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with the end technique the complexity is O(V + EO(routing)) (instead of O(P(V + EO(routing)))) and with the insertion technique it is O(V2 + E2 O(routing)) (instead of O(V2 + PE2 O(routing))) Remember, the third alternative of edge scheduling, where the leaving edges of a node are scheduled on the processors and links, is only meaningful with the insertion technique (Section 832) 843 Experimental Results As for the contention model (Section 754), the question arises as to how much the accuracy of scheduling and the execution times of the produced schedules bene t from the involvement contention model As argued in Section 754, only an experimental evaluation on real parallel systems can help answer this question Such an experimental evaluation is performed in Sinnen [172] and Sinnen et al [180] The employed methodology is based on the one described in Section 754: a large set of graphs is generated and scheduled by algorithms under the different models to several target systems Because the focus is on the comparison of the different scheduling models, the same list scheduling algorithm is employed under each model From the produced schedules, code is generated using C and MPI and executed on the real parallel systems Modeling of the target machines as topology graphs is relatively simple, as discussed in 7 For the involvement contention model it is additionally necessary to specify for each target system the overhead and involvement of the processors in communication Due to the lack of a deep insight into the target systems communication mechanisms and their MPI implementations, 100% involvement is assumed, that is, the source and destination processors are involved during the entire communication time on the rst and last link, respectively: is (e, L1 ) = (e, L1 ) and ir (e, Ll ) = (e, Ll ) (820)
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The overhead is intuitively set to an experimentally measured setup time: os (e, P) = or (e, P) = setup_time (821)
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While it is clear that this de nition of the overhead and the involvement is probably not an accurate description of the target systems communication behavior, it is very simple The idea is to demonstrate that accuracy and ef ciency of scheduling can be improved even with a rough but simple estimate of the overhead and involvement functions Results The experiments demonstrated that the involvement contention model achieves profoundly more accurate schedules and signi cantly shorter execution times Thus, considering processor involvement in communication further improves the already improved accuracy under the contention model The length of a schedule is now in a region where it can be seriously considered an estimate of the real execution
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PROCESSOR INVOLVEMENT IN COMMUNICATION
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time Under the classic model and for some topology graphs under the contention model, the schedule lengths are only small fractions of the real execution times, in particular, for medium (CCR = 1) and high communication (CCR = 10), which can hardly be considered execution time estimations For example, on the Cray T3E-900 (Section 754), the estimation error under the involvement contention model was on average below 20%, while the estimation error under the classic model goes up to 1850% Still, the scheduling accuracy under the involvement contention model is not perfect, especially for low communication (CCR = 01) A possible explanation might be the blocking communication mechanisms used in MPI implementations (White and Bova [199]), which does not match the assumption of nonblocking communication made in the involvement contention model Furthermore, the employed overhead and involvement functions are very rough estimates; a better approximation of these functions has the potential to further increase the accuracy In any case, it is in the nature of any model that there is a difference between prediction and reality The profoundly improved accuracy under the involvement contention model allows more than just the reduction of execution times: it also allows one to evaluate algorithms and their schedules without execution of the schedules Hence, new algorithms can be developed and evaluated without the large efforts connected with an evaluation on real parallel systems In the experiments conducted, the involvement contention model also clearly demonstrated its ability to produce schedules with signi cantly reduced execution times The bene t of the high accuracy is apparent in the signi cantly improved execution times with speedup improvements of up to 82% Despite the very good results, the ef ciency improvement lags behind the accuracy improvementA possible explanation lies in the heuristic employed in the experiments List scheduling is a greedy algorithm, which tries to reduce the nish time of each node to be scheduled Thus, it does not consider the leaving communications of a node, which may impede an early start of following nodes The high importance of communication under the involvement contention model seems to demand research of more sophisticated algorithms in order to exploit the full potential of this new model
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85 CONCLUDING REMARKS This chapter investigated processor involvement in communication and its integration into task scheduling The motivation originated from the unsatisfying accuracy results obtained under the classic and the contention model as referenced in Section 754, which indicated a shortcoming of these scheduling models Thus, the chapter began by thoroughly analyzing the various types of processor participation in communication and their characteristics Based on this analysis, another scheduling model was introduced that abandons the idealizing assumption of a dedicated communication subsystem and instead integrates the modeling of all
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