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By Henri Casanova, Arnaud Legrand, Yves Robert

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1. The objective is to find a sorting network architecture that only depends on the length of the input sequence and that is independent of the values of the sequence. Therefore, the main difference between sorting networks and traditional comparison-based sorting algorithms is that the sequence of comparisons is set in advance, regardless of the outcome of previous comparisons. 1: A comparator. In this chapter, we present two sorting networks. 4. The second network uses an odd-even transposition scheme that can easily be mapped to a one-dimensional network of processors.

Lastly, let us look at the root of the tree. At step t = 7, it receives {8} and {4} and computes val(7) = MERGE WITH HELP({8}, {4}, ∅). At step t = 8, it receives {6, 8} and {2, 4} and computes val(8) = MERGE WITH HELP({6, 8}, {2, 4}, val(7)). At step t = 9, it receives {5, 6, 7, 8} and {1, 2, 3, 4} and computes val(9) = MERGE WITH HELP({5, 6, 7, 8}, {1, 2, 3, 4}, val(8)). On can easily check that val(t) is a GS of X(t + 1) and Y (t + 1) for t = 6, 7, 8. At the end of step t = 9, the root is complete and all values are sorted.

For any vertex j that is not a p-vertex, C(j) denotes the p-vertex of j. An invariant of the algorithm is that a p-vertex i is the smallest index of vertices labeled by i and that all these vertices belong to the same connected component. This holds true when initializing C(i) = i for all i ∈ V = {1, . . , n}. In other words, at the beginning for the algorithm each PU considers itself as the reference vertex of its component. The algorithm then iteratively changes this rather egocentric perspective.

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