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Algorithm 2 DFS(G) for each v V do if v not discovered then DFS-Visit(v) end if end for Algorithm 3 DFS-Visit(u) for each v adj(u) do if v not discovered then Mark v as discovered DFS-Visit(v) end if end for Mark u as nished adjacent vertices is O(E) (see also Section 311) For more details and an in-depth analysis of the properties of the two algorithms please refer to Cormen et al [42] Topological Order Now an important concept for directed acyclic graphs is considered the topological order of their vertices (Cormen et al [42]) Directed acyclic graphs build an essential class of graphs for task scheduling, because they are utilized for the representation of programs (Section 35) in scheduling algorithms The topological order is de ned as follows De nition 36 (Topological Order) A topological order of a directed acyclic graph G = (P, E) is a linear ordering of all its vertices such that if E contains an edge euv , then u appears before v in the ordering To illustrate De nition 36, consider the small directed acyclic graph in Figure 34 The topological order of the graph s vertices can be interpreted as a horizontal arrangement of the vertices (ie, a linear order), in such a way that all edges are directed from left to right This arrangement of the graph in Figure 34 is depicted in Figure 35 The acyclic property is crucial for the topological order; otherwise no such order exists Lemma 33 (Topological Order and Directed Graphs) A directed graph G = (P, E) is acyclic if and only if there exists a topological order of its vertices Proof : Suppose no topological order exists for G Thus, for any ordering of the vertices of G, there is at least one edge evu with u appearing before v in the list Consequently, there must be a path p(u v) from u to v; otherwise v and all of its ancestors that lie between u and v in the ordering could be inserted just before u,
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GRAPH REPRESENTATIONS
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Figure 34 A directed acyclic graph
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Figure 35 The directed acyclic graph of Figure 34 arranged in topological order; note that all directed edges go from left to right
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making the edge evu comply with the topology order without making a new edge violating it With the path p(u v), however, G is cyclic, since edge evu builds a cycle together with the path p(u v) : Suppose G is cyclic; hence, it has at least one simple cycle pc = v0 , v1 , , vk 1 , v0 Consider the distinct vertices v0 , v1 , , vk 1 of pc In a topological order, vertex vi must appear before vi+1 , 0 i < k 1, imposed by the edge ei,i+1 pc That implies v0 comes before vk 1 , but then edge ek 1,0 does not comply with the condition of the topological order Consequently, no topology order exists for the vertices of pc Since no topology order can be found for the vertices of pc , no topology order exists for G The vertices of a directed acyclic graph can be sorted into topological order by a simple DFS-based algorithm, which is outlined in Algorithm 4 (Cormen et al [42]) As soon as a vertex is marked nished in the DFS (see DFS-Visit, Algorithm 3), it is inserted onto the front of a list and upon termination of DFS, the list holds the topologically ordered vertices Algorithm 4 Topological-Sort(G) Execute DFS(G) with the following addition: Insert each vertex of G onto the front of a list L as soon as it is marked nished Return L The correctness of Algorithm 4 can be veri ed by the following considerations It is ensured by the main part of DFS (Algorithm 2) that every vertex is discovered; therefore, every vertex is eventually marked as nished and inserted onto the front of
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