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The Hopcroft-Karp algorithm goes like this. We first initialize the matching to the empty set, then we repeat the following. First, we build an alternating level graph rooted at the unmatched vertices on the left part of the partition using breadth-first search. Let’s pause for a moment here and see how this works in an

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Now we turn to the concept of a vertex cover, which will play a role analogous to the one played by the concept of minimum cut in our discussion of maximum flows. Given a graph, G, we say that S is a vertex cover if every edge is incident on a vertex in S. Thus

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in this lesson, we will see how to deduce and calculate running time of an algorithm and analyze the time complexity of it the actual time taken by an algorithm or what we also call the running time of the algorithm may depend upon a number of factors and let us see what those factors

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We’ve defined P as a set of languages, but ultimately we want to talk about it as a set of problems. Unfortunately, this isn’t as easy as it might seem. The encoding rules we use for turning an abstract problem into a string can effect whether or not the language is in P. Let’s see

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We turn now to the key part of the analysis where we show that each phase of the Dinic algorithm takes V times E time. As with Edmonds-Karp, we will use a level graph. In this case, however, the algorithm actually builds the graph, whereas in Edmonds-Karp we simply used it for the analysis. The

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the mode of research right now in theoretical computer science is you think of a problem and you come up with an algorithm that solve this problem this is Boaz Barak, a professor at Harvard University I think we are starting to see more general themes that kind of repeat themselves the same tool or

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This idea that we should prefer heavier flows brings us to the scaling algorithm. One idea is to find the heaviest possible flow. We could do this by starting with an empty graph and then adding an edge with the largest remaining residual capacity until there was an ST path. But this would be unnecessarily

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