GRAPH REPRESENTATIONS

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Section 351 At the end, the motivations for the task graph model were presented and its limitations were analyzed The chapter s focus on the task graph resulted in the omission of several other, less common, graph models, some of which are analyzed by Sinnen and Sousa [173]:

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Task Interaction Graph (TIG) (Stone [182]) An undirected graph model only capturing the communication relations between entire processes as opposed to tasks The TIG is used for mapping of processes onto parallel processors (eg, Sadayappan et al [163], Stone [182]) Temporal Communication Graph (TCG) (Lo [129]) A model that integrates the task and process oriented view of a parallel program The TCG is based on the space time diagram introduced by Lamport [116] and can be interpreted for mapping and scheduling algorithms as a TIG as well as a task graph (Lo et al [130]) Control Flow Graph (CFG) and Control Dependence Graph (CDG) (Allen and Kennedy [12], Wolfe [204]) These two closely related directed graph models represent the control ow and control dependence of a program As a consequence, they re ect conditional execution paths (branching, see Section 32), as opposed to all other graph models discussed in this chapter Both are used in compilers to analyze and handle the control ow of a program

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For the fundamental problem of integrating iterative and noniterative computations into one ef cient graph model, the idea of a hierarchical graph (Sinnen and Sousa [175]) emerged, where a node of a higher level can represent an entire subgraph With the end of this chapter, the foundation for the discussion of task scheduling in the next chapter has been laid

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37 EXERCISES 31 Section 31 reviews basic graph concepts for undirected and directed graphs Consider the following directed graph G = (V, E) consisting of 6 vertices and 9 edges:

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(a) Which vertex has the highest degree (b) What is the average in-degree of the vertices What is the average outdegree

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EXERCISES

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(c) Is vertex y an ancestor or a descendant of vertex x (d) Is vertex v reachable from vertex w Is vertex w reachable from vertex v 32 Paths and cycles are important and powerful concepts in graphs (Section 31) Consider again the directed graph of Exercise 31 (a) How many simple cycles has the graph Specify them (b) What is the length of the longest simple path, in terms of the number of edges, in this graph Specify a path with such a length Commonly, there are two different approaches to the representation of a graph in a computer (Section 311) Consider again the directed graph of Exercise 31 (a) Give an adjacency matrix representation of this graph (b) Give an adjacency list representation of this graph

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34 The topological order is an important concept for directed acyclic graphs (Section 312) Find a topological order for the following directed acyclic graph G = (V, E) consisting of 10 vertices:

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Which vertices are source (entry) vertices and which are sink (exit) vertices 35 36 37 Construct the dependence graph for the code fragment of Exercise 29 Construct the iteration dependence graph for the double loop of Exercise 211 Construct the ow graph (Section 34) for the following iterative computation: for i = 1 to n do S: A(i) = B(i) - 1 T: B(i+1) = (A(i-1)-A(i))/2 U: C(2 i) = B(i-1) + E(i+2) V: D(i) = C(2 i-2) + B(i+1) end for Construct a task graph (Section 35) for the code below Each line shall be represented by one task, named by its line number, and the costs shall be assumed as follows: Computation Assignment alone: 1 unit; add/subtract operation: 2 units; multiply operation: 3 units; divide operation: 4 units Communication Communicating a variable with a small letter and with a capital letter costs 1 unit and 2 units, respectively (imagine variables with capital letters to have higher precision)

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