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graph The indegree is computed for each vertex Vertex V, has indegree 0, so it is first in the topological order If there were several vertices of indegree 0, we could choose any one of them When V 2 and its edges are removed from the graph, the indegrees of V,,, V,, and V5 are all decremented by 1 Now V has indegree 0, so it is next in the topological order, and V, and V, , have their indegrees lowered The algorithm continues, and the remaining vertices are examined in the order V1, V3, V4 V6,and Vs To reiterate, we do not physically delete edges from the graph; removing edges just makes it easier to see how the indegree count is lowered Two important issues to consider are correctness and eficiencj Clearly, any ordering produced by the algorithm is a topological order The question is whether every acyclic graph has a topological order, and if so, whether our algorithm is guaranteed to find one The answer is yes to both questions If at any point there are unseen vertices but none of them have an indegree of 0, we are guaranteed that a cycle exists To illustrate we can pick any vertex A,, Because A has an incoming edge, let A, be the vertex connected to it And as A , has an incoming edge, let A , be the vertex connected to it We repeat this process N times, where N is the number of unprocessed vertices left in the graph Among Ao, A ,, ,A,, there must be two identical vertices (because there are N vertices but N + 1 A,'s) Tracing backward between those identical A, and A, exhibits a cycle We can implement the algorithm in linear time by placing all unprocessed indegree 0 vertices on a queue Initially, all vertices of indegree 0 are placed on the queue To find the next vertex in the topological order, we merely get and remove the front item from the queue When a vertex has its indegree lowered to 0, it is placed on the queue If the queue empties before all the vertices have been topologically sorted, the graph has a cycle The running time is clearly linear, by the same reasoning used in the unweighted shortest-path algorithm.



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This code first retrieves the customer data into a data set, as shown earlier in the appendix If you were really retrieving data solely for the purpose of manipulating it as XML, you would be better off using the XML query capability of SQL Server, returning the results as XML, and stuffing the results into an XPathDocument Once that data set is populated with the appropriate schema and the data, that is passed to the constructor of the XmlDataDocument An XPath query is performed against the XML document using the SelectNodes method, and this method returns a list of XML nodes that then iterate through for constructing a simple display of the customer names An important feature to realize about an XmlDataDocument is that it can load and hold a lot of XML content that isn't exposed through the DataSet property, and that additional content can be accessed through the normal document object model of the base XmlDocument class So you could load a document containing a set of data that matches a data set schema that you want to expose for data binding, and the document could also contain a bunch of other nodes that don't match the schema The parts of the XML document that match the data set schema would be added to the data set, but the parts of the XML document that don't match would simply be contained within the XmlDocument base class' object model and could be accessed through normal XML navigation of the document So if you need to jump back and forth between dealing with a set of data as relational and dealing with it as an XML object hierarchy, the XmlDataDocument is one of the first places to look





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An important application of topological sorting is its use in solving the shortest-path problem for acyclic graphs The idea is to have the eyeball visit vertices in topological order This idea works because, when the eyeball visits vertex v, we are guaranteed that D,, can no longer be lowered; by the topological ordering rule, it has no incoming edges emanating from unvisited nodes Figure 1532 shows the stages of the shortest-path algorithm, using topological ordering to guide the vertex visitations Note that the sequence of vertices visited is not the

6, 7

Figure 1532 The stages of acyclic graph algorithmThe conventions are the same as those in Figure 1522

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same as in Dijkstra's algorithm Also note that vertices visited by the eyeball prior to its reaching the starting vertex are unreachable from the starting vertex and have no influence on the distances of any vertex We do not need a priority queue Instead, we need only to incorporate the topological sort into the shortest-path computation Thus we find that the algorithm runs in linear time and works even with negative edge weights

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The implementation combines a topological sort calculation and a shortest-path calculationThe indegree information is stored in the scratch data member

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