Convex hull of backtracking operation

Meaning of the questions: a tree, point to ask each ancestor node$v$To$u$of$c[v]-c[u]/dis(v,u)$of$min$value.

The third question exam yesterday, is already seen on the convex hull maintenance, but for$dfs$Back when the convex hull of how to maintain the violence will only get stack maintenance (but it seems that for the time complexity of random data$O(n log_2 n$)), However positive solution approach is to use a persistent stack.

In fact, the persistence stack is a tree, back to back operation is equivalent to the ancestors, while adding a new node$u$, Following a directly connected state of the node, and then multiplying the direct-half (new position), to find the optimal node$v$, And can persist in the stack$u$To$v$The section of the path, in fact, the addition of normal$u$The need to pop those elements of the stack, so that you can directly multiply-half to find the optimal solution, directly connected to that point below.

Multiplying two points:
sweep from high to binary status, the node compares the current optimal multiplication node$u$versus$u$Father$v$Who better, if$v$Better, remains unchanged binary (this period because he could still be the optimal solution), then assign it as the optimal node$v$Otherwise, it is a bit lower, continue to$u$Doubled.