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By Ingold L.

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For the moment, let’s not worry about how many terms we use C = I ∨ A ∨ A2 ∨ A3 ∨ A4 ∨ · · · Ak has the property that Ak (i, j) > 0 if and only if there is a path from node i to node j in exactly k steps. If there are m ways to go from node i to node j in exactly k steps, then Ak (i, j) = m. Ak (i, j) = 0 when there is no such path with exactly k steps. The resulting matrix C has a nonzero in position i, j if and only if there are paths from node i to node j in any number of steps—including 0 steps, because we included I in the series.

Bellman 1952] R. Bellman. On the theory of dynamic programming. Proceedings of the National Academy of Sciences, 38:716–719, 1952. 245. php Chapter 4 Some Graph Algorithms in an Array-Based Language Viral B. Shah∗, John Gilbert†, and Steve Reinhardt‡ Abstract This chapter describes some of the foundations of linear algebraic graph algorithms and presents a number of classic graph algorithms using Matlab style syntax. These algorithms are implicitly parallel, provided the underlying parallel matrix operations are supported in the array-based language.

1), we saved only the minimum A(i, k) + B(k, j) and did not record the k which achieved it. + calculation B = A min . + B, we simply maintain a second matrix D whose i, j element is the k for which the minimum was achieved. There may sometimes be ties, but if a tie is arbitrarily broken, then one of the minimum cost paths will be identified. Now we return to prove the claim that for any nonnegative p and q, A (p+q) = p A min . + A q . But this is simply another statement of Bellman’s principle. For any k, R = A p has the optimum p-step path cost R(i, k) between node i and node k.

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Associated Types of Linear Connection by Ingold L.

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