# Lab 7: Recursive Objects lab07.zip

Due at 11:59pm on Friday, 10/12/2018.

Lab Check-in 4 questions here.

## Starter Files

Download lab07.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the Ok autograder.

## Submission

By the end of this lab, you should have submitted the lab with python3 ok --submit. You may submit more than once before the deadline; only the final submission will be graded. Check that you have successfully submitted your code on okpy.org.

• To receive credit for this lab, you must complete Questions 2 through 8 in lab07.py and submit through OK.
• Questions 9 through 12 are extra practice. They can be found in the lab07_extra.py file. It is recommended that you complete these problems on your own time.

# Topics

## Check-Off

### Q1: A-Okay

Sign up for checkoffs in your lab if you'd like to get credit for this week's checkoff.
What happens in the following code? Make sure you can explain why each line works or breaks.

>>> class A:
...    y = 1
...    def __init__(self, y):
...        self.y = y
...    def f(self, x):
...        self.y += x
...
>>> a = A(0) # Ok or not okay?
>>> a.f(6)   # Ok or not okay?
>>> a.f(A, 9) # Ok or not okay?
>>> A.f(a, 4) # Ok or not okay?
>>> A.f(A, 4) # Ok or not okay?

We've learned that a Python list is one way to store sequential values. Another type of list is a linked list. A Python list stores all of its elements in a single object, and each element can be accessed by using its index. A linked list, on the other hand, is a recursive object that only stores two things: its first value and a reference to the rest of the list, which is another linked list.

We can implement a class, Link, that represents a linked list object. Each instance of Link has two instance attributes, first and rest.

>>> s.first
1
True
>>> s.second
3
>>> s.first = 5
>>> s.second = 6
>>> s                                    # Displays the contents of repr(s)
>>> s
>>> print(s)                             # Prints str(s)
<5 7 <8 9>>
"""
empty = ()

def __init__(self, first, rest=empty):
self.first = first
self.rest = rest

@property
def second(self):
return self.rest.first

@second.setter
def second(self, value):
self.rest.first = value

def __repr__(self):
rest_repr = ', ' + repr(self.rest)
else:
rest_repr = ''
return 'Link(' + repr(self.first) + rest_repr + ')'

def __str__(self):
string = '<'
string += str(self.first) + ' '
self = self.rest
return string + str(self.first) + '>'

A valid linked list can be one of the following:

2. A Link object containing the first value of the linked list and a reference to the rest of the linked list

What makes a linked list recursive is that the rest attribute of a single Link instance is another linked list! In the big picture, each Link instance stores a single value of the list. When multiple Links are linked together through each instance's rest attribute, an entire sequence is formed.

Note: This definition means that the rest attribute of any Link instance must be either Link.empty or another Link instance! This is enforced in Link.__init__, which raises an AssertionError if the value passed in for rest is neither of these things.

We've also defined a pseudo-attribute second with the @property decorator that will return the second element in the linked list as well as a corresponding setter. Note that the second element of a linked list is really just the first attribute of the Link instance stored in rest. Don't worry too much about the syntax of the setter function for now. See the docstring for a closer look at how to use this property.

To check if a linked list is empty, compare it against the class attribute Link.empty. For example, the function below prints out whether or not the link it is handed is empty:

else:
print('This linked list is not empty!')

Since you are already familiar with Python's built-in lists, you might be wondering why we are teaching you another list representation. There are historical reasons, along with practical reasons. Later in the course, you'll be programming in Scheme, which is a programming language that uses linked lists for almost everything.

For now, let's compare linked lists and Python lists by looking at two common sequence operations: inserting an item and indexing.

Python's built-in list is like a sequence of containers with indices on them:

Linked lists are a list of items pointing to their neighbors. Notice that there's no explicit index for each item.

Suppose we want to add an item at the head of the list.

• With Python's built-in list, if you want to put an item into the container labeled with index 0, you must move all the items in the list into its neighbor containers to make room for the first item;

• With a linked list, you tell Python that the neighbor of the new item is the old beginning of the list.

We can compare the speed of this operation by timing how long it takes to insert a large number of items to the beginning of both types of lists. Enter the following command in your terminal to test this:

python3 timing.py insert 100000

Now, let's take a look at indexing. Say we want the item at index 3 from a list.

• In the built-in list, you can use Python list indexing, e.g. lst[3], to easily get the item at index 3.
• In the linked list, you need to start at the first item and repeatedly follow the rest attribute, e.g. link.rest.rest.first. How does this scale if the index you were trying to access was very large?

To test this, enter the following command in your terminal

python3 timing.py index 10000

This program compares the speed of randomly accessing 10,000 items from both a linked list and a built-in Python list (each with length 10,000).

What conclusions can you draw from these tests? Can you think of situations where you would want to use one type of list over another? In this class, we aren't too worried about performance. However, in future computer science courses, you'll learn how to make performance tradeoffs in your programs by choosing your data structures carefully.

## Trees (Again)

Recall that a tree is a recursive abstract data type that has a label (the value stored in the root of the tree) and branches (a list of trees directly underneath the root).

We saw one way to implement the tree ADT -- using constructor and selector functions that treat trees as lists. Another, more formal, way to implement the tree ADT is with a class. Here is part of the class definition for Tree, which can be found in lab07.py:

class Tree:
"""
>>> t = Tree(3, [Tree(2, [Tree(5)]), Tree(4)])
>>> t.label
3
>>> t.branches[0].label
2
>>> t.branches[1].is_leaf()
True
"""
def __init__(self, label, branches=[]):
for b in branches:
assert isinstance(b, Tree)
self.label = label
self.branches = list(branches)

def is_leaf(self):
return not self.branches

Even though this is a new implementation, everything we know about the tree ADT remains true. That means that solving problems involving trees as objects uses the same techniques that we developed when first studying the tree ADT (e.g. we can still use recursion on the branches!). The main difference, aside from syntax, is that tree objects are mutable.

Here is a summary of the differences between the tree ADT implemented using functions and lists vs. implemented using a class:

- Tree constructor and selector functions Tree class
Constructing a tree To construct a tree given a label and a list of branches, we call tree(label, branches) To construct a tree object given a label and a list of branches, we call Tree(label, branches) (which calls the Tree.__init__ method)
Label and branches To get the label or branches of a tree t, we call label(t) or branches(t) respectively To get the label or branches of a tree t, we access the instance attributes t.label or t.branches respectively
Mutability The tree ADT is immutable because we cannot assign values to call expressions The label and branches attributes of a Tree instance can be reassigned, mutating the tree
Checking if a tree is a leaf To check whether a tree t is a leaf, we call the convenience function is_leaf(t) To check whether a tree t is a leaf, we call the bound method t.is_leaf(). This method can only be called on Tree objects.

# Required Questions

## What Would Python Display?

Read over the Link class in lab07.py. Make sure you understand the doctests.

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

Enter Function if you believe the answer is <function ...>, Error if it errors, and Nothing if nothing is displayed.

If you get stuck, try drawing out the box-and-pointer diagram for the linked list on a piece of paper or loading the Link class into the interpreter with python3 -i lab07.py.

>>> from lab07 import *
______
1000
______
True
______
AssertionError
______
TypeError
>>> from lab07 import *
______
1
______
2
______
True
______
9001
______
3
______
1
______
1
______
2
>>> from lab07 import *
______
6
______
7
______
______
8
______
<5 <8 9> 7>

### Q3: WWPD: Trees

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

python3 ok -q trees -u

Enter Function if you believe the answer is <function ...>, Error if it errors, and Nothing if nothing is displayed. Recall that Tree instances will be displayed the same way they are constructed.

>>> from lab07 import *
>>> t = Tree(1, Tree(2))
______
Error
>>> t = Tree(1, [Tree(2)]) >>> t.label
______
1
>>> t.branches[0]
______
Tree(2)
>>> t.branches[0].label
______
2
>>> t.label = t.branches[0].label >>> t
______
Tree(2, [Tree(2)])
>>> t.branches.append(Tree(4, [Tree(8)])) >>> len(t.branches)
______
2
>>> t.branches[0]
______
Tree(2)
>>> t.branches[1]
______
Tree(4, [Tree(8)])

## Coding Practice

Write a function link_to_list that takes in a linked list and returns the sequence as a Python list. You may assume that the input list is shallow; none of the elements is another linked list.

Try to find both an iterative and recursive solution for this problem!

"""Takes a linked list and returns a Python list with the same elements.

[1, 2, 3, 4]
[]
"""

Use Ok to test your code:

### Q5: Store Digits

Write a function store_digits that takes in an integer n and returns a linked list where each element of the list is a digit of n.

def store_digits(n):
"""Stores the digits of a positive number n in a linked list.

>>> s = store_digits(1)
>>> s
>>> store_digits(2345)
>>> store_digits(876)
"""
result = Link.empty while n > 0: result = Link(n % 10, result) n //= 10 return result

Use Ok to test your code:

python3 ok -q store_digits

### Q6: Cumulative Sum

Write a function cumulative_sum that mutates the Tree t so that each node's label becomes the sum of all labels in the subtree rooted at the node.

def cumulative_sum(t):
"""Mutates t so that each node's label becomes the sum of all labels in
the corresponding subtree rooted at t.

>>> t = Tree(1, [Tree(3, [Tree(5)]), Tree(7)])
>>> cumulative_sum(t)
>>> t
Tree(16, [Tree(8, [Tree(5)]), Tree(7)])
"""
for b in t.branches: cumulative_sum(b) t.label = sum([b.label for b in t.branches]) + t.label

Use Ok to test your code:

python3 ok -q cumulative_sum

## Midterm Review (required)

### Q7: Is BST

Write a function is_bst, which takes a Tree t and returns True if, and only if, t is a valid binary search tree, which means that:

• Each node has at most two children (a leaf is automatically a valid binary search tree)
• The children are valid binary search trees
• For every node, the entries in that node's left child are less than or equal to the label of the node
• For every node, the entries in that node's right child are greater than the label of the node

Note that, if a node has only one child, that child could be considered either the left or right child. You should take this into consideration. Do not use the BST constructor or anything from the BST class.

Hint: It may be helpful to write helper functions bst_min and bst_max that return the minimum and maximum, respectively, of a Tree if it is a valid binary search tree.

def is_bst(t):
"""Returns True if the Tree t has the structure of a valid BST.

>>> t1 = Tree(6, [Tree(2, [Tree(1), Tree(4)]), Tree(7, [Tree(7), Tree(8)])])
>>> is_bst(t1)
True
>>> t2 = Tree(8, [Tree(2, [Tree(9), Tree(1)]), Tree(3, [Tree(6)]), Tree(5)])
>>> is_bst(t2)
False
>>> t3 = Tree(6, [Tree(2, [Tree(4), Tree(1)]), Tree(7, [Tree(7), Tree(8)])])
>>> is_bst(t3)
False
>>> t4 = Tree(1, [Tree(2, [Tree(3, [Tree(4)])])])
>>> is_bst(t4)
True
>>> t5 = Tree(1, [Tree(0, [Tree(-1, [Tree(-2)])])])
>>> is_bst(t5)
True
>>> t6 = Tree(1, [Tree(4, [Tree(2, [Tree(3)])])])
>>> is_bst(t6)
True
>>> t7 = Tree(2, [Tree(1, [Tree(5)]), Tree(4)])
>>> is_bst(t7)
False
"""
def bst_min(t): """Returns the min of t, if t has the structure of a valid BST.""" if t.is_leaf(): return t.label return min(t.label, bst_min(t.branches[0])) def bst_max(t): """Returns the max of t, if t has the structure of a valid BST.""" if t.is_leaf(): return t.label return max(t.label, bst_max(t.branches[-1])) if t.is_leaf(): return True if len(t.branches) == 1: c = t.branches[0] return is_bst(c) and (bst_max(c) <= t.label or bst_min(c) > t.label) elif len(t.branches) == 2: c1, c2 = t.branches valid_branches = is_bst(c1) and is_bst(c2) return valid_branches and bst_max(c1) <= t.label and bst_min(c2) > t.label else: return False

Use Ok to test your code:

python3 ok -q is_bst

### Q8: In-order traversal

Write a function that returns a generator that generates an "in-order" traversal, in which we yield the value of every node in order from left to right, assuming that each node has either 0 or 2 branches.

def in_order_traversal(t):
"""
Generator function that generates an "in-order" traversal, in which we
yield the value of every node in order from left to right, assuming that each node has either 0 or 2 branches.

For example, take the following tree t:
1
2       3
4     5
6  7

We have the in-order-traversal 4, 2, 6, 5, 7, 1, 3

>>> t = Tree(1, [Tree(2, [Tree(4), Tree(5, [Tree(6), Tree(7)])]), Tree(3)])
>>> list(in_order_traversal(t))
[4, 2, 6, 5, 7, 1, 3]
"""
if t.is_leaf(): yield t.label else: left, right = t.branches yield from in_order_traversal(left) yield t.label yield from in_order_traversal(right)

Use Ok to test your code:

python3 ok -q in_order_traversal

# Optional Questions

The following questions are for extra practice -- they can be found in the the lab07_extra.py file. It is recommended that you complete these problems on your own time.

### Q9: Remove All

Implement a function remove_all that takes a Link, and a value, and remove any linked list node containing that value. You can assume the list already has at least one node containing value and the first element is never removed. Notice that you are not returning anything, so you should mutate the list.

"""Remove all the nodes containing value. Assume there exists some
nodes to be removed and the first element is never removed.

>>> print(l1)
<0 2 2 3 1 2 3>
>>> remove_all(l1, 2)
>>> print(l1)
<0 3 1 3>
>>> remove_all(l1, 3)
>>> print(l1)
<0 1>
"""

Use Ok to test your code:

python3 ok -q remove_all

### Q10: Mutable Mapping

Implement deep_map_mut(fn, link), which applies a function fn onto all elements in the given linked list link. If an element is itself a linked list, apply fn to each of its elements, and so on.

Hint: The built-in isinstance function may be useful.

True
>>> isinstance(s, int)
False
"""Mutates a deep link by replacing each item found with the
result of calling fn on the item.  Does NOT create new Links (so

Does not return the modified Link object.

>>> deep_map_mut(lambda x: x * x, link1)
<9 <16> 25 36>
"""

Use Ok to test your code:

python3 ok -q deep_map_mut

### Q11: Cycles

The Link class can represent lists with cycles. That is, a list may contain itself as a sublist.

>>> s.rest.rest.rest = s
>>> s.rest.rest.rest.rest.rest.first
3

Implement has_cycle,that returns whether its argument, a Link instance, contains a cycle.

Hint: Iterate through the linked list and try keeping track of which Link objects you've already seen.

"""Return whether link contains a cycle.

>>> s.rest.rest.rest = s
>>> has_cycle(s)
True
>>> has_cycle(t)
False
>>> has_cycle(u)
False
"""

Use Ok to test your code:

python3 ok -q has_cycle

As an extra challenge, implement has_cycle_constant with only constant space. (If you followed the hint above, you will use linear space.) The solution is short (less than 20 lines of code), but requires a clever idea. Try to discover the solution yourself before asking around:

"""Return whether link contains a cycle.

>>> s.rest.rest.rest = s
>>> has_cycle_constant(s)
True
>>> has_cycle_constant(t)
False
"""
if link is Link.empty: return False slow, fast = link, link.rest while fast is not Link.empty: if fast.rest == Link.empty: return False elif fast is slow or fast.rest is slow: return True else: slow, fast = slow.rest, fast.rest.rest return False

Use Ok to test your code:

python3 ok -q has_cycle_constant

## Tree Practice

### Q12: Reverse Other

Write a function reverse_other that mutates the tree such that labels on every other (odd-depth) level are reversed. For example, Tree(1,[Tree(2, [Tree(4)]), Tree(3)]) becomes Tree(1,[Tree(3, [Tree(4)]), Tree(2)]). Notice that the nodes themselves are not reversed; only the labels are.

def reverse_other(t):
"""Mutates the tree such that nodes on every other (odd-depth) level
have the labels of their branches all reversed.

>>> t = Tree(1, [Tree(2), Tree(3), Tree(4)])
>>> reverse_other(t)
>>> t
Tree(1, [Tree(4), Tree(3), Tree(2)])
>>> t = Tree(1, [Tree(2, [Tree(3, [Tree(4), Tree(5)]), Tree(6, [Tree(7)])]), Tree(8)])
>>> reverse_other(t)
>>> t
Tree(1, [Tree(8, [Tree(3, [Tree(5), Tree(4)]), Tree(6, [Tree(7)])]), Tree(2)])
"""