Sub-niche tree-forming algorithm

For a undirected band-edge connected graph G (v,e), we can certainly extract the smallest spanning tree, then how to get the sub-niche into a tree? In graph G, the effective spanning tree set is Z, and T is the lowest total weight in g, so the tree with the smallest total weight in G\{t} is the sub-niche tree.

We may as well first consider such a problem, remember T is the smallest spanning tree in Figure G, because the spanning tree is | The v|-1 edge is the only decision, so we can assume that T represents the set of edges in the smallest spanning tree. It is easy to find that the sub-niche into a tree must contain an edge of the e\t, we can enumerate the edge e in each e\t, and the end point of E is merged, on this basis run the Kruskal or prim algorithm, find a new spanning tree, and add e to the spanning tree, We get all the least weight spanning trees in the spanning tree with E, and we call this tree the smallest spanning tree with E . It is possible to find the smallest spanning tree in these trees as a secondary niche tree, but the efficiency is very moving and the time complexity is O (| e|-| v|) *| V|LOG2 (| v|)).

In fact can be in O (| E|LOG2 (| v|)) Time complexity found in the sub-niche into a tree.

First we need to find the minimum spanning tree T. Then we enumerate each edge in E\t E, add E to T, and there must be a ring in T, and the ring contains E (| v| Edge Connected graph must have a ring), in order to make t become a legitimate tree, we need to remove an edge in the ring, obviously we should not remove E (otherwise why join it?). ), we want to remove the most weighted edge T in the ring, and then T becomes a generation tree T ' again. The Guaranteed Tree T ' is a minimum spanning tree with E, in fact we have merged the two endpoints of E to get a new figure G ', the smallest spanning tree in Figure G ' must be T ' \{e}, this is because we run the Kruskal algorithm on it, all weights less than t edge will be selected as usual, And after the completion of the edge processing of the weight less than T, the original E ring is also missing an edge T, because the spanning tree is not allowed to have a ring, so T will not be joined, to this isolated connected sub-map and establish a minimum spanning tree on G no two. So T ' \{e} is the smallest spanning tree on G '. If T ' is not a G containing the e minimum spanning tree, then there is an e spanning Tree f ', but F ' \e is also a spanning tree of G ', which means that the total weight of f ' \e is less than T ' \e, which is inconsistent with the minimum spanning tree of T ' \e is G ', so t ' is the smallest spanning tree in G

So how is the O (| E|LOG2 (| v|)) Time complexity to complete all of the above operations, we just for each side of E, can be log2 (| v|) Time complexity quickly calculates the largest edge on the path between the ends of E in the smallest spanning tree. Such algorithms are present and can be implemented using LCT (dynamic tree).

The time complexity for finding the smallest spanning tree is O (| V|LOG2 (| v|), and the build time complexity of establishing LCT on the minimum spanning tree is O (| V|LOG2 (| v|)), the time complexity of finding the minimum spanning tree with E for all sides E on the e\t is O (| e|-| v|) LOG2 (| v|)), so the total time complexity is O (| E|LOG2 (| v|).

Sub-niche tree-forming algorithm