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Algorithm A linear clustering can be achieved with a imple and intuitive algorithm proposed by Kim and Browne [104] Consider as a basis the generic clustering Algorithm 15: in each re nement step a new longest path p, consisting of previously unexamined edges, is selected in task graph G The nodes of p are merged into one cluster and the edges incident on these nodes are marked examined This is repeated until all edges of G have been examined Clearly, such an algorithm belongs to the path type in terms of the multiplicity of edge zeroing Algorithm 16 outlines the above described procedure Eunex E is the set of unexamined edges and Gunex = (Vunex , Eunex ) is the task graph spawned by these edges and the nodes Vunex V, which are incident on these edges Hence, each Gunex is a subgraph of G In each step the critical path cp of the current unexamined graph Gunex is determined and its edges are merged into one cluster Observe that all edges incident on the nodes of cp, {eij Eunex : ni Vcp nj Vcp }, are marked as examined at the end of each step, and not only those edges eij Ecp that are part of cp This is important, because otherwise, in a later step, one of the edges eij Ecp / not being part of cp could be zeroed This would mean that the edge s two incident nodes ni and nj , of which at least one is part of a cluster with more than one node, are merged into the same cluster, potentially resulting in a nonlinear cluster Algorithm 16 Linear Clustering Algorithm (G = (V, E, w, c)) Create initial clustering C0 : allocate each node n V to a distinct cluster C C, |C| = |V| Eunex E Eunex : set of unexamined edges, Eunex E while Eunex = do there is at least one unexamined edge Vunex : set of nodes Find a critical path cp of graph Gunex = (Vunex , Eunex ) n V on which edges Eunex are incident Merge all nodes Vcp of cp into one cluster Mark edges that are Eunex Eunex {eij Eunex : ni Vcp nj Vcp } incident on nodes of Vcp as examined end while In comparison to the general structure of a clustering algorithm, as outlined in Algorithm 15, there is no check of whether the schedule length of the new clustering is not worse than the current one The check is unnecessary, as Lemma 52 guarantees that this is the case Finding the critical path of the task graph Gunex in each step is essentially the determination of the nodes top and bottom levels This can be performed using the algorithms presented in Section 442, with a complexity of O(V + E) Merging the nodes and marking the corresponding nodes and edges as examined has lower complexity than O(V + E) As there are at most O(V) steps, Algorithm 16 s total complexity is thus O(V(V + E)) As a last comment on linear clustering, note the similarity between linear projections of dependence graphs as analyzed in Section 351 and linear clustering The linear projection of DGs for example, is used in VLSI array processor design (Kung [109]). asprise ocr c# example A9T9/Free-OCR-API-CSharp: Web API test app for the OCR ... - GitHub
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