A document from MCS 275 Spring 2024, instructor Emily Dumas. You can also get the notebook file.

MCS 275 Spring 2024 Worksheet 2

  • Course instructor: Emily Dumas

Topics

This worksheet corresponds to material from lectures 1-3, which contained the Python tour and a bit of object-oriented programming. Thus it contains:

  • General Python review exercises at MCS 260 level (but a bit more challenging than worksheet 1)
  • Practice with classes, instances, and methods

Instructions

  • This isn't collected or graded. Work on it during your lab on Jan 16 or 18.
  • This worksheet prepares you for Homework 2. (Worksheet n always prepares for Homework n)
  • These instructions will be essentially the same for every worksheet, so we'll stop writing them out explicitly. (Homework will always have full, detailed instructions.)

Resources

These things might be helpful while working on the problems. Remember that for worksheets, we don't strictly limit what resources you can consult, so these are only suggestions.

1. Increasing run length

Suppose you have a list of integers. Even if the list is not entirely in increasing order, there are sometimes parts of the list that are in increasing order.

For example, consider L = [1,21,4,17,38,39,35,22,0,6,2,5]*. The list is not in increasing order, but the first two elements (i.e. L[0:2] or [1,21]) are in increasing order. Also, if you take the slice L[2:6], which is [4,17,38,39], that is in increasing order.
Let's call a slice of a list that is in increasing order an increasing run. A given list of integers can have many increasing runs.

In the example given above, the longest increasing run you can find in L is L[2:6], and it has length 4.

Write a function IRL(L) that takes a list L and returns the length of the longest increasing run within L. In order for that to make sense, you should assume the elements of L can be compared to one another using > and < and ==, but it shouldn't matter exactly what their types are. If it helps in thinking about the problem, you could imagine they are integers.

Special cases (not necessarily requiring special handling, but explained here to clarify the extremes):

  • If L is empty, the function should return 0
  • If L is in decreasing order, the function should return 1 because a slice with one element can be considered to be in "increasing order", and that would be the longest increasing run.

* The sample list L shown above had a typo which was corrected on Jan 16.

Here are some sample outputs:

In [47]:
IRL([40, 1, 63, 20, 53, 9])  # [1,63] and [20,53] are the longest, each has length 2
Out[47]:
2
In [48]:
IRL([22, 40, 47, 46, 26, 45, 25, 30, 47, 40]) # [22,40,47] and [25, 30, 47] are the longest
Out[48]:
3
In [49]:
IRL([78, 26, 1, 20, 72, 19, 44, 50, 59, 67]) # [19, 44, 50, 59, 67] is the longest
Out[49]:
5
In [53]:
IRL([1,2,3,4,5,6,7,8]) # The entire list, of length 8, is increasing!
Out[53]:
8
In [80]:
IRL([]) # no elements = no increasing runs = max increasing run length 0
Out[80]:
0
In [55]:
IRL([275]) # [275] is the longest increasing run
Out[55]:
1
In [81]:
IRL([10,8,6,4,2]) # [10] and [8] and [6] and [4] and [2] are the longest, each has length 1
Out[81]:
1

Hint: Here is one possible strategy: Use a for loop to scan through the list, keeping track of the current element and the previous one. As you go, make note of how many successive elements have been increasing. Use another variable to hold the largest number of increasing elements seen so far, and update it as needed on each iteration.

2. Zitterbewegung

Zitterbewegung is a German word meaning "jittery motion".

This is an exercise in object-oriented programming.

Make a class ZBPoint that stores a point (x,y) in the plane with integer coordinates.

The constructor should take (x,y) as arguments and store them as attributes.

Add a __str__ method Printing an instance of this class should produce a nice string like

ZBPoint(17,-2)

Also include a method .jitter() that will move the point up, down, left or right by one unit. It should be a mutation (modifying the coordinates, not returning anything). The direction should be chosen at random (e.g. using random.choice or random.randint from the Python random module that you can import with import random.)

The code below demonstrates how you would use this class to create a point at (20,22) and then let it jitter about for 30 steps, printing the location each time.

In [61]:
P = ZBPoint(20,22)
for _ in range(30):
    print(P)
    P.jitter()

ZBPoint(20,22)
ZBPoint(20,21)
ZBPoint(20,20)
ZBPoint(19,20)
ZBPoint(19,19)
ZBPoint(18,19)
ZBPoint(18,18)
ZBPoint(18,19)
ZBPoint(18,18)
ZBPoint(19,18)
ZBPoint(19,19)
ZBPoint(20,19)
ZBPoint(19,19)
ZBPoint(19,20)
ZBPoint(19,21)
ZBPoint(18,21)
ZBPoint(18,22)
ZBPoint(18,23)
ZBPoint(17,23)
ZBPoint(17,22)
ZBPoint(17,23)
ZBPoint(18,23)
ZBPoint(17,23)
ZBPoint(16,23)
ZBPoint(17,23)
ZBPoint(17,22)
ZBPoint(17,23)
ZBPoint(17,22)
ZBPoint(17,23)
ZBPoint(17,24)

3. Text circles

In this problem it's important to use a big terminal window, at least 80 characters wide and 40 lines tall.

Write a program circle.py that takes one command line argument, r, and then prints 40 lines of text, each containing 80 characters. The output should make a picture of a circle, using characters as pixels that can be either "on" (@ character) or "off" (` character). Whenr=20the circle should be slightly too big to fit in the block of text, and whenr=1` it should be nearly too small to have any empty space inside.

For example, here is possible output for r=5. image of terminal showing a small circle made of text

And here is possible output for r=15: image of a terminal showing a large circle made of text

This exercise is deliberately a little vague about how to go about this, but here is a hint: Each character you print corresponds to a certain point (x,y) in the plane, with character 40 on line 20 corresponding to (20,20). The factor of 2 in the y coordinate is because haracters in a terminal are usually twice as tall as they are wide. Calculating the coordinates of the next character to be printed inside of the main loop will be useful. Then, how would you decide whether to print a " " or a "@"?

4. Recurrence time histogram

This builds on the ZBPoint class from problem 2.

Suppose you start a ZBPoint at (0,0) and then repeatedly call the method .jitter() until the first time it returns to (0,0). The number of steps it takes to do this is called the recurrence time. It's possible that this may never happen, but that is very unlikely. (It can be shown that the probability it will eventually return to (0,0) is 100%.)

Write a program that runs this process 1000 times and records the recurrence times. (It's OK to give up on a point if it's taken 200 steps and hasn't yet returned to (0,0).) The program should then print a table showing how many times each recurrence time was seen (a histogram). Here's an example of what it might look like:

Ran 1000 random walks starting at (0,0).  Of these:
 235 returned to (0,0) after   2 steps
  76 returned to (0,0) after   4 steps
  51 returned to (0,0) after   6 steps
  31 returned to (0,0) after   8 steps
  24 returned to (0,0) after  10 steps
  17 returned to (0,0) after  12 steps
  18 returned to (0,0) after  14 steps
   7 returned to (0,0) after  16 steps
   6 returned to (0,0) after  18 steps
  13 returned to (0,0) after  20 steps
   7 returned to (0,0) after  22 steps
   8 returned to (0,0) after  24 steps
   9 returned to (0,0) after  26 steps
   4 returned to (0,0) after  28 steps
   2 returned to (0,0) after  30 steps
   5 returned to (0,0) after  32 steps
   3 returned to (0,0) after  34 steps
   5 returned to (0,0) after  36 steps
   2 returned to (0,0) after  38 steps
   2 returned to (0,0) after  40 steps
   3 returned to (0,0) after  42 steps
   2 returned to (0,0) after  44 steps
   1 returned to (0,0) after  46 steps
   4 returned to (0,0) after  48 steps
   2 returned to (0,0) after  50 steps
   1 returned to (0,0) after  52 steps
   7 returned to (0,0) after  54 steps
   1 returned to (0,0) after  58 steps
   3 returned to (0,0) after  60 steps
   4 returned to (0,0) after  62 steps
   4 returned to (0,0) after  64 steps
   2 returned to (0,0) after  66 steps
   1 returned to (0,0) after  68 steps
   2 returned to (0,0) after  70 steps
   1 returned to (0,0) after  72 steps
   3 returned to (0,0) after  76 steps
   2 returned to (0,0) after  78 steps
   1 returned to (0,0) after  80 steps
   2 returned to (0,0) after  82 steps
   1 returned to (0,0) after  84 steps
   2 returned to (0,0) after  86 steps
   2 returned to (0,0) after  92 steps
   2 returned to (0,0) after  94 steps
   1 returned to (0,0) after  96 steps
   1 returned to (0,0) after  98 steps
   2 returned to (0,0) after  106 steps
   1 returned to (0,0) after  108 steps
   2 returned to (0,0) after  110 steps
   4 returned to (0,0) after  116 steps
   2 returned to (0,0) after  118 steps
   1 returned to (0,0) after  122 steps
   1 returned to (0,0) after  124 steps
   1 returned to (0,0) after  126 steps
   1 returned to (0,0) after  128 steps
   3 returned to (0,0) after  130 steps
   1 returned to (0,0) after  134 steps
   2 returned to (0,0) after  136 steps
   2 returned to (0,0) after  138 steps
   1 returned to (0,0) after  142 steps
   1 returned to (0,0) after  144 steps
   2 returned to (0,0) after  146 steps
   1 returned to (0,0) after  156 steps
   1 returned to (0,0) after  158 steps
   1 returned to (0,0) after  160 steps
   3 returned to (0,0) after  162 steps
   1 returned to (0,0) after  166 steps
   1 returned to (0,0) after  174 steps
   2 returned to (0,0) after  178 steps
   1 returned to (0,0) after  180 steps
   3 returned to (0,0) after  184 steps
   1 returned to (0,0) after  190 steps
   2 returned to (0,0) after  198 steps
 376 didn't return to (0,0) after 200 steps

Extra challenges

I recommend you don't work on these unless you finish the other problems and have extra time in lab or want an additional challenge on your own time.

A. Efficiency of IRL

This doesn't have a solution per se, but it asks you to evaluate and possibly improve your solution to problem 1.

There are lots of ways to solve problem 1, and some are more efficient than others. That is, if L is very long, there are some approaches where IRL(L) will return quickly, while a valid but different way of writing the function would take a very long time to return an answer.

Try your IRL function on a list of 10,000 or 100,000 integers in some sort of random order. An efficient solution should handle this very quickly (less than a second). If yours takes much longer, can you think of ways to make it faster (but still correct)?

The code below could be used for the necessary testing, if you add it to a file that contains your IRL function.

In [17]:
import random
import time

test_length = 100_000
test_input = list(range(test_length))
random.shuffle(test_input)

t0 = time.time()
k = IRL(test_input)
delta_t = time.time() - t0

print("IRL for a list of {} random integers returned {} after {:.2f} seconds".format(
    test_length,
    k,
    delta_t
))

IRL for a list of 100000 random integers returned 8 after 0.01 seconds

B. Prettier circles

Make a program circle2.py so that it makes circles with softer edges, like this: image of a terminal showing a circle made of text with soft edges

Revision history

  • 2024-01-15 Initial release
  • 2024-01-16 Important typo fixed: sample list L in problem 1 was missing element 38