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Rolph, J E (1976) Choosing shrinkage estimators for regression problems Communications in Statistics A, vol 5, pp 789-802 Rousseeuw, P J, and A M Leroy (1987) Robust Regression and Outlier Detection Wiley, New York Seber, G A F (1977) Linear Regression Analysis Wiley, New York Seber, G A F, and C J Wild (1989) Nonlinear Regression Wiley, New York Weisberg, S (1985) Applied Linear Regression, 2nd ed Wiley, New York Yohai, V J, and R H Zamar (1988) High breakdown-point estimates of regression by means of the minimization of an efficient scale Journal of the American Statistical Association, vol 83, pp 406-413..

In a simple algorithm for performing a topological sort we first find any vertex v that has no incoming edges Then we print the vertex and logically remove it, along with its edges, from the graph Finally, we apply the same strategy to the rest of the graph More formally, we say that the indegree of a vertex v is the number of incoming edges (u, v) We compute the indegrees of all vertices in the graph In practice, logica11y remove means that we lower the count of incoming edges for each vertex adjacent to v Figure 1531 shows the algorithm applied to an acyclic

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The indegreeof a vertex is the number of incoming edges A topological sort can be performed in linear time by repeatedly and logically removing vertices that have no incoming edges

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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.

The table contains two-tailed values ttab such that Prob[ltl 2 ttab] = a where t is a random variable having a t distribution with IJ degrees of freedom.

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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

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

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