Day 8


For next time

Reading Journal Debrief

Now that we’ve read about Dictionaries and Tuples, we’ve seen almost all built-in Python types. As an activity, let’s compare and contrast the types and their uses.

With the person sitting next to you, review your solutions to the most_frequent exercise. Pay particular attention to your strategies for iteration and sorting.

“Recursive” problem analysis from 5 Whys, introduction and Examples and An Introduction to 5-why: What is the value in continuing to ask “why”? How do you know when you’ve reached a root cause?

Recursion Practice

Levenshtein Distance

Let’s circle back on some of the recursion practice problems from last time. We’ll start by implementing Levenshtein distance together as a class. Here is the description of the problem from last time.

Write a function called levenshtein_distance that takes as input two strings and returns the Levenshtein distance between the two strings. Intuitively, the Levenshtein distance is the minimum number of edit operations to transform one string into the other (for this reason Levenshtein distance is sometimes called “edit distance”). These edits can either be insertions, deletions, or substitutions. Note that Levenshtein distance is similar to Hamming distance, but works for strings of differing lengths

Here are some examples of these operations:

  1. kittensitten (substitution of s for k)
  2. sittensittin (substitution of i for e)
  3. sittinsitting (insertion of g at the end).

While this function seems initially daunting, it admits a very compact recursive solution. You can either work on your own to see the recursive solution, or use the recursive solution given in the Wikipedia article.

To get a better handle on this, let’s consider some more examples.

levenshtein_distance('kitten', 'smitten') -> 2 (see below for steps)

  1. kitten → sitten (k gets replaced by s)
  2. sitten → smitten (insert between s and i)

levenshtein_distance('beta', 'pedal') -> 3 (see below for steps)

  1. beta → peta (b gets replaced by p)
  2. peta → petal (l gets inserted at the end)
  3. petal → pedal (t gets replaced by d)

levenshtein_distance('battle', 'bet') -> 4 (see below for steps)

  1. battle → bettle (a gets replaced by e)
  2. bettle → bettl (the last e gets deleted)
  3. bettl → bett (delete l)
  4. bett → bet (delete t)

Base Cases

Let’s consider the base cases when one of the two strings is empty. What should the Levenshtein distance be in this case?

Recursive Step

Let’s consider the different ways in which we can make the first character of string a equal to the first character of string b. Here are the possible cases.

For each of these steps we have to consider two things:

Let’s write a recursive implementation of this function.


Last class, a lot of you had the chance to do this exercise:

Write a function called choose that takes two integer, n and k, and returns the number of ways to choose k items from a set of n (this is also known as the number of combinations of k items from a pool of n). Your solution should be implemented recursively using Pascal’s rule.

Here is a sample solution:

def nchoosek(n, k):
    """ returns the number of combinations of size k
        that can be made from n items.

        >>> nchoosek(5,3)
        >>> nchoosek(1,1)
        >>> nchoosek(4,2)
    if k == 0:
        return 1
    if n == k:
        return 1
    return nchoosek(n - 1, k - 1) + nchoosek(n - 1, k)

if __name__ == '__main__':
    import doctest

It passes all the unit tests!!! Hooray!

Unfortunately, this code is going to be quite slow. To get a sense of it, let’s draw a tree that shows the recursive pattern of the function.

You can even visualize the call graph within Jupyter notebook using an extension. In Linux:

$ sudo apt-get install graphviz
$ pip install callgraph

Then within your notebook, you can draw a call graph as a function is called:

%load_ext callgraph
%callgraph nchoosek(4, 2)

As you can see, the function is recursively called multiple times with the same arguments.

In order to improve the speed of this code, we can make use of a pattern called memoization. The basic idea is to transform a recursive implementation of a function to make use of a cache (in this case a Python dictionary) that remembers all previously computed values for the function. Here’s is a skeleton of a memoized recursive function (we are being a little fast and loose with the mixing pseudo code and Python, but this should become clearer as you do a concrete example.

def recursive_function(input1, input2):
    if input1, input2 is a base case:
        return base case result
    if input1, input2 is in the list of already computed answers
       return precomputed answer
    return recursive step on a smaller version of input1, input2

Add memoization to your implementation of nchoosek (and/or levenshtein_distance) and study its impact on performance. Think Python Chapter 11.6 will also be helpful.

Turtle World

We’re devoting a bit more time to Turtle World today - with the remainder of the time, choose at least one of the fractal drawings below to pursue.

One additional bit of fun: now that we know about tuples, we’re not limited to a tiny color palette. Colors in computer graphics are typically expressed as a (red, green, blue) tuple, which you can pass to turtle.color to paint the entire rainbow!

Teleportation, Cloning, and Other Unethical Experiments on Turtles

In addition to the what you did on your day 5 reading journal and in the day 6 class activity, a few other Turtle Tricks that may prove useful.

A Turtle is a Python object, which we will learn more about next week. Turtles have methods, which we can call to inspect change their behavior. One trick that will be useful here, which you saw in but may not have thought about much, is the speed() function. The speed() function can be used to speed up slowpoke Turtles. While it seems weird that a speedy turtle would have a speed of 0, in this case the input 0 is reserved for having the turtle go as fast as possible (remember, when in doubt, check the documentation).

import turtle
speedy = turtle.Turtle()

Other important Turtle methods include xcor() and ycor() position, and heading().

Read more about turtles here.

Since Turtles are simple creatures, mainly defined by their current position and heading, we can “clone” them by reading these values and using them to direct a new Turtle.

leo = turtle.Turtle()
# leo does some arbitrary drawing (e.g., makes a 45 degree angle)

# Create a new Turtle with the same attributes as the first
don = turtle.Turtle()
if leo.isdown():
# don.bandana_color = "purple" # TODO: Ninja functionality not yet implemented

As an exercise, encapsulate this functionality in a clone function that takes a Turtle argument and returns a new Turtle with the same position and heading, leaving the original Turtle untouched.


Fractals are geometrical constructions that display self-similar repeated patterns at every scale as you zoom in. They are often extremely beautiful, and are found throughout nature. Fractals are also useful across many fields, from antenna engineering to poetry to finance. Check out Yale’s Panorama of Fractals and their Uses for more examples.

Today, we will teach our turtles to draw fractal shapes using recursion. A very cool recursive drawing we can create is called the snowflake curve (or Koch snowflake). To get started, let’s write a function called snow_flake_side with the following signature:

def snow_flake_side(turtle, length, level):
    """Draw a side of the snowflake curve with side length length and recursion
    depth of level"""

The snow_flake_side function should have a base case that draws the following image:

The recursive step should replace each of the line segments above with a snow_flake_side with size length / 3.0 and recursion depth level - 1. Take some time to work on this and then we’ll discuss as a group.

Once you have completed your snow_flake_side function, create a function called snow_flake that draws the whole snowflake.

Recursive Trees

Next, we will draw a tree using recursion. Define a function called recursive_tree that takes as input a turtle, a branch length, and a recursion depth and draws the recursive tree to the canvas.

def recursive_tree(turtle, branch_length, level):
    """Draw a tree with branch length branch_length and recursion depth of level

The base case is:

This structure is given by moving forward branch_length steps (assuming the turtle has the correct orientation).

For the recursive step, you should:

  1. Draw the line as above
  2. Clone your turtle
  3. Turn the new turtle left 30 degrees
  4. Recurse using the cloned turtle to draw a tree with branch length branch_length * 0.6 and depth `level - 1
  5. Hide the cloned turtle using the hideturtle method
  6. Back the original turtle up branch_length / 3.0
  7. Clone your turtle
  8. Turn the new turtle right 40 degrees
  9. Recurse using the cloned turtle to draw a tree with branch length branch_length * 0.64 and depth level - 1
  10. Hide the cloned turtle using the hideturtle method

After implementing the recursive step, if you set level to 1 more than the base case (which will either be 1 or 2 depending on what level you consider the base case), you will get the following picture:

Once you’ve built your recursive_tree function, try making a few enhancements:

More Recursion

The Koch snowflake and our recursive tree are both part of a more general class of curves called L-systems (Lindenmayer Systems). Next, read the linked Wikipedia article on L-systems and try to implement Sierpinski’s triangle and fractal plant.

Hint 1: For Sierpinski’s triangle you will want to create a function to generate both symbols A and B and have them call each other.

Hint 2: For the fractal plant you should create the following functions to save and then restore then Turtle’s state (symbols [ and ] respectively):

def save_turtle_state(turtle_states, t):
    turtle_states.append((t.xcor(), t.ycor(), t.heading()))

def restore_turtle_state(turtle_states, t):
    s = turtle_states.pop()