2 Broadcasting Techniques on Irregular Networks
2.4 Location Aware Broadcast Scheme (LABS)
2.4.2 Contention-Free Broadcast
The performance of broadcasting is affected by link contention. Many studies have been done using up*/down* routing to reduce link contention in order to get a better performance. In this study, three ways for making a routing link contention-free were explored. In the first step, all senders and receivers were connected to the same switch, as shown in Figure 2.7(a). In the second, all senders were connect to the same switch and receivers to other switches, as shown in Figure 2.7(b), the senders at the same time send the messages to the receivers without link contention. In the last instance, two messages were sent from different sender switches to different receiver switches, as shown in Figure 2.7(c). Since both messages were passed through switch 1 at the same time, it did not cause any link contention in switch 1.
switch 1 sender
receiver receiver sender
sender sender
sender sender
receiver
receiver receiver receiver
(a) (b) (c)
Figure 2.7: Three contention-free links examples
After constructing the LST, some information was obtained from it for the construction of scheduling tree. For instance, the postorder list was obtained from the LST. Also, the workstation with the fastest speed could be determined so that a switch could be established as a group head, and then the postings could be grouped accordingly.
The first step in the construction of a scheduling tree, is acquiring the postorder of the workstations from the LST. The message routing in an ordered chain does contend in simultaneous links between different workstations. The postorder list of workstations from the LST is an ordered chain; this is used to construct the scheduling tree with contention-free links. In the second step, the fastest workstation for each switch is taken as the group head. The members of a group are the workstations connected to the same switch. Every group has its sub-tree group size that is the number of workstations in the sub-tree. In the third step, these sub-tree groups were adjusted according to their group size, decreasing from left to right. The adjusted list placed the large size groups in the upper level of the scheduling tree and the width and depth of the scheduling tree closer to the binomial-like scheduling tree. After adjusting the postorder list of workstations, the first level of the scheduling tree was constructed with all group heads using binomial-like scheduling. Then, each group constructed its scheduling tree, i.e. the second level scheduling tree, using the binomial-like scheduling until the scheduling tree construction is completed. In the last step, all the fast workstations were arranged to be the father workstations of the slow workstations; these workstations were then responsible to send the messages to the other slow workstations. This made the scheduling tree broadcasting more efficient. The links on the first level of the scheduling tree was still contention-free, even if the postorder list was adjusted. Because the workstations on the first level of the scheduling tree all were connected to the different switches, it was impossible to generate link contention between any two workstations sending the messages at the same time.
The links of the second level of the scheduling tree were also contention-free because the workstations in the same group were connected to the same switch. See Figure 2.7(a).
The workstations in this scheduling tree had only two connections: one was to those connected to neighboring switches and the others were connected to the same switch.
This conclusively indicated that the workstations in the scheduling tree had the shortest paths between all workstations. Thus, the links used in broadcasting in this scheduling tree was not only contention-free, but the paths between all workstations were also the shortest.
Algorithm 2.2 //Broadcast Scheduling Tree
Step 1. The Postorder List (PL) of workstations is obtained from the LST.
W = {w1, w2, w3,…, wn} be all workstations in PL, where n is the number of workstations in PL.
Step 2. Choose wi with the fastest speed in every switch to be the group head. Every group head groups PL.
Step 3. Every group has its Sub-tree Group size (SG) that is the number of workstations in its sub-tree group. Adjust all groups according to SG decreasing from left to right and obtain an adjusted list AL.
Step 4. Construct the First Level scheduling Tree (FLT) with all group heads using the binomial-like scheduling.
Step 5. For each group head Do
Construct the Second Level scheduling Tree (SLT) using the
binomial-like scheduling. Every group head is the root of SLT and every group of members form the tree nodes of SLT.
end for Step 6. For each wi in SLT, Do
While wi has children Do
a. Find a workstation wj that has no children and i≠j, in which the speed of wj is faster than the speed of wi, and both of wj and wi connect to the same switch.
b. Exchange wj and wi.
c. wj becomes the father of wi’s children.
end while end for
Step 1 and step 2 show the information from the LST for constructing a broadcast scheduling tree. In step 3, the postorder from the LST is adjusted and the adjusted list is used for constructing the scheduling tree. Step 4 constructs the first level scheduling tree according to all group heads. In step 5, every group constructs its second level scheduling tree according to its group members. Step 6 rearranges all fast workstations to send the messages to the slow ones.
Below is an example to demonstrate the LABS technique. Figure 2.8 gives a graph representation of the network topology. Every workstation has an id and the symbols represent the speed of the different workstations. The symbols representing the speed are rhomb, circle and hexagon; from fast to slow.
S
1S
2S
8S
7S
3S
5S
6S
41 3 2
4 5
6
7
8
9 10
11
12
13
14 15
16 17 18
19 20 21
22 23
24 25
Figure 2.8: Representational graph with node id.
Assume that workstation 9 connected to S4 is the source workstation. Using LABS, it constructs an LST for the source workstation in the first step. In Figure 2.9, the LST rooted by S4 is constructed. Moreover, all workstations in each switch are sorted in ascending order according to speed in Figure 2.9.
The postorder of workstations {9, 10, 11, 12, 7, 8, 1, 2, 3, 22, 23, 24, 25, 13, 14, 15, 4, 5, 6, 16, 17, 18, 19, 20, 21} is obtained from the LST. The fastest workstation of each switch is selected as the group head; they are 9, 7, 1, 22, 13, 4, 16 and 19. By checking the size of sub-trees, the three groups, {4, 5, 6}, {16, 17, 18} and {19, 20, 21} are adjusted in order to obtain a better scheduling tree. The adjusted spanning tree is shown in Figure 2.10.
S2 S6
9 11 12
7 8
1 2 3 13 14 15 19 20 21
16 18
4 5 6
22 23 source
10
17 S7
S4
S3
S1 S8
S5
24 25
Figure 2.9: LST rooted by S4.
9 11 12
7 8
1 2 3 13 14 15 19 20 21
16 18 4 5 6
22 23 24 25 source
10
17
S
4S
3S
6S
2S
1S
5S
7S
8Figure 2.10: Adjusted LST rooted by S4.
The gray rhombs in Figure 2.11 represent the first level of the scheduling tree. The group head constructs the binomial-like scheduling tree according to its group members as the second level of the scheduling tree. The faster workstations send the messages to the slower ones in the last step. Figure 2.12 shows complete scheduling tree that is constructed for the source workstation 9. It takes five steps to complete the broadcasting of the scheduling tree.
9
16 7
13
1 19
22
24 23
25
5 6
8 17 18
11
12 10
20 21
2 3
1 2 3
4 5
2 3 3
3 4
4
4 4 4
4
5
5
5 5 5
5
5
4
14 15
4 5
Figure 2.11: Completed scheduling tree rooted by workstation 9.
9
16 7
13
1 19
22
24 23
25
5 6
8 17 18 12
20 21
2 3
1 2 3
4 5
2 3 3
3 4
4
4 4 4
4
5
5
5 5 5
5
5
4
14 15
4 5
11 10
Figure 2.12: Completed scheduling tree after exchanging nodes 10 and 11.