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Weighted matroid : ウィキペディア英語版 | Weighted matroid In combinatorics, a branch of mathematics, a weighted matroid is a matroid endowed with function with respect to which one can perform a greedy algorithm. A ''weight function'' ''w'' : ''E'' → R+ for a matroid ''M''=(''E'', ''I'') assigns a strictly positive weight to each element of ''E''. We extend the function to subsets of ''E'' by summation; ''w''(''A'') is the sum of ''w''(''x'') over ''x'' in ''A''. A matroid with an associated weight function is called a weighted matroid. ==Spanning forest algorithms== As a simple example, say we wish to find the maximum spanning forest of a graph. That is, given a graph and a weight for each edge, find a forest containing every vertex and maximizing the total weight of the edges in the tree. This problem arises in some clustering applications. If we look at the definition of the forest matroid above, we see that the maximum spanning forest is simply the independent set with largest total weight — such a set must span the graph, for otherwise we can add edges without creating cycles. But how do we find it?
抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Weighted matroid」の詳細全文を読む
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