feature of tree generation with variant-fanout to minimize the expected delay of the corresponding broadcast program. The execution scenario of algorithm VFKis illustrated in section 3.2. 3.1. Design of algorithm VFK
With the problem transformation described above, we devise algorithm VFK (standing for Variant-Fanout with the con-straint K) to explore variant-fanout in the allocation tree gen-eration to minimize the cost of this tree.
Algorithm VFKis greedy in nature and builds the alloca-tion tree in a top down manner. Algorithm VFK starts with attaching all data records to the root node. Then, after some evaluation, algorithm VFK groups nodes with small access frequencies and moves them to one level lower so as to reduce the cost of the tree. Figure 3 shows the scenario of grouping a set of nodes and moving them to a lower level.
Definition 1. Suppose that level v in the allocation tree has
j− i + 1 data nodes, Ri, Ri+1, . . . , Rj. The cost of level v is
defined as Ci,j = j−i+i k=1 (j− i + 1) − k j− i + 1 j q=i Pr(Rq),
which is equal to wv·PLv, where wvand PLvare, respectively,
the weight of leaf nodes and the aggregate access frequency for the leaf nodes in level v of the allocation tree.
In essence, the value of Ci,j is related to the average
ex-pected delay of leaf nodes in level v. It can be seen from figure 3 that moving down those nodes to the next level de-creases the corresponding expected delay of leaf nodes, and thus the cost of the allocation tree will be reduced. In order to evaluate the reduction of partitioning, we have the following definition.
Definition 2. Suppose that node R has j− i + 1 child data nodes, Ri, Ri+1, . . . , Rj, which are sorted according to the
descending order of Pr(Rq), i q j, i.e., Pr(Rq) Pr(Ry)iff q y. The reduction gain achieved by grouping
nodes Rp+1, Rp+2, . . . , Rj and attaching them under a new
child node, denoted by δ(p), can be formulated as δ(p) = Ci,j − (Ci,p+ Cp+1,j).
Algorithm VFK.
Input. Assume that R1, . . . , Rnhave been sorted according to the
descending order of Pr(Rj), 1 j n, i.e.,Pr(Rq) Pr(Ry)iff q y.
Kis the number of broadcast disks in a broadcast disk array.
Output. The resulting allocation tree. begin
1. Create table AT with K rows;
2. AT(1).B= 1; /∗AT(1).B records the beginning of level 1∗/ 3. AT(1).E= n; /∗AT(1).E records the end of level 1∗/ 4. AT(1).LC= C1,n; /∗AT(1).LC records the cost of level 1∗/ 5. for each row i in table AT and i 2
6. begin
7. AT(i).B= 0; /∗AT(i).B records the beginning of level i∗/ 8. AT(i).E= 0; /∗AT(i).E records the end of level i∗/ 9. AT(i).LC= 0; /∗AT(i).LC records the cost of level i∗/ 10. end
11. pivot= 1; 12. repeat 13. begin
14. Choose row i from table AT such that AT(i).LC is maximal among all unmarked rows;
15. if (i== 1 or i == pivot) /∗Upward and downward partition∗/
16. begin
17. j= Partition(RAT(i).B, RAT(i).B+1, . . . , RAT(i).E);
18. {Update table AT accordingly and unmark all rows; 19. pivot+ +;}
20. end
21. else /∗Middle partition∗/
22. begin
23. j= Partition(RAT(i).B, RAT(i).B+1, . . . , RAT(i).E);
24. if (AT (i− 1).E − AT (i − 1).B) <(j − AT (i).B)
25. {Update table AT accordingly and unmark all rows;
26. pivot+ +;}
27. else {
28. Mark row i;
29. Merge (RAT(i).B, RAT(i).B+1, . . . , Rj)and
(Rj+1, Rj+2, . . . , RAT(i).E) together;}
30. end
31. end
32. until (pivot== K)
end
In light of definition 2, we devise algorithm VFK which contains a procedure Partition to identify the group of nodes to be moved downward in each execution level so as to maxi-mize the reduction gain in each step.
Procedure Partition(Ri, Ri+1, . . . , Rj).
1. Determine p∗such that
δ(p∗)= max
∀p∈{i,i+(j−i+1)/2−1}{δ(p)}.
2. Attach nodes Rp∗+1, Rp∗+2, . . . , Rj under a new node I
in the tree. 3. Return p∗.
To generate the allocation tree, algorithm VFK first starts with a configuration where all nodes are attached to the root. Table AT (standing for Auxiliary Table) is created to record the status of the allocation tree as stated from line 1 to line 10 in algorithm VFK. Note that since there are three kinds of
(a)
(b)
Figure 4. The tree representations by using TLand TU. (a) A tree represen-tation by using TLsubtree. (b) A tree representation by using TUsubtree. partitioning, i.e., upward, middle and downward partitions, judiciously applying these partitions is able to reduce the to-tal cost of the allocation tree. To facilitate the description of algorithm VFK, we define two subtrees, i.e., TU and TL,
where TU (respectively, TL) is left-upper (respectively,
right-lower) subtree and does not contain the top level (respectively, the last level) of the tree. Consider the tree representations by TU and TL in figure 4 where the original tree is the one
in figure 3(b). Figure 5(a) shows the scenario of an upward partition for the allocation tree in figure 4(a). On the other hand, figure 5(b) illustrates the scenario of a downward par-tition for the allocation tree in figure 4(b). From table AT, algorithm VFK chooses the level with the maximal cost to partition (line 14 of algorithm VFK). Let pivot be the number of levels expanded in the allocation tree thus far. Explicitly, as can be seen from line 15 to line 20 in algorithm VFK, if the cost of the top (respectively, the last) level is larger than that of other levels, algorithm VFK performs upward (respectively, downward) partition. Otherwise, VFK performs the middle partition according to the operations from line 21 to line 31 in algorithm VFK. Table AT is then updated accordingly. As can be seen from line 24 to line 29 in algorithm VFK, al-gorithm VFK makes sure that each partition will satisfy the feature of hierarchy. That is, the number of leaf nodes (i.e., data items) in the upper level is always smaller than that in the lower level. According to definition 2, the candidate set of nodes with the maximal reduction gain δ(p∗)is chosen. When such a set of nodes is identified and moved to the next level, those nodes will be evaluated by themselves to see if any further partition for some of them is necessary in line 32 (the condition (pivot == K) means that the number of the channels has been reached). Algorithm VFK partitions the nodes iteratively with the objective of minimizing the
aver-(a)
(b)
Figure 5. Illustrations of upward and downward partitions. (a) A scenario of upward partition. (b) A scenario of downward partition.
age costKi=1wi· PLi until the depth of the tree reaches the
number of broadcast disks in the broadcast disk array. As such, the allocation tree is expanded and constructed in a top down manner.
It can be verified that algorithm VFK is of polynomial time complexity. With n sorted data items and a broadcast disk array of K broadcast disks given, the complexity of al-gorithm VFK can be expressed by K· (O(K log K) + O(n)). Specifically, the complexity of choosing the partition with the maximal cost from table AT is K · O(K log K) and that of partitioning is K · O(n). When the number of data items is significantly larger than the number of broadcast disks, i.e., n K, the value of K · O(n) is the dominating factor of the complexity of VFK.
3.2. An example execution scenario VFK
For example, consider the profile in table 3 where the number of data items n is 11 and the number of broadcast disks K is 4. The initial tree configuration is shown in figure 6(a), where all data records are attached to the root. Procedure Partition then determines the optimal partition of nodes to be moved to the next level. The values in table AT and their changes made in accordance with the execution of algorithm VFK are shown in table 4, where in table 4(a)–4(c), the first table is table AT and the second table is the one to determine the cut point p∗ for the partition selected where the maximal value of δ(p) is marked with an ‘*’. From the calculation shown in table 4(a), we obtain p∗= 4, and therefore group nodes R5, R6, . . . ,and
Table 3
The profile of an illustrative example.
Data record R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11
Pr(Ri) 0.237 0.211 0.132 0.132 0.08 0.05 0.05 0.027 0.027 0.027 0.027
Figure 6. An execution scenario of algorithm VFK: (a)–(d) the generation of the allocation tree, and (e) the resulting broadcast program.
R11together and move them to the next level, resulting in the configuration shown in figure 6(b). Note that in table AT, the value of AT(i).LC is the cost of nodes in level i, i.e., wi· PLi,
and explicitly, equal to CAT(i).B, AT (i).E. In table 4(b), since AT(1).LC > AT(2).LC, the upward partition is performed. Following the calculation shown at the bottom of table 4(b), we have p∗ = 2 and figure 6(b), in turn, leads to figure 6(c). Similarly, we obtain AT(1).LC = 0.224 and AT(2).LC = 0.132 shown in table 4(c). Since AT(3).LC = 0.864 is the maximal in table 4(c), algorithm VFK performs the down-ward partition. Following the calculation in table 4(c), we have p∗ = 7. Then, the configuration shown in figure 6(d) and in table 4(d) follows. As the depth of the allocation tree equals the number of broadcast disks, algorithm VFK
completes. It can be seen that the very advantage of algo-rithm VFK is that even when the number of broadcast chan-nels K changes dynamically, algorithm VFK can reach the new configuration very efficiently with the number of data items required to be moved around broadcast channels min-imized. Also, as validated in our experiments in section 5, when the number of data items to be broadcast is large, al-gorithm VFK can expand the allocation tree very efficiently. According to the configuration in figure 6(d), we can have the hierarchical broadcast program shown in figure 6(e) where it can be verified that the average expected delay is the sum of costs of all levels in the tree shown in figure 6(d), i.e., 0.224+ 0.132 + 0.18 + 0.162 = 0.698.
Table 4
Determining from table AT the set of nodes to be grouped together as a partition for allocation tree generation.
(a) Partition (R1, R2, . . . , R11)is selected and decomposed to (R1, . . . , R4) and (R5, . . . , R11).
Level i AT(i).B AT(i).E AT(i).LC
1 1 11 5∗ 2 0 0 0 3 0 0 0 4 0 0 0 p 1 2 3 4 5 C1,11 5 5 5 5 5 C1,p+ Cp+1,11 3.4335 2.484 2.05 1.932 2.104 δ(p) 1.5665 2.516 2.95 3.068∗ 2.896
(b) Partition (R1, R2, . . . , R4)is selected and decomposed to (R1, R2)and
(R3, R4).
Level i AT(i).B AT(i).E AT(i).LC
1 1 4 1.068∗ 2 5 11 0.864 3 0 0 0 4 0 0 0 p 1 2 C1,4 1.068 1.068 C1,p+ Cp+1,4 0.475 0.356 δ(p) 0.593 0.712∗
(c) Partition (R5, R6, . . . , R11) is selected and decomposed to (R5, . . . , R7) and (R8, . . . , R11).
Level AT(i).B AT(i).E AT(i).LC
1 1 2 0.224 2 3 4 0.132 3 5 11 0.864∗ 4 0 0 0 p 5 6 7 C5,11 0.864 0.864 0.864 C5,p+ Cp+1,11 0.52 0.381 0.342 δ(p) 0.344 0.483 0.522∗
(d) The final result of table AT.
Level i AT(i).B AT(i).E AT(i).LC
1 1 2 0.224
2 3 4 0.132
3 5 7 0.18
4. Algorithm OPT: obtaining the optimal broadcast program
In order to compare the solution quality obtained by algo-rithm VFK with the optimal one, by utilizing the concept of A∗search [19], we implement in section 4.1 algorithm OPT which constructs a solution tree and is able to determine the optimal solution. The execution scenario of algorithm OPT is presented in section 4.2.
4.1. Design of algorithm OPT
Finding solutions for the problem of partitioning data items and allocating them into each broadcast disk can be rep-resented by a search problem based on state transition. In essence, algorithm OPT is a best-first state transition search [16,19]. The search made by algorithm OPT can be represented by a solution tree where each node is associated with a state of partitioning. Note that the solution tree in algo-rithm OPT should not be confused with the allocation tree in algorithm VFK. Figure 7 shows part of an example state
tran-sition search which corresponds to the case of allocating 6 data items into 3 broadcast disks. Starting from the root in the level one, algorithm OPT generates nodes in level i to explore the possible partitions of data items into i group. Al-gorithm OPT is mainly a guided search and is similar to the well-known A∗ search by using a cost function to guide the search and to ensure the optimality of the goal node reached. However, algorithm OPT we employed in this study is differ-ent from a typical A∗search in that our interest here is only limited to obtain the optimal partition efficiently so that we can evaluate the solution quality obtained by algorithm VFK. Yet, in this specific application, we do not have to use the op-timal path from the root to the goal node found by the search. As shown in figure 7, a node in the solution tree contains the set of partitions where the data items in the same partition are allocated to the same broadcast disk. A node in the level j has j partitions, which is associated with one allocation for a broadcast disk array of j broadcast disks. The set of partitions for the node i in the level j is denoted by SPi and the
parti-tions belonging to SPi are denoted by P1, P2, . . . , Pj. We use
|SPi| to represent the number of partitions in that set. Let the
data items in the beginning and the end of Pj be denoted by Pj.B and Pj.E, respectively. The cost of the partition Pj,
expressed by c(Pj), is then CPj.B,Pj.E which can be
deter-mined by definition 1. The cost of the node can be obtained by summing up the costs of partitions within that node. Also, let Pmaxrefer to the partition having the maximal cost within
Figure 7. The solution tree with the number of data items to be 6 for K= 3.
the node. Consider the node marked with a star in figure 7 as an example. In that node, there are three partitions P1, P2 and P3. It can be verified that c(P1)= C1,1 = 0, c(P2) = C2,2 = 0, c(P3) = C3,6 and the cost of that node is equal to the value of3i=1c(Pi). P3is then the Pmaxof that node with Pmax.B = 3 and Pmax.E = 6. With a given solution tree of its depth K, the problem of generating a hierarchical broadcast program for K broadcast disks becomes a search problem based on state transition in which a node in level K with the minimal cost is to be found (i.e., the goal node).
Similarly to an A∗search, the search in algorithm OPT is controlled by an evaluation function f (·). The node i chosen for further partition (i.e., whose immediate successors will be generated in the solution tree) is the one which has the small-est value of f (·) among all generated nodes which have not been partitioned so far. Algorithm OPT stops the expansion until the goal node is found. Hence, the function f (i) that guides the search in algorithm OPT consists of two compo-nents: the cost of reaching node i from the root, i.e., g(i), and the expected cost of arriving at the goal node from node i, i.e., h(i). Accordingly, f (i)= g(i) + h(i).
The function g(i) is calculated by the cost from the root to node i along the minimal cost path found so far in the so-lution tree. Assume that the number of broadcast disks in a broadcast disk array is K and the node i is in level j . Thus, we have, g(i)= j−1 x=1 c(Px),
where Px ∈ SPi − Pmaxfor the guided search in algorithm OPT.
In addition, the function h(i) is the estimated cost from node i to the goal node. In order to obtain the minimal cost, Pmax that has the maximal cost is chosen for further parti-tioning so as to reduce the total cost from the root to the nodes generated from node i. By utilizing the procedure Partition described in section 3, we have the following pro-cedure Partition_Cost which returns the minimal cost for par-titioning Pmax.
Procedure Partition_Cost(Pmax).
1. p∗= Partition(RPmax·B, RPmax·B+1, . . . , RPmax·E).
2. Return ((CPmax·B,Pmax·p∗)+ (CPmax·p∗+1,Pmax·E)).
Consequently, the function h(i) for node i in level j can be defined as below:
Case 1. If the level of node i equals the number of broadcast disks, i.e., j = K, meaning that no further partition is needed, h(i)= c(Pmax).
Case 2. If the level of node i is smaller than the number of broadcast disks, i.e., j < K, meaning that further parti-tions are needed, h(i)= Partition_Cost(Pmax)/(K− j). In light of the evaluation function, algorithm OPT can be outlined below.
Figure 8. An execution scenario of algorithm OPT.
Algorithm OPT.
Input. Assume that R1, R2, . . . , Rn have been sorted
ac-cording to the descending order of Pr(Ri), 1 i n, i.e., Pr(Rp) Pr(Rq)iff p q. K is the number of broadcast
disks in the broadcast disk array. Output. Report the goal node. begin
1. Construct a heap by the value of the evaluation function f (·);
/∗A heap is a data structure which returns the element with the minimal value when one is to remove an element from the heap [7]∗/
2. Insert the root node into the heap; 3. repeat
4. begin
5. Remove the node i from the heap; 6. Expand the child nodes of node i;
7. Calculate the descendants of node i and insert these descendants into the heap;
8. end
9. until (|SPi| == K);
10. Return the node i; end
Algorithm OPT first constructs the heap [7] and inserts the root node into the heap (line 1 to line 2 in algorithm OPT). Al-gorithm OPT removes the node i from the heap (line 5 in al-gorithm OPT) and expands the descendants of node i (line 6). The values of function f (·) for all descendants are calculated and these descendants are inserted into the heap (line 7). Al-gorithm OPT expands the nodes iteratively until|SPi| = K
and the node i with the minimal cost is removed form the heap (line 10).
Let h∗(i)denote the cost of a search path from node i to the goal node. Theorem 1 states that for any node i, h(i) h∗(i).
Similarly to that in an A∗search [19], this inequality ensures the optimality of the solution found by algorithm OPT. Theorem 1. Given a node i in level j , h(i) is a lower bound of h∗(i).
Proof. Clearly, h∗(i)can be decomposed into
K−1 q=j
Partition_CostPmaxq
+ cPmaxK ,
where Pmaxq is the partition with the maximal cost in level q. Note that if (j <K), we have
h∗(i)− h(i) = (K − j)−1 × (K− j) K−1 q=j+1 Partition_CostPmaxq + cPmaxK + (K − j − 1) · Partition_CostPmaxj 0. Also, if j equals K, we have h∗(i)− h(i) =Kq=j−1 Parti-tion_Cost(Pmaxq ) 0. Consequently, we have h∗(i) h(i).
This theorem follows.
4.2. An example execution scenario of OPT
Consider the profile in table 3 where the number of broadcast disks is 4, i.e., K = 4. The execution scenario of algorithm OPT is shown in figure 8, where the number next to each link represents the sequence of node expansion. Algorithm OPT starts with the root node and then expands the descendants. The values of functions g(·), h(·) and f (·) are calculated ac-cordingly. Consider the node marked with a star in figure 8 as an example. In that node, the value of g(·) is C1,2= 0.224 and that of h(·) equals (C3,4 + C5,11)/2 = 0.431. Thus,
f (·) is obtained accordingly, i.e., f (·) = g(·) + h(·) = 0.224+ 0.431 = 0.655.
Following the same procedure, we can obtain the values of function f (·) for each descendant node. After expanding the root node, all these descendant nodes are inserted to the heap. Since the level of the root node is smaller than K, al-gorithm OPT removes from the heap the node with the min-imal cost to expand. Thus, the node marked with a star in figure 8 is selected for further expansion. Algorithm OPT stops expanding after the level number reaches K. It can be seen that the resulting broadcast program, with the partition (R1, R2), (R3, R4), (R5, R6, R7)and (R8, . . . , R11)and the corresponding average expected delay of 0.698, obtained by algorithm OPT is indeed the same as the one obtained by al-gorithm VFKin section 3.2. As the number of data items and the value of K increase, the number of nodes in the solution tree increases drastically. It can be seen that the number of nodes expanded in this illustrative example is 15. With the proper design of the guide function, OPT can obtain the opti-mal solution very efficiently.
Note that procedure Partition is the most time consum-ing procedure. Performance of algorithm VFKand algorithm OPT can be compared in terms of the number of procedure calls for Partition. For a broadcast disk array of K disks, the number of procedure calls for Partition in algorithm VFK is K − 1, whereas that in algorithm OPT is the number of internal nodes expanded by algorithm OPT in the solution tree. As will be shown in section 5, the solution obtained by algorithm VFK is of high quality and is in fact very close to that of algorithm OPT. As the number of broadcast disks and the number of data items increase, the advantage of algo-rithm VFKover algorithm OPT becomes more prominent.
5. Performance evaluation
In order to evaluate the performance of algorithm OPT and VFK, we have implemented a simulation model of the
broad-cast environment. Specifically, the simulation model is de-scribed in section 5.1. Then, we examine the impact of em-ploying hierarchical broadcast programs in section 5.2. Per-formance of algorithm VFK and algorithm OPT is
compara-tively analyzed in section 5.3. 5.1. Simulation model
Table 5 summarizes the definitions for some primary simula-tion parameters. The number of data items to be broadcasted in a broadcast disk array is denoted by n and the number of broadcast disks in a broadcast disk array is K. The access fre-quencies of broadcast data items are modelled by the Zipf dis-tribution. Let Pr(Ri)= ((N − i)/N)θ/
n
j=1((N− j)/N)θ,
where θ is the parameter of Zipf distribution [10]. It can be verified that the access frequencies become increasingly skewed as the value of θ increases. For comparison purposes, a scheme that evenly allocates data items to each broadcast disk, referred to as FLAT, is implemented. An example
broad-Table 5
The parameters used in the simulation. Notation Definition
n Total number of data items to be broadcast K Number of broadcast disks in a broadcast disk array θ Zipf distribution parameter
FLAT Scheme to generate a flat broadcast program
Figure 9. The average expected delay of FLAT, OPT and VFKwith the value of θ varied.
cast program generated by FLAT is the one shown in fig-ure 1(a). Experimental results for algorithm VFK, algorithm OPT and FLAT have been obtained and comparatively evalu-ated.
5.2. The impact of employing hierarchical broadcast programs
To show the advantage of generating hierarchical broadcast program, we set the value of n to 50 and the value of K to 4. The expected delay of data items under FLAT, OPT and VFK are examined with the value of θ varied. Without loss of gen-erality, assume that all the data items are of the same size which is used as one unit of waiting time. The resulting ex-pected delay of data items by running FLAT, OPT and VFK are shown in figure 9. It can be seen from figure 9 that the ac-cess frequencies become increasingly skewed as the value of θincreases and the difference between expected delay of OPT and that of VFKis almost negligeable, showing the very high
quality of the solutions obtained by algorithm VFK. Note
that as the access frequencies become increasingly skewed, the hierarchical broadcast programs generated by OPT and VFKperform significantly better than the flat broadcast pro-gram, indicating the very advantage of exploiting the feature of variant-fanout for the allocation tree generation in algo-rithm VFK.
5.3. Comparative analysis for VFK and OPT
We now examine the impact of varying the values of n and K to the performance of VFK and OPT. First, in order to evalu-ate the impact of increasing the value of K, we set the value
Figure 10. The expected delay of OPT and VFKwith the value of K varied.
Figure 11. The execution time incurred by OPT and VF with the value of K varied.
of n to 50 and the value of θ to 3. Figures 10 and 11 show the performance results of OPT and VFK.
As can be seen in figure 10, the expected delay of OPT and VFK decreases as the value of K increases. This agrees
with our intuition, since as the number of broadcast channels increases, the number of data items in each broadcast chan-nel decreases, thereby reducing the expected delay of data items. Notice that the difference between the expected delay of VFK and that of OPT is very small when the value of K varies, again showing the good quality and the robustness of solutions obtained by VFK. From figure 11, the execution time incurred by OPT increases sharply as the value of K increases. This means that the advantage of algorithm VFK over algorithm OPT becomes even more prominent when the number of K is large, since the expected delays resulted by the solutions from both algorithms are all very close to each other.
Next, the experiments of varying the value of n for OPT and VFK are conducted where we set the value of K to 4 and
the value of θ to 3. Figure 12 shows the performance com-parison between OPT and VFK when the value of n varies.
In figure 12(a), as the number of data items to be broadcast increases, the expected delays of data items resulted by OPT and VFK increase linearly as we anticipate. Also, the
dif-(a)
(b)
Figure 12. The performance comparison between OPT and VFKwith the value of n varied. (a) The expected delay of OPT and VFKwith the value of
nvaried. (b) The execution time incurred by OPT and VFKwith the value
of n varied.
ference between the expected delays resulted by OPT and VFK is negligible, again suggesting that algorithm VFK be able to find a solution of very high quality efficiently when the value of n increases. Note that the execution time in-curred by OPT is also significantly larger than that inin-curred by VFK in figure 12(b), showing the increasing advantage of algorithm VFKover algorithm OPT when the number of data items increases.
It is shown by our experimental results that algorithm VFK is not only able to produce the solutions of very high quality but also of good scalability which is important for VFK to be of practical use to generate hierarchical broadcast programs dynamically in a mobile computing environment.
6. Conclusions
In this paper, we explored the issue of generating hierarchi-cal broadcast programs with the data access frequencies and the number of broadcast disks in a broadcast disk array given. Specifically, we first formulated the problem of generating hi-erarchical broadcast programs and transformed this problem into the one of constructing a channel allocation tree with
variant-fanout. By exploiting the feature of tree generation with variant-fanout, we developed a heuristic algorithm VFK
to minimize the expected delay of the corresponding broad-cast program. In order to evaluate the solution quality ob-tained by algorithm VFK and compare its resulting broad-cast program with the optimal one, we devised an algorithm OPT based on a guide search to obtain the optimal solution. Performance of these algorithms was comparatively analyzed and sensitivity analysis on several parameters, including the number of data items and the number of broadcast disks, was conducted. It was shown by our simulation results that by ex-ploiting the feature of variant-fanout in constructing the chan-nel allocation tree, the solution obtained by algorithm VFK is of very high quality and is in fact very close to the opti-mal one resulted by algorithm OPT. With its polynomial time complexity, algorithm VFK incurs a much shorter execution time than algorithm OPT. As the number of broadcast disks and the number of data items increase, the advantage of algo-rithm VFKover algorithm OPT becomes more prominent. By
exploring the feature of variant-fanout tree construction, algo-rithm VFK is not only able to produce the solutions of very high quality but also of good scalability which is important for algorithm VFK to be of practical use to generate hierarchical broadcast programs dynamically in a mobile computing envi-ronment.
Appendix. Average expected delay of data items
The broadcast channel is divided into slots of same size that is equal to the size of data item. Assume that the number of data items to be broadcast is Ni and the size of slots for
data items is one. Let Li(j )denote the number of slots from
the end of slot j for data item i. Without loss of generality, the arrival requests for data items are triggered in the middle of slots. An illustrative example of a request for data item x while data item x is being pushed is shown in figure 13. Note that the latency of the request for data item x is equal to 1/2+Lx(x). Since the residual time, i.e., 1/2, from the arrival
point of the request until the end of the slot is independent of the program studied in this paper, we ignore it and use Lx(x)
as the measure of the expected delay. It can be verified that the Lx(x)equals Ni− 1 in our illustrative example shown below.
While the request of data item x may occur at any time slots, we can formulate the average expected delay of data item x asNi
j=1Lx(j )/Ni =
Ni
j=1(Ni− j)/Ni. For more detailed
formulations, interested readers are referred to [1] and [23].
Figure 13. An illustrative example of the expected delay for data item x.
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Wen-Chih Peng was born in Hsinchu, Taiwan,
R.O.C in 1973. He received the BS and MS de-grees from National Chiao Tung University, Taiwan, in 1995 and 1997, respectively, and the Ph.D. de-gree in electrical engineering from the University of National Taiwan University, Taiwan, R.O.C. in 2001. During his Ph.D. program, he was mainly in-volved in the projects related to mobile computing, telecommunication databases and data mining. His research interests include mobile computing, mobile data management, and data mining. He is a member of the Phi Tau Phi Scholastic Honor Society.
E-mail: wcpeng@arbor.ee.ntu.edu.tw
Ming-Syan Chen received the B.S. degree in
elec-trical engineering from National Taiwan University, Taipei, Taiwan, and the M.S. and Ph.D. degrees in computer, information and control engineering from The University of Michigan, Ann Arbor, MI, USA, in 1985 and 1988, respectively. Dr. Chen is currently a Professor in Electrical Engineering Department, National Taiwan University, Taipei, Taiwan. He was a research staff member at IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA from 1988 to 1996. His research interests include database systems, data min-ing, mobile computing systems, and multimedia networkmin-ing, and he has pub-lished more than 120 papers in his research areas. In addition to serving as program committee members in many conferences, Dr. Chen is an asso-ciate editor of IEEE Transactions on Knowledge and Data Engineering on data mining and parallel database areas from 1997 to 2001, an editor of Jour-nal of Information Science and Engineering, a distinguished visitor of IEEE Computer Society for Asia–Pacific from 1998 to 2000, and program chair of PAKDD-02 (Pacific Area Knowledge Discovery and Data Mining), Program vice-Chair of VLDB-2002 (Very Large Data Bases), general chair of Real-Time Multimedia System Workshop in 2001, program chair of IEEE ICDCS Workshop on Knowledge Discovery and Data Mining in the World Wide Web in 2000, and program co-chair of the International Conference on Mobile Data Management in 2003, and also of International Computer Symposium on Computer Networks, Internet and Multimedia in 1998 and 2000. He was a keynote speaker on Web Data Mining in International Computer Congress in Hong Kong, 1999, a tutorial speaker on Web Data Mining in DASFAA-1999 and on Parallel Databases in the llth IEEE International Conference on Data Engineering in 1995 and also a guest co-editor for IEEE Transactions on Knowledge and Data Engineering on a special issue for Data Mining in December 1996. He holds, or has applied for, eighteen U.S. patents and seven ROC patents in the areas of data mining, Web applications, interactive video playout, video server design, and concurrency and coherency control proto-cols. He received the Outstanding Innovation Award from IBM Corporate in 1994 for his contribution to parallel transaction design and implementation for a major database product, and numerous awards for his research, teach-ing, inventions and patent applications. Dr. Chen is a senior member of IEEE and a member of ACM.
