Biostatistics 615/815
Problem Set 1
Due Wednesday September 22
Union – Find Algorithms
1. Write a program that generates random connections between pairs of objects.
Estimate how many random connections must be sampled before all objects are
connected to each other in a set of 1000 objects.
2. The Weighted Quick Union algorithm generates a tree that summarizes
connectability between different objects. When all objects are connected, they all
lead to the same “root” object. Develop and implement an algorithm that
computes the average distance between each node and the root of the tree.
Modify the program you generated for exercise 1, so that it outputs the average
distance between each node and the root of the tree after enough random
connections have been generated for all objects to be connected.
What is the average distance between each node and the root of the tree? What is
the maximum distance between each node and the root of the tree?
3. Propose a modified version of the Weighted Quick Union algorithm that uses the
height of trees (longest path to any node from root) rather than the weight (total
number of nodes in tree) to decide where to add the new pointer. Provide your
example code.
Which version do you expect to be more efficient? [You can make a theoretical
argument, conduct empirical studies or even use the code you developed in
problem 2 to justify your answer.]
Please indicate how much time it took you to complete this assignment. This will help me
calibrate future assignments.
