CS 112 Project 9: Binary Search Trees

Objectives:

  1. Practice building BST operations.
  2. Practice building recursive operations.
  3. Learn more about binary search trees.

Introduction

This week's project is a single person project. The project is to:

As usual, you should use test-drive development to accomplish these tasks.

The Experiment

The height of a BST determines the speed of operations like insert() and contains(). If we build a "large" BST of N items (e.g., ~220), and that tree is balanced, then its height should be the minimal O(lg(N)) (e.g., 20).

This week's problem is to conduct a simple experiment to see how "tall" BSTs get, given random values. Our objective is to see how close we come to lg(N) height if we build large trees of N randomly ordered values.

In the ulab, the folder:

/home/cs/112/proj/09
contains ten data files (randomInts01.txt, randomInts02.txt, ..., randomInts10.txt), each containing 220-1 = 1,048,575 randomly generated integer values. If these values were ordered in exactly the right way, we could build trees of height 20. (On the other hand, if they were ordered in exactly the wrong way, we could build trees of height 1,048,575!)

Your task is to write a simple program that prompts the user for the name of an input file, reads each value from that file, and inserts it into a BST. Your program should then display the height of the resulting BST.

There are three minor details to keep in mind:

Since there are ten input files, you will need to run your program ten times. (Or find a clever way to automate this.) For each file, record the corresponding tree-height in a spreadsheet, and the number of duplicate values that were found. Given all ten heights, use the spreadsheet to calculate the minimum height, the maximum height, the average height, the median height, and the standard deviation of the ten heights. (Your spreadsheet will make each of these computations easy.) Finally, use your spreadsheet to create a bar-chart showing each of the ten heights, as a visualization of your data. Save this chart in the worksheet where you recorded and analyzed your data.

Finally

Answer the following questions in your spreadsheet, below the chart:
  1. Is lg(N) or N a better approximation for your measured heights? Why?
  2. How much variance is there in the height of these trees? Is this surprising? Why or why not?
  3. How many duplicate values were there (on average)? Is this surprising? Why or why not?
  4. The "big Oh" notation O(f(n)) implies that there are constants a and b, such that a * f(n) + b describes the reality for which O(f(n)) is an approximation. A balanced BST of N items has height O(lg(N)). Theoretically, there should be values a and b such that a * lg(N) + b describes the actual height of a BST built using random data. Using your experimental results, what is a reasonable value for a?

Submit


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