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algorithms

A collection of algorithms and data structures

booth_multipliers

Parameterized Booth Multiplier in Verilog 2001

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hopcroft--karp-algorithm-in-java

A vertex that is not the endpoint of an edge in some partial matching M M is called a free vertex. The basic concept that the algorithm relies on is that of an augmenting path, a path that starts at a free vertex, ends at a free vertex, and alternates between unmatched and matched edges within the path. It follows from this definition that, except for the endpoints, all other vertices (if any) in augmenting path must be non-free vertices. An augmenting path could consist of only two vertices (both free) and single unmatched edge between them. If M M is a matching, and P P is an augmenting path relative to M M, then the symmetric difference of the two sets of edges, M ⊕ P M \oplus P, would form a matching with size | M | + 1 |M| + 1. Thus, by finding augmenting paths, an algorithm may increase the size of the matching. Conversely, suppose that a matching M M is not optimal, and let P P be the symmetric difference M ⊕ M ∗ M \oplus M^* where M ∗ M^* is an optimal matching. Because M M and M ∗ M^* are both matchings, every vertex has degree at most 2 in P P. So P P must form a collection of disjoint cycles, of paths with an equal number of matched and unmatched edges in M M, of augmenting paths for M M, and of augmenting paths for M ∗ M^*; but the latter is impossible because M ∗ M^* is optimal. Now, the cycles and the paths with equal numbers of matched and unmatched vertices do not contribute to the difference in size between M M and M ∗ M^*, so this difference is equal to the number of augmenting paths for M M in P P. Thus, whenever there exists a matching M ∗ M^* larger than the current matching M M, there must also exist an augmenting path. If no augmenting path can be found, an algorithm may safely terminate, since in this case M M must be optimal. An augmenting path in a matching problem is closely related to the augmenting paths arising in maximum flow problems, paths along which one may increase the amount of flow between the terminals of the flow. It is possible to transform the bipartite matching problem into a maximum flow instance, such that the alternating paths of the matching problem become augmenting paths of the flow problem.[3] In fact, a generalization of the technique used in Hopcroft–Karp algorithm to arbitrary flow networks is known as Dinic's algorithm.

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