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Chapter 3: Q18E (page 109)

You are given a tree T=(V,E) (in adjacency list format), along with a designated root node r鈭�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 鈥渋s u an ancestor v?鈥� can be answered in constant time. The pre-processing itself should take linear time. How can this be done?

Short Answer

Expert verified

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

Step by step solution

01

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 .

02

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

Consider the tree T=V,E, in adjacency list format, along with a designated root node r. Perform a depth-first search on T=V,E, mark the root node as 1. Label the node L, if the left child is labelled L0and its right child is labelled L1.

Consider any two nodes u,v , If the node u is marked as the L0and 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 鈥渋s u an ancestor v鈥�, is done by performing the Depth first search with pre and post numbering and it has been done in constant time.

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Most popular questions from this chapter

A bipartite graph is a graph G=(V,E)whose vertices can be partitioned into two sets (V=V1V2andV1V2=蠒) such that there are no edges between vertices in the same set (for instance, if , then there is no edge between and ).

(a) Give a linear-time algorithm to determine whether an undirected graph is bipartite.

(b) There are many other ways to formulate this property. For instance, an undirected graph is bipartite if and only if it can be colored with just two colors. Prove the following formulation:

an undirected graph is bipartite if and only if it contains no cycles of odd length.

(c) At most how many colors are needed to color in an undirected graph with exactly one odd length?

Rewrite the explore procedure (Figure 3.3) so that it is non-recursive (that is, explicitly use a stack). The calls to pre visit and post visit should be positioned so that they have the same effect as in the recursive procedure.

Two paths in a graph are called edge-disjointif they have no edges in common. Show that in any undirected graph, it is possible to pair up the vertices of odd degree and find paths between each such pair so that all these paths are edge-disjoint.

Give a linear-time algorithm for the following task.
Input: A directed acyclic graph G

Does G contain a directed path that touches every vertex exactly once?

The police department in the city of Computopia has made all streets one-way. The mayor contends that there is still a way to drive legally from any intersection in the city to any other intersection, but the opposition is not convinced. A computer program is needed to determine whether the mayor is right. However, the city elections are coming up soon, and there is just enough time to run a linear-time algorithm.

a) Formulate this problem graph-theoretically, and explain why it can indeed be solved in linear time.

(b) Suppose it now turns out that the mayor鈥檚 original claim is false. She next claims something weaker: if you start driving from town hall, navigating one-way streets, then no matter where you reach, there is always a way to drive legally back to the town hall. Formulate this weaker property as a graph-theoretic problem, and carefully show how it too can be checked in linear time.
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