algorithmsdynamic programminggraph theory
Give a linear time algorithm that, given an undirected tree T = (V,E) with edges weights
w, returns the weight of the heaviest path in T.
This is basically asking to do the exact opposite of what the Floyd-Warshall algorithm does. Any suggestions (or even solution) as to how to approach/solve this question?
Consider the following algorithm, where we start at terminal nodes and keep removing them while accounting for the corresponding paths. For each node $v$, we store $2$ longest paths we have obtained so far (say of length $l_1(v)$ and $l_2(v)$, with $l_1 \geq l_2$), which end at that node. Initially set all these values to zero.
When all edges have been removed, we can show that maximum path length is given by $\max {(l_1(v) + l_2(v))}$, over all nodes $v$ in the original tree. Each vertex and edge is visited exactly once and all the operations are constant time, so the algorithm is linear in the number of vertices.
I assumed here that an empty path (path with no edges) is admissible and has length $0$ (for example, when all edge weights are negative, empty path is the longest). But if you only want non-empty paths, you could change the initialization of $l_1$ and $l_2$ values for all nodes to $-\infty$ instead of $0$.
Your assumption is false, this is a counterexample:
Check, that the longest path (all black edges) has weight $6$, while any path containg the red edge has at most weight $5$.
Best Answer
You should use Dynamic Programming to solve this problem. Make the tree rooted from arbitrary vertex. Then for each subtree find the maximum weighted path from root of that subtree to other nodes.