**6.3 Applicabiligy of** *δopt*(*φ*0)

16 Will-be-set-by-IN-TECH

A. At first, the lower bound of the cluster size is derived as *δopt*(*φ*0). Then the task clustering algorithm in fig. 4 is performed, while processor assignment policy is based on CHP(C.

B. The lower bound of the cluster size is derived as *δopt*(*φ*0). Then the task clustering policy is based on "load balancing" (J. C. Liou, 1997), while processor assignment policy is based on CHP(C. Boeres, 2004), in which merging step for generating one cluster is proceeded

C. The lower bound of the cluster size is derived as *δopt*(*φ*0). Then the task clustering policy is random-basis, i.e., two clusters smaller than *δopt*(*φ*0) are selected randomly to merge into one larger cluster, while processor assignment policy is based on CHP(C. Boeres, 2004), in which merging step for generating one cluster is proceeded until the cluster size exceeds

The difference between A, B and C is how to merge clusters, while they have the common lower bound for the cluster size and the common processor assignment policy. We compared

former is the result in the case of random DAGs. On the other hand, the latter is the result in the case of FFT DAGs. In both tables, *α* corresponds to max-min ratio for processing speed in *P*, and *β* corresponds to max-min ratio for communication bandwidth in *P*. "*slw*(*G<sup>R</sup>*

degree of heterogeneity (i.e., *α* and *β*). The same results hold to table 4. From those results,

SL Ratio

*cls*, *φR*) Ratio" correspond to ratios to "A", i.e., a value larger than 1 means

*cls*, *<sup>φ</sup>R*) and *sl*(*G<sup>R</sup>*

 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

*cls*, *φR*) is larger than that of "A". In table 3, it can be seen that both

*cls*, *<sup>φ</sup>R*) and *sl*(*G<sup>R</sup>*

*cls*, *φR*) leads to minimizing the schedule length as

0 2 4 6 8 10

CCR

(b) (*α*, *β*)=(10, 10)

*cls*, *φR*) in "A" are better than "B" and "C" as a whole. Especially, the

*cls*, *φR*) and the schedule length. The

*cls*, *φR*) in "A" become. It can not be

*cls*, *φR*) with varying the

 0.25 0.5 1.5 2.0

opt 0 δ φ( ) opt 0 δ φ( ) opt 0 δ φ( ) opt 0 δ φ( ) opt 0 δ φ( ) *cls*, *φR*)

*cls*, *φR*) and the schedule length by averaging them in 100 random DAGs.

Boeres, 2004).

*δopt*(*φ*0).

Ratio" and "*sl*(*G<sup>R</sup>*

*cls*, *<sup>φ</sup>R*) or *sl*(*G<sup>R</sup>*

larger CCR becomes, the better both *slw*(*G<sup>R</sup>*

it can be concluded that minimizing *slw*(*G<sup>R</sup>*

theoretically proved by theorem 4.1 and 4.2.

seen that noteworthy characteristics related to *slw*(*G<sup>R</sup>*

 0.25 0.5 1.5 2.0

opt 0 δ φ( ) opt 0 δ φ( ) opt 0 δ φ( ) opt 0 δ φ( ) opt 0 δ φ( )

0 2 4 6 8 10

CCR

Fig. 5. Optimality for the Lower Bound of the Cluster Size.

(a) (*α*, *β*)=(5, 5)

*cls*, *<sup>φ</sup>R*) and *sl*(*G<sup>R</sup>*

that *slw*(*G<sup>R</sup>*

 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

SL Ratio

*slw*(*G<sup>R</sup>*

*slw*(*G<sup>R</sup>*

until the cluster size exceeds *δopt*(*φ*0).

Table 3 and 4 show comparison results in terms of *slw*(*G<sup>R</sup>*

In this experiment, we confirmed that how optimal the lower bound of the cluster size, *δopt*(*φ*0) derived by eq. (25). Comparison targets in this experiment are based on "A" at sec. 6.2, but only the lower bound of the cluster size is changed, i.e., *δopt*(*φ*0), 0.2*δopt*(*φ*0), 0.5*δopt*(*φ*0), 1.5*δopt*(*φ*0), and 2.0*δopt*(*φ*0). The objective of this experiment is to confirm the range of applicability of *δopt*(*φ*0), due to the fact that *δopt*(*φ*0) is not a value when *slw*(*G<sup>s</sup> cls*, *φs*) can be minimized for 1 ≤ *s*. Fig. 5 shows comparison results in terms of the optimality of *δopt*(*φ*0). (a) corresponds to the case of the degree of heterogeneity (*α*, *β*)=(5, 5), and (b) corresponds to (10, 10). From (a), it can be seen that *δopt*(*φ*0) takes the best schedule length than other cases during CCR takea from 0.1 to 5.0. However, when CCR is 7 or more, 1.5*δopt*(*φ*0) takes the best schedule length. This is because *δopt*(*φ*0) may be too small for a data intensive DAG. Thus, it can be said that 1.5*δopt*(*φ*0) is more appropriate size than *δopt*(*φ*0) when CCR exceeds a certain value. On the other hand, in (b), the larger CCR becomes, the better the schedule length by case of 1.5*δopt*(*φ*0) becomes. However, during CCR is less than 3.0, *δopt*(*φ*0) can be the best lower bound of the cluster size. As for other lower bounds, 2.0 *δopt*(*φ*0) has the local maximum value of the schedule length ratio when CCR takes from 0.1 to 2.0 in both figures. Then in larger CCR, the schedule length ratio decreases because such size becomes more appropriate for a data intensive DAG. On the other hand, in the case of 0.25*δopt*(*φ*0), the schedule length ratio increases with CCR. This means that 0.25*δopt*(*φ*0) becomes smaller for a data intensive DAG with CCR increases.

From those results, it can be said that the lower bound for the cluster size should be derived according to the mapping state. For example, if the lower bound can be adjusted as a function of each assigned processor's ability (e.g., the processing speed and the communication bandwidth), the better schedule length may be obtained. For example in this chapter the lower bound is derived by using the mapping state of *φ*0. Thowever, by using the other mapping state, we may be obtain the better schedule length. To do this, it must be considered that which mapping state has good effect on the schedule length. This point of view is an issue in the future works.
