Introduction to Algorithms
Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 21.3, Problem 1E
Program Plan Intro

To redo the exercise 21.2-2 with a disjoint-sets forest by using the rank and path compression method.

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Develop a topological sort implementation thatmaintains a vertex-indexed array that keeps track of the indegree of each vertex. Initialize the array and a queue of sources in a single pass through all the edges Then, perform the following operations until the source queue is empty:■ Remove a source from the queue and label it.■ Decrement the entries in the indegree array corresponding to the destinationvertex of each of the removed vertex’s edges.
Your second function is called “isTree". Its input is a graph G, which is a dictionary whose keys are the vertices, and whose values are lists of vertices that are adjacent to the given vertex. Its output is True if G is a tree and False if G is not a tree. Hint: You may want to make use of your "connected" function from the last coding assignment.
Consider the implementation of disjoint sets using forests. Assume the union is done by weight, i.e., the root of the tree with lesser nodes points to the root of the tree with more nodes: The following operations are applied on an initial set of elements {x, x2, X3, X4, X5} : union (x, x2); union (x1, x3); union (x4, xg); union (x4, x1). Show the forest after each operation.
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