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are discussed in the following Roughly, contention can be divided into end-point and network contention
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711 End-Point Contention End-point contention refers to the contention in the interfaces that connect the processors with the communication network Only a limited number of communications can pass from the processor into the network and from the network into the processor at one instance in time Beaumont et al [19] extend the classic model by associating a communication port with each processor, called one-port model At any instance in time, only one incoming and one outgoing communication can be carried out through this port For example, multiple leaving communications of a node, which are ready as soon as the task nishes execution (node strictness, De nition 38), are serialized through this approach A port can be regarded as the processor s network interface, allowing bidirectional full duplex transfers Communication between disjoint pairs of processors can still be performed concurrently Thus, the one-port model captures end-point contention but does not re ect network contention A similar, but more general, approach is accomplished with the parameter g of the LogP model (Section 213) A few scheduling algorithms have been proposed for the LogP model (eg, Boeres and Rebello [25], Kalinowski et al [98], L we et al [131]) In these algorithms, the number of concurrent communications that can leave or enter a processor is limited, since the time interval between two consecutive communications must be at least g (see Section 213) This approach is more general, because the gap can be set to a value smaller than o + L (overhead + latency), thus allowing several communications to overlap All of the above referenced algorithms neglect the network capacity de ned in LogP of at most L/g simultaneous message transfers in the network End-point contention is only a partial aspect of contention in communication The limited number of resources within the network (Section 22) also leads to con icts Furthermore, both of the above approaches suppose completely homogeneous systems and are tied to systems where each processor has one full duplex communication port
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712 Network Contention To successfully handle communication contention in scheduling, an accurate model of the network topology is required Static networks, that is, networks with xed connections between the units of the system, are commonly represented as undirected graphs This representation form was informally used during the description of static networks in Section 221 In the following it is de ned formally De nition 71 (Undirected Topology Graph) The topology of a static communication network is modeled as an undirected graph UTG = (P, L) The vertices in P represent the processors and the undirected edges in L the communication links
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between them Link Lij L represents the bidirectional communication link between the processors Pi and Pj and Pi , Pj P Examples of such undirected topology graphs are omitted here, but a large number of them can be found in Section 221 The undirected graph model is a simple and intuitive representation of communication networks In task scheduling it is employed for various purposes Wu and Gajski [207] apply it to map scheduled task graphs onto physical processors, trying to balance the network traf c Other algorithms utilize it for a more accurate determination of communication costs, for example, by counting the hops of a communication (ie, the transitions between links) (eg, Coli and Palazzari [39], Sandnes and Megson [164]) Contention Awareness with Undirected Topology Graph Primarily, the undirected topology graph is the model employed in the few existing network contention aware scheduling heuristics El-Rewini and Lewis [63] propose a contention aware scheduling algorithm called MH (mapping heuristic) The target system is represented as an undirected topology graph and a table is associated with each processor This table maintains for each processor (1) the number of hops on the route, (2) the preferred outgoing link, and (3) the communication delay due to contention With this information, the communication route between processors is adapted during the execution of the list scheduling based MH However, the tables are only updated at each start and at each nish of a communication, which is a trade-off between complexity and a real view of the traf c Furthermore, such a scheduling approach is only meaningful if the target system can be forced to use the routes determined by the algorithm, which is usually not the case (Section 722) Concept of Edge Scheduling A more realistic view of the network traf c is gained when, for each communication, its start and nish times on every utilized communication link are recorded This corresponds to scheduling the edges on the links, like the nodes are scheduled on the processors, which is illustrated in the Gantt chart of Figure 71 The idea of edge scheduling emerged with the DLS algorithm proposed by Sih and Lee [169] They proposed representing the target system as an undirected topology
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