Suppose a hash table is constructed so that collisions are resolved through chaining. The hash table has \(10\) bins, and 100 items are inserted into the hash table.

True or False: it is possible for one of the bins to contain all 100 items.

Suppose a hash table in constructed so that collisions are resolved through chaining. The hash table has a number of bins \(k\) that is \(\Theta(n)\), where \(n\) is the number of elements being stored in the hash table; that is, the hash table grows as more elements are inserted.

Suppose the hash function uses only the first \(k / 2\) bins of the hash table, but appears to hash items into these bins evenly.

What is the expected time complexity of a membership query on this hash table? State your answer in asymptotic notation in terms of the number of elements in the hash table, \(n\).

Consider the code below:

def intersection(A, B):
    common = set()
    for x in A:
        if x in B:
            common.add(x)
    return common

What is the expected time complexity of this code if A and B are Python sets (hash tables) with \(n\) elements each? State your answer as a function of \(n\) using asymptotic notation. You may assume that the hash function used is ``good'' in that it hashes evenly (that is, it satisfies the simple universal hashing assumption).

What is the expected time complexity of the code shown below, assuming that numbers is a Python list? Express your answer as a function of \(n\) using asymptotic notation (e.g., \(\Theta(n)\)).

def foo(numbers):
    """Assume that `numbers` is a list containing n numbers."""
    # counts is a Python dictionary, which is implemented using hash table
    counts = {}
    for x in numbers:
        for y in numbers:
            diff = abs(x - y)
            if diff not in counts:
                counts[diff] = 1
            else:
                counts[diff] = counts[diff] + 1
    return counts

foo([1, 2, 3, 4])

Suppose a hash table with \(n/2\) bins stores \(n\) elements, and suppose that when the next element is inserted into the hash table, a ``grow'' will be triggered. That is, the insertion will cause the hash table to resize to \(n\) bins containing the same \(n + 1\) elements, plus the newly-added element.

What will be the time complexity of this single insertion operation that causes the table to grow?

Suppose \(T\) is a hash table of size 100 (that is, it has 100 bins). 300 elements are inserted into the hash table using a hash function that satisfies the uniform hashing assumption (that is, any element is equally-likely to hash into any bin). What is the expected number of elements in the first bin?

Suppose a hash table is constructed using linked lists to store the data within each bin, and that collisions are resolved by chaining. What is the worst case time complexity of inserting a new element into the hash table? You may assume that the hash table will not need to be resized, and that evaluating the hash function takes \(\Theta(1)\) time.