## Description

- Instructions

Submit your work through Canvas. You should submit a tar le containing all source les and a README for running your project. Don’t submit any other les (e.g., test case or pyc les).

More precisely, submit on Canvas a tar le named lastname.tar (where lastname is your last name) that contains:

All source les. You can choose any language that builds and runs on ix-dev.

A le named README that contains your name and the exact commands for building and running your project on ix-dev. If the commands you provide don’t work on ix-dev, then your project can’t be graded and there will be a signi cant penalty.

Here is an example of what to submit:

hampton.tar

README

my class1.py

my class2.py

my class3.py

problem.py

another problem.py

…

README

Andrew Hampton

Problem 1: python problem1.py <input_filename>

Problem 1: python problem1.py <input_filename>

…

Note that Canvas might change the name of the le that you submit to something like lastname-N.tar. This is totally ne!

The grading for the project will be roughly as follows:

Task | Points | ||

Problem 1 | 15 | ||

pass given sample test case | 5 | ||

pass small grading test case | 5 | ||

pass large grading test case | 5 | ||

Problem 2 | 20 | ||

pass given sample test case | 5 | ||

pass small grading test case | 5 | ||

pass large grading test case | 10 | ||

Problem 3 | 15 | ||

pass given sample test case | 5 | ||

pass small grading test case | 5 | ||

pass large grading test case | 5 | ||

TOTAL | 50 | ||

- Applications

Problem 1. Prioritizing HTTP Requests

Suppose you are working on a web application that receives HTTP requests over a LAN in batches. When you receive a batch of requests, they have already been preprocessed with an estimate of how long it will take your application to complete each of them.

You want to serve the requests in order of the estimated service time, with the shortest requests being served rst.

Additionally, your application has two tiers of service: A and B. All of the requests in the A tier should be served before any of the requests in the B tier.

Write a program that will give the correct service order according to the above criteria. Your program must use one or more priority queues to accomplish this!

In the event of a tie in service time (and tier), the request that appears rst in the input list should be served rst (that is, your sort should be stable).

Your program should take a single command-line argument, which will be a lename. The input le will contain request strings. The rst line of the input le will be an integer 0 N 10^{6} giving the number of requests. Following will be N lines, each containing a string having the following format:

IP_ADDR TIER ESTIMATE

IP ADDR is an IP address in decimal IPv4 format. TIER is either A or B. ESTIMATE is an integer 0 < X 10^{4} representing a time estimate for processing the request. The separator is a single space.

Output the IP addresses in the order the requests will be served, separated by newlines. Again, in the event of a tie in service time (and tier), the request that appears rst in the input list should be served rst.

Example input le:

8

10.31.99.245 B 30

10.16.0.105 A 150

10.16.115.160 B 60

10.30.111.90 B 65

10.16.0.105 A 20

10.30.100.100 A 25

10.16.100.115 A 150

10.111.111.119 B 60

Example output:

10.16.0.105

10.30.100.100

10.16.0.105

10.16.100.115

10.31.99.245

10.16.115.160

10.111.111.119

10.30.111.90

Problem 2. Rolling Median

Suppose you have streaming (integer) data and want to compute some summary statistics. It’s easy to calculate the cumulative rolling average: this is a constant time operation (look up the formula if you’re interested!). What about the cumulative rolling median?

In this problem, you’ll develop an algorithm to compute the cumulative rolling median and test it on simu-lated streaming data.

We will use the most common de nition of median, as described on the Wikipedia page. We can simulate streaming data by giving a list of integers L of length n and calculating the median on the slice L[1 : i] for every 0 < i n.

Your program should take a single command-line argument, which will be a lename. The input le will contain integers, one per line. The rst line of the input le will be an integer 2 N 10^{5} giving the number of integers in the list L. Following will be N lines, each containing an integer 0 x 10^{6}.

You should output the median of the slice L[1 : i] for every 0 < i N, with a newline between each result. If a median is not an integer, it should be printed to one decimal place. If a median is an integer, it should be printed as an integer (i.e., without a decimal point). See the sample output below.

Your solution should have runtime complexity O(n log n).

Hint: use two binary heaps, a maxheap to hold the smaller half of the data and a minheap to hold the larger half of the data. The median of the data is either at the top of one of the heaps or it’s the average of those two values.

Example input le:

5

1

8

4

3

2

Example output:

1

4.5

4

3.5

3

The rst line of the sample input says that the le contains 5 integers. So, we will read in 5 integers and with each new integer compute the median of those we have seen so far.

The rst integer is 1 and the median of f1g is 1. The next integer is 8 and the median of f1; 8g is 4:5. The

next integer is 4 and the median of f1; 8; 4g is 4. The next integer is 3 and the median of f1; 8; 4; 3g is 3:5.

The nal integer is 2 and the median of f1; 8; 4; 3; 2g is 3.

- Implementation

Problem 3. Binary Search Tree

For this problem, you will implement a binary search tree with integer keys. Do not use any builtin tree structures that your language might have. You must implement your own tree class that satis es the binary search tree property as described in Chapter 12 (p. 287) of the textbook.

Your binary search tree data structure should implement (at least) the following methods with speci ed runtime, where h refers to the height of the tree:

insert(X): Inserts a node X into the tree. O(h)

remove(X): Removes a node X from the tree. This method should be implemented as described in the

textbook (pp. 295-298). In particular, use the in-order successor as the replacement node. Runtime: O(h)

search(X, K): Returns a node in the subtree rooted at node X having key K, if present. Runtime: O(h)

maximum(X): Returns the node in the subtree rooted at node X having the largest key. Runtime: O(h)

minimum(X): Returns the node in the subtree rooted at node X having the smallest key. Runtime: O(h)

to list preorder(): Returns a list of the keys in the tree ordered by a pre-order traversal. Runtime: O(n)

to list inorder(): Returns a list of the keys in the tree ordered by an in-order traversal. Runtime: O(n)

to list postorder(): Returns a list of the keys in the tree ordered by a post-order traversal. Runtime: O(n)

(Depending on the programming language you use, replace a node with a pointer to a node as appropriate.)

Note: It’s important that you implement the remove method as described. The pre- and post-order traversals will not match the reference output if implemented di erently.

Note: The behavior of these methods in exceptional cases is unspeci ed. You should think about what these cases might be, and raise appropriate exceptions.

Note: No test case will insert duplicate keys into the tree.

Write a driver program that takes a single command-line argument, which will be a lename. The input le will contain instructions for tree operations. The rst line of the input le will be an integer 0 N 10^{6} giving the number of instructions. Following will be N lines, each containing an instruction. The possible instructions are:

insert K, where 10^{5} K 10^{5} is an integer: insert a node with key K into the tree. There is no output.

remove K, where 10^{5} K 10^{5} is an integer: remove a node with key K from the tree. If such a node exists, there is no output. If no such node exists, output TreeError.

search K, where 10^{5} K 10^{5} is an integer: output Found if a node exists with key K. If no such node exists, output NotFound.

max: output the maximum key in the tree. If the tree is empty, output Empty.

min: output the minimum key in the tree. If the tree is empty, output Empty.

preprint: print the keys of the tree according to a pre-order traversal, separated by a single space. If the tree is empty, output Empty.

inprint: print the keys of the tree according to an in-order traversal, separated by a single space. If the tree is empty, output Empty.

postprint: print the keys of the tree according to a post-order traversal, separated by a single space. If the tree is empty, output Empty.

Example input le:

20

inprint

remove 2

max

search 5

insert 1

insert 2

search 1

search 2

insert 3

inprint

insert 10

insert 5

inprint

preprint

postprint

search 5

remove 2

inprint

max

min

Example output:

Empty

TreeError

Empty

NotFound

Found

Found

1 2 3

1 2 3 5 10

1 2 3 10 5

5 10 3 2 1

Found

1 3 5 10

10

1

6