Consider the code below, which shows binary_search as discussed in lecture, but with one additional line: a print statement.

import math
def binary_search(arr, t, start, stop):
    if stop - start <= 0:
        return None
    middle = math.floor((start + stop)/2)
    print(arr[middle]) # <---- this line is new
    if arr[middle] == t:
        return middle
    elif arr[middle] > t:
        return binary_search(arr, t, start, middle)
    else:
        return binary_search(arr, t, middle+1, stop)

Suppose this version of binary search is called on an array and with a target of t = 7. Which of the following statements is true?

Suppose we modify binary_search so that instead of using the floor operation to find the middle element of the array, we use the ceiling operation (recall that the ceiling of a real number \(x\) is the smallest integer that is \(\geq x\)). The full code is shown below for convenience:

import math

def new_binary_search(arr, start, stop, target):
    if stop <= start:
        return None

    middle = math.ceil((start + stop) / 2)

    if arr[middle] == target:
        return middle
    elif arr[middle] < target:
        return new_binary_search(arr, middle + 1, stop, target)
    else:
        return new_binary_search(arr, start, middle, target)

Which of the following statements about the correctness of new_binary_search is true? You may assume that arr will be a list containing at least one number, and that the initial call to new_binary_search will be with start = 0 and stop = len(arr) - 1.

Suppose binary_search has been called on an array arr with a target t that is not necessarily in the array, and with start and stop indices start and stop, respectively (remember, our convention is that the start index is included, while the stop index is excluded from the search). Which of the following must be true? Mark all which apply. You may assume that stop - start\(\geq 1\).

Note: remember that any statement about an empty array is considered to be automatically true. Also, you cannot assume that this call to binary_search is the root call (it could be a recursive call).

Suppose binary_search is called on a sorted array of size \(n\). In total, how many recursive calls to binary_search are made in the worst case?

Consider the problem of searching for an element t in a sorted array, with the added requirement that if t is in the array multiple times, we return the index of the first occurrence. For example, if arr = [3, 4, 4, 4, 5, 8], and t = 4, the code should return 1, since the first 4 is at index one.

binary_search as implemented in lecture may not necessarily return the index of the first occurrence of t, but it can be modified so that it does.

Fill in the blanks below so that the function returns the index of the first occurence of t. Hint: last_seen should keep track of the index of the last known occurrence of t.

import math
def mbs(arr, t, start, stop, last_seen=None):
    if start - stop <= 0:
        return last_seen

    middle = math.floor((stop - start) / 2)

    if arr[middle] == t:
        # _________ <-- fill this in
    elif arr[middle] < t:
        # _________ <-- fill this in
    else:
        # _________ <-- fill this in

if arr[middle] == t:
    return mbs(arr, t, start, middle, middle)
elif arr[middle] < t:
    return mbs(arr, t, middle + 1, stop, last_seen)
else:
    return mbs(arr, t, start, middle, last_seen)

Consider the version of binary search shown below:

import math
def binary_search(arr, t, start, stop):
    """
    Searches arr[start:stop] for t.
    Assumes arr is sorted.
    """
    if stop - start <= 0:
        return None
    middle = math.floor((start + stop)/2)
    if arr[middle] == t:
        return middle
    elif arr[middle] > t:
        return binary_search(arr, t, start, middle)
    else:
        return binary_search(arr, t, middle+1, stop)

True or false: if the target element t occurs multiple times in the array, binary_search is guaranteed to return the first occurrence.

You may assume that arr is sorted and that binary_search(arr, t, 0, len(arr)) is called.

Consider the iterative implementation of binary search shown below:

import math

def iterative_binary_search(arr, target):

    start = 0
    stop = len(arr)

    while (stop - start) > 0:
        print(arr[start])
        middle = math.floor((start + stop) / 2)
        if arr[middle] == target:
            return middle
        elif arr[middle] > target:
            stop = middle
        else:
            start = middle + 1

    return None

Which of the following loop invariants is true, assuming that arr is sorted and non-empty, and target is not in the array? Select all that apply.

Consider iterative_binary_search below and note the print statement in the while-loop:

import math

def iterative_binary_search(arr, target):

    start = 0
    stop = len(arr)

    while (stop - start) > 0:
        print(arr[start])
        middle = math.floor((start + stop) / 2)
        if arr[middle] == target:
            return middle
        elif arr[middle] > target:
            stop = middle
        else:
            start = middle + 1

    return None

Suppose iterative_binary_search is run on the array:

[-202, -201, -200, -50, -20, -10, -4, -3, 0, 1, 3, 5, 6, 7, 9, 10, 12, 15, 22]

with target 11.

What will be the last value of arr[start] printed?

Suppose we modify binary_search so that instead of using the floor operation to find the middle element of the array, we use the ceiling operation (recall that the ceiling of a real number \(x\) is the smallest integer that is \(\geq x\)). The full code is shown below for convenience:

import math

def new_binary_search(arr, start, stop, target):
    if stop <= start:
        return None

    middle = math.ceil((start + stop) / 2)

    if arr[middle] == target:
        return middle
    elif arr[middle] < target:
        return new_binary_search(arr, middle + 1, stop, target)
    else:
        return new_binary_search(arr, start, middle, target)

You may assume that arr will be a sorted list containing at least one number, and that the initial call to new_binary_search will be with start = 0 and stop = len(arr).