Question

You are given a tree T=(V,E) (in adjacency list format), along with a designated root node $\mathit{r}\mathbf{\in }\mathit{V}$. Recall that u is said to be an ancestor of v in the rooted tree if the path from r to v in T passes through u.

You wish to reprocess the tree so that queries of the form “is u an ancestor v?” can be answered in constant time. The pre-processing itself should take linear time. How can this be done?

Short answer

To reprocess the tree to fit the query “is u an ancestor v ”, by performing the Depth first search with pre and post numbering the tree T=(V,E).

Step-by-step solution

Explain the information given.

Consider the tree T=(V,E), in adjacency list format, along with a designated root node r . The node u is said to be an ancestor of v in the rooted tree if the path from r to v in T passes through .

Step 2: Explain how the reprocess is done and in linear time

Consider the tree $T=\left(V,E\right)$, in adjacency list format, along with a designated root node $r$. Perform a depth-first search on $T=\left(V,E\right)$, mark the root node as 1. Label the node L, if the left child is labelled $L0$and its right child is labelled $L1$.

Consider any two nodes u,v , If the node u is marked as the $L0$and the other node v that is labelled as $L0$. Then the u is the ancestor of v.

The search and the labelling can be done in constant time.

Therefore, the reprocess to fit the tree into the query “is u an ancestor v”, is done by performing the Depth first search with pre and post numbering and it has been done in constant time.