问题
I am trying to modify the minimum spanning tree if the weight of an edge in the graph not belonging to MST is decreased.I read on stackoverflow that first connect the edge to the MST now there is exactly one cycle in MST and by cycle property the edge whose weight is maximum in cycle can be deleted from MST? How to find the max weight edge in that cycle?
回答1:
Let the new edge added be between node i and j.There will be exactly one cycle containing all nodes between node i and j, including them. Also as before it was a tree only one path is there from node i to j. So you can use DFS/BFS to traverse the graph and calculate the maximum weight of any edge occurring in the path from i to j.If the maximum weight is less than that of new edge weight, don't add the new one.Else remove the previous one and add this one.The complexity would be O(V). Following is the pseudo code , here ans[k][0],ans[k][1] store the nodes such that the edge between these nodes is of maximum weight if the source node is i and destination k and ans[k][2] as weight of that edge.
for all nodes
mark them unvisited
mark ans[node][2] as -1
/*i is the node which is one of the nodes of two of the new edge (i--j) */
Push node i in queue Q
mark node i visited
while Q is not empty
assign current_node as front element of Q
pop Q
for all neighbors of current_node
if neighbor is unvisited
mark neighbor visited
assign w to be maximum of weight of edge (current_node---neighbor) and ans[current_node]
if w is greater than ans[neighbor]
ans[neighbor][2] = w
##Depending on which was max in the the if condition
ans[neighbor][0] = current_node/ans[current_node][0]
ans[neighbor][1] = neighbor/ans[current_node][1]
push neighbor in Q
if weight of edge (i--j) is greater than ans[j][2]
don't add the new edge
else
remove edge (ans[j][0]---ans[j][1]) and add edge (i--j)
来源：https://stackoverflow.com/questions/27740400/maximum-weight-of-an-edge-in-a-cycle-in-graph