Hello! I have come across a tree problem where number of nodes(V<=1e5) and number of edges(E<=1e5) and edge weights are given .There are Q (Q<=1e5) queries of two types:
Query 1 asks to change the weight of edge i to W (given in query) Query 2 asks to print the path length between nodes a and b (given in query)
//As I know, LCA can be used to find the distance when edge weights are not changing (But does not seem to be useful here)
Can anyone suggest any method on how to solve it efficiently ?
Using heavy-light decomposition should be ok
Thanks man! Got the ROOT.
Instead of doing HLD, you can do Euler Tour Tree+fenwick, which will work faster and will be much easier to write
How on a path? I can see it on subtree not a path
Let F[u] be the sum of the weights of edges going from u to the root node
then an update to the edge connecting node k and it's parent would change all F[v] where v is in subtree k
with euler tour, all the v values are going to be in a range [l..r]
to calculate distance, it's just: F[u] + F[v] — F[lca(u,v)]
Okay I will explore that too. Thanks !
sqrt decomposition (keep lazy updates and rebuild distance after each B queries)
maybe sqrt decomposition will work
Simply change the query to the root to the vertex $$$a$$$. Then you can use DFS index and fenwick tree to solve this in $$$O(n \log n)$$$ time, which is fast enough.
Detaily, let $$$id_u$$$ be the DFS id of vertex $$$u$$$, and $$$sz_u$$$ denotes the size of the subtree rooted at vertex $$$u$$$. Note that a subtree rooted at $$$u$$$ is $$$[id_u, id_u+sz_u-1]$$$, so in fact you need to solve segment add and single point query. Use fenwick tree to solve it!