n p,V

Decode QR Code In VS .NETUsing Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.

w(n) +

Drawing Quick Response Code In .NET FrameworkUsing Barcode encoder for VS .NET Control to generate, create QR image in .NET framework applications.

e p,E

Reading QR Code JIS X 0510 In Visual Studio .NETUsing Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications.

c(e)

Barcode Generation In .NETUsing Barcode printer for .NET framework Control to generate, create barcode image in VS .NET applications.

(420)

Recognizing Barcode In Visual Studio .NETUsing Barcode reader for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.

The computation length of a path p in G is the sum of the weights of its nodes: lenw ( p) =

QR Code JIS X 0510 Encoder In C#.NETUsing Barcode drawer for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET framework applications.

n p,V

Generate QR Code 2d Barcode In Visual Studio .NETUsing Barcode creator for ASP.NET Control to generate, create QR-Code image in ASP.NET applications.

w(n)

Printing QR In VB.NETUsing Barcode creation for .NET framework Control to generate, create QR Code JIS X 0510 image in .NET framework applications.

(421)

Making Data Matrix In Visual Studio .NETUsing Barcode drawer for Visual Studio .NET Control to generate, create Data Matrix 2d barcode image in Visual Studio .NET applications.

Note that the de nition of the path length in a task graph differs from the general De nition 32 in Section 31, where the length of a path equals the number of edges The path length len( p) can be interpreted as the time the path p takes for its execution if all communications between its nodes are interprocessor communications, which happens, for instance, when each node of p is allocated to a different processor Due to the sequential order inherent in the path, none of the nodes is executed concurrently with any other node The computation path length lenw ( p) can be interpreted as the execution time of the path p when all communications between its nodes are local, that is, they have zero costs Consequently, all nodes of p are executed on the same processor In a task graph G = (V, E, w) without communication costs, the path length is necessarily de ned as it is in Eq (421) When the processor allocations of the nodes of p are known, the path length can be determined taking into account that local communications are cost free In a scheduling algorithm it might be desirable to calculate a path length based on a partial schedule: that is, some processor allocations are given and others are not The path length determined for a given (partial) processor allocation A (De nition 41)

Generate GTIN - 12 In .NET FrameworkUsing Barcode creation for .NET Control to generate, create UPC A image in .NET framework applications.

44 TASK GRAPH PROPERTIES

Encoding Code 128 In Visual Studio .NETUsing Barcode creator for Visual Studio .NET Control to generate, create Code 128C image in Visual Studio .NET applications.

is denoted by len( p, A), the allocated path length Communications between nodes whose processor allocations are unknown are assumed to be remote The allocated path length len( p, A) is thus in between the path length len( p) and the computation path length lenw ( p): len( p) len( p, A) lenw ( p) (422)

Drawing NW-7 In Visual Studio .NETUsing Barcode generation for .NET Control to generate, create Ames code image in Visual Studio .NET applications.

The above scheme for the distinction between the different path lengths is used throughout this text All de nitions based on path lengths will only be formulated for len( p) but are implicitly valid for len( p, A) and lenw (p), too The corresponding de nitions use the same scheme for distinction: that is, the allocated path length is parameterized with A and the subscript w is used for the computation length 441 Critical Path An important concept for scheduling is the critical path the longest path in the task graph De nition 418 (Critical Path (CP)) A critical path cp of a task graph G = (V, E, w, c) is a longest path in G len(cp) = max{len( p)}

Scanning Bar Code In .NETUsing Barcode decoder for .NET framework Control to read, scan read, scan image in VS .NET applications.

(423)

Generating Bar Code In .NET FrameworkUsing Barcode creation for ASP.NET Control to generate, create bar code image in ASP.NET applications.

The computation critical path cpw and the allocated critical path cp(A) for a processor allocation A are de ned correspondingly The nodes of a critical path cp, consisting of l nodes, are denoted by ncp,1 , ncp,2 , , ncp,l Clearly, there might be more than one critical path as several paths can have the same maximum length Note that in general cp = cpw = cp(A), from which follows for their path lengths len(cpw ) len(cp) and len(cp(A)) len(cp) (425) (424)

Code-39 Maker In JavaUsing Barcode printer for Java Control to generate, create Code 39 Full ASCII image in Java applications.

Equivalent inequalities hold for the computation path length and the allocated path length and the corresponding critical paths Lemma 43 (Critical Path: From Source to Sink) Let G = (V, E, w, c) be a task graph A critical path cp of G always starts in a source node and nishes in a sink node of G

Barcode Drawer In Visual C#.NETUsing Barcode creation for .NET Control to generate, create barcode image in Visual Studio .NET applications.

TASK SCHEDULING

Make Barcode In VB.NETUsing Barcode creation for Visual Studio .NET Control to generate, create barcode image in .NET framework applications.

Proof By contradiction: Suppose cp does not start in a source node Then, per de nition, the rst node of cp, here denoted by n1 , has at least one predecessor n0 pred(n1 ); hence, there is the edge e01 E A new path q can be concatenated from n0 , e01 , and cp, whose length is len(q) = w(n0 ) + c(e01 ) + len(cp) As w(n0 ) > 0, it follows that len(q) > len(cp) a contradiction Likewise for the sink node The critical path gains its importance for scheduling from the fact that its length is a lower bound for the schedule length Lemma 44 (Critical Path Bound on Schedule Length) Let G = (V, E, w, c) be a task graph and cpw a computation critical path of G For any schedule S of G on any system P, sl lenw (cpw ) (426) Proof Due to their precedence constraints (Condition 42), the nodes of cpw can only be executed in sequential order, which takes lenw (cpw ) time, independently of the schedule or the number of processors Thus, the duration of G s execution is at least lenw (cpw ) In the worst case that all communications among nodes are remote, sl len(cp), (427)

UPC - 13 Creator In JavaUsing Barcode maker for Java Control to generate, create EAN13 image in Java applications.

yet Eq (426) is also ful lled For the special case of scheduling without communication costs on an unlimited number of processors (Section 432), the lower bound of the schedule length established by Eq (426) is tight In other words, the length of the optimal schedule is the length of the critical path Lemma 45 (Optimal sl on Unlimited Processors Cost-Free Communication) Let G = (V, E, w) be a task graph, cpw a computation critical path of G, and Pc0, a parallel system, with |Pc0, | |V| For an optimal length schedule Sopt of G on system Pc0, , sl(Sopt ) = lenw (cpw ) (428) Proof Theorem 44 establishes that Algorithm 6 produces an optimal schedule Sopt of G on Pc0, In this algorithm, the start time ts (n) of each node n is set to its DRT ts (n) = tdr (n) n V (429) By De nition 416, node n s DRT is the maximum of the nish times of its predecessor nodes (Eq (417)) Together with tf (n) = ts (n) + w(n) (Eq (42) of De nition 45), it holds that tdr (n) = max {tdr (ni } + w(ni )} (430)

Code 128A Encoder In JavaUsing Barcode creation for Java Control to generate, create Code 128C image in Java applications.

ni pred(n)

UPC-A Supplement 2 Recognizer In .NETUsing Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications.