CS 61A Lab 7

Sets

Starter Files

We've provided a set of starter files with skeleton code for the exercises in the lab. You can get them in the following places:

• sets.py

Sets

A set is an unordered collection of distinct objects that supports membership testing, union, intersection, and adjunction.

Construction

>>> s = {3, 2, 1, 4, 4}
>>> s
{1, 2, 3, 4}

Adjunction

>>> s.add(5)
>>> s
{1, 2, 3, 4, 5}

Operations

>>> 3 in s
True
>>> 7 not in s
True
>>> len(s)
5
>>> s.union({1, 5})
{1, 2, 3, 4, 5}
>>> s.intersection({6, 5, 4, 3})
{3, 4, 5}

For more detail on Sets you can go to the following link: PythonSets

Question 1

Implement the union function for sets. Union takes in two sets, and returns a new set with elements from the first set, and all other elements that have not already have been seen in the second set.

def union(s1, s2):
    """Returns the union of two sets.

    >>> r = {0, 6, 6}
    >>> s = {1, 2, 3, 4}
    >>> t = union(s, {1, 6})
    >>> t
    {1, 2, 3, 4, 6}
    >>> union(r, t)
    {0, 1, 2, 3, 4, 6}
    """
    "*** YOUR CODE HERE ***"

def union(s1,s2):
    union_set = set()
    for elem in s1:
        union_set.add(elem)
    for elem in s2:
        union_set.add(elem)
    return union_set

Question 2

Implement the intersection function for two sets. Intersection takes in two sets and returns a new set of only the elements in both sets.

def intersection(s1, s2):
    """Returns the intersection of two sets.

    >>> r = {0, 1, 4, 0}
    >>> s = {1, 2, 3, 4}
    >>> t = intersection(s, {3, 4, 2})
    >>> t
    {2, 3, 4}
    >>> intersection(r, t)
    {4}
    """
    "*** YOUR CODE HERE ***"

def intersection(s1, s2):
    intersection_set = set()
    for elem in s1:
        if elem in s2:
            intersection_set.add(elem)
    return intersection_set

Orders of Growth

One really convenient thing about sets is that many operations on sets (adding elements, removing elements, checking membership) run in O(1) (constant) time.

Question 3

Write the following function so it runs in O(n) time.

def extra_elem(a,b):
    """B contains every element in A, and has one additional member, find
    the additional member.

    >>> extra_elem(['dog', 'cat', 'monkey'], ['dog', 'cat', 'monkey', 'giraffe'])
    'giraffe'
    >>> extra_elem([1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6])
    6
    """
    "*** YOUR CODE HERE ***"

def extra_elem(a,b):
    """B contains every element in A, and has one additional member, find
    the additional member.

    >>> extra_elem(['dog', 'cat', 'monkey'], ['dog', 'cat', 'monkey', 'giraffe'])
    'giraffe'
    >>> extra_elem([1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6])
    6
    """
    return list(set(b) - set(a))[0]

Question 4

Write the following function so it runs in O(n) time.

def add_up(n, lst):
    """Returns True if any two non identical elements in lst add up to any n.

    >>> add_up(100, [1, 2, 3, 4, 5])
    False
    >>> add_up(7, [1, 2, 3, 4, 2])
    True
    >>> add_up(10, [5, 5])
    False
    """
    "*** YOUR CODE HERE ***"

def add_up(n, lst):
    """Returns True if any two non identical elements in lst add up to any n.

    >>> add_up(100, [1, 2, 3, 4, 5])
    False
    >>> add_up(7, [1, 2, 3, 4, 2])
    True
    >>> add_up(10, [5, 5])
    False
    """
    check_set = set()
    for elem in lst:
        if n - elem != elem:
            check_set.add(n - elem)
    return bool(intersection(check_set, set(lst)))

Question 5

Write the following function so it runs in O(n) time.

def find_duplicates(lst):
    """Returns True if lst has any duplicates and False if it does not.

    >>> find_duplicates([1, 2, 3, 4, 5])
    False
    >>> find_duplicates([1, 2, 3, 4, 2])
    True
    """
    "*** YOUR CODE HERE ***"

def find_duplicates(lst):
    """Returns True if lst has any duplicates and False if it does not.

    >>> find_duplicates([1, 2, 3, 4, 5])
    False
    >>> find_duplicates([1, 2, 3, 4, 2])
    True
    """
    return len(set(lst)) != len(lst)

Question 6

Write the following function so it runs in O(n) time.

def find_duplicates_k(k, lst):
    """Returns True if there are any duplicates in lst that are within k
    indices apart.

    >>> find_duplicates_k(3, [1, 2, 3, 4, 1])
    False
    >>> find_duplicates_k(4, [1, 2, 3, 4, 1])
    True
    """
    "*** YOUR CODE HERE ***"

def find_duplicates_k(k, lst):
    """Returns True if there are any duplicates in lst that are within k
    indices apart.

    >>> find_duplicates_k(3, [1, 2, 3, 4, 1])
    False
    >>> find_duplicates_k(4, [1, 2, 3, 4, 1])
    True
    """
    prev_set = set()
    for i, elem in enumerate(lst):
        if elem in prev_set:
            return True
        prev_set.add(elem)
        if i - k >= 0:
            prev_set.remove(lst[i - k])
    return False

Question 7

Write the following function so it runs in O(log n) time.

def pow(n,k):
    """Computes n^k.

    >>> pow(2, 3)
    8
    >>> pow(4, 2)
    16
    """
    "*** YOUR CODE HERE ***"

def pow(n,k):
    """Computes n^k.

    >>> pow(2, 3)
    8
    >>> pow(4, 2)
    16
    """
    if k == 1:
        return n
    if k % 2 == 0:
        return pow(n*n,k//2)
    else:
        return n * pow(n*n, k//2)

Question 8

Write the following function so it runs in O(n) time.

def missing_no(lst):
    """lst contains all the numbers from 0 to n for some n except some
    number k. Find k.

    >>> missing_no([1, 0, 4, 5, 7, 9, 2, 6, 3])
    8
    >>> missing_no(list(filter(lambda x: x != 293, list(range(2000)))))
    293
    """
    "*** YOUR CODE HERE ***"

def missing_no(lst):
    """lst contains all the numbers from 0 to n for some n except some
    number k. Find k.

    >>> missing_no([1, 0, 4, 5, 7, 9, 2, 6, 3])
    8
    >>> missing_no(list(filter(lambda x: x != 293, list(range(2000)))))
    293
    """
    return sum(range(max(lst) + 1)) - sum(lst)

Question 9: Extra for Experts

Write the following function so it runs in O(n) time.

def find_duplicates_k_l(k, l, lst):
    """Returns True if there are any two values who in lst that are within k
    indices apart AND if the absolute value of their difference is less than
    or equal to l.

    >>> find_duplicates_k_l(4, 0, [1, 2, 3, 4, 5])
    False
    >>> find_duplicates_k_l(4, 1, [1, 2, 3, 4, 5])
    True
    >>> find_duplicates_k_l(4, 0, [1, 2, 3, 4, 1])
    True
    >>> find_duplicates_k_l(2, 0, [1, 2, 3, 4, 1])
    False
    >>> find_duplicates_k_l(1, 100, [100, 275, 320, 988, 27])
    True
    >>> find_duplicates_k_l(1, 100, [100, 199, 275, 320,988,27])
    True
    >>> find_duplicates_k_l(1, 100, [100, 23, 199, 275,320,988,27])
    True
    >>> find_duplicates_k_l(2, 100, [100, 23, 199, 275,320,988,27])
    True
    """
    "*** YOUR CODE HERE ***"