# Topics

Consult this section if you need a refresher on the material for this lab. It's okay to skip directly to the questions and refer back here should you get stuck.

## Recursion

A recursive function is a function that calls itself in its body, either directly or indirectly. Recursive functions have three important components:

1. Base case(s), the simplest possible form of the problem you're trying to solve.
2. Recursive case(s), where the function calls itself with a simpler argument as part of the computation.
3. Using the recursive calls to solve the full problem.

Let's look at the canonical example, `factorial`.

Factorial, denoted with the `!` operator, is defined as:

``n! = n * (n-1) * ... * 1``

For example, `5! = 5 * 4 * 3 * 2 * 1 = 120`

The recursive implementation for factorial is as follows:

``````def factorial(n):
if n == 0:
return 1
return n * factorial(n - 1)``````

We know from its definition that 0! is 1. Since `n == 0` is the smallest number we can compute the factorial of, we use it as our base case. The recursive step also follows from the definition of factorial, i.e., ```n! = n * (n-1)!```.

The next few questions in lab will have you writing recursive functions. Here are some general tips:

• Paradoxically, to write a recursive function, you must assume that the function is fully functional before you finish writing it; this is called the recursive leap of faith.
• Consider how you can solve the current problem using the solution to a simpler version of the problem. The amount of work done in a recursive function can be deceptively little: remember to take the leap of faith and trust the recursion to solve the slightly smaller problem without worrying about how.
• Think about what the answer would be in the simplest possible case(s). These will be your base cases - the stopping points for your recursive calls. Make sure to consider the possibility that you're missing base cases (this is a common way recursive solutions fail).
• It may help to write an iterative version first.

## Tree Recursion

A tree recursive function is a recursive function that makes more than one call to itself, resulting in a tree-like series of calls.

A classic example of a tree recursion function is finding the nth Fibonacci number:

``````def fib(n):
if n == 0 or n == 1:
return n
return fib(n - 1) + fib(n - 2)``````

Calling `fib(6)` results in the following call structure (where `f` is `fib`): Each `f(i)` node represents a recursive call to `fib`. Each recursive call makes another two recursive calls. `f(0)` and `f(1)` do not make any recursive calls because they are the base cases of the function. Because of these base cases, we are able to terminate the recursion and beginning accumulating the values.

Generally, tree recursion is effective when you want to explore multiple possibilities or choices at a single step. In these types of problems, you make a recursive call for each choice or for a group of choices. Here are some examples:

• Given a list of paid tasks and a limited amount of time, which tasks should you choose to maximize your pay? This is actually a variation of the Knapsack problem, which focuses on finding some optimal combination of different items.
• Suppose you are lost in a maze and see several different paths. How do you find your way out? This is an example of path finding, and is tree recursive because at every step, you could have multiple directions to choose from that could lead out of the maze.
• Your dryer costs \$2 per cycle and accepts all types of coins. How many different combinations of coins can you create to run the dryer? This is similar to the partitions problem from the textbook.

# Required Questions

## Recursion

Write a function `skip_add` that takes a single argument `n` and computes the sum of every other integer between `0` and `n`. Assume `n` is non-negative.

``````this_file = __file__

""" Takes a number n and returns n + n-2 + n-4 + n-6 + ... + 0.

>>> skip_add(5)  # 5 + 3 + 1 + 0
9
>>> skip_add(10) # 10 + 8 + 6 + 4 + 2 + 0
30
>>> # Do not use while/for loops!
>>> from construct_check import check
>>> # ban iteration
...       ['While', 'For'])
True
"""
if n <= 0:
return 0
return n + skip_add(n - 2)``````

Use Ok to test your code:

``python3 ok -q skip_add``

### Q2: Summation

Now, write a recursive implementation of `summation`, which takes a positive integer `n` and a function `term`. It applies `term` to every number from `1` to `n` including `n` and returns the sum of the results.

``````def summation(n, term):

"""Return the sum of the first n terms in the sequence defined by term.
Implement using recursion!

>>> summation(5, lambda x: x * x * x) # 1^3 + 2^3 + 3^3 + 4^3 + 5^3
225
>>> summation(9, lambda x: x + 1) # 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
54
>>> summation(5, lambda x: 2**x) # 2^1 + 2^2 + 2^3 + 2^4 + 2^5
62
>>> # Do not use while/for loops!
>>> from construct_check import check
>>> # ban iteration
>>> check(this_file, 'summation',
...       ['While', 'For'])
True
"""
assert n >= 1
if n == 1:
return term(n)
else:
return term(n) + summation(n - 1, term)
# Base case: only one item to sum, so we return that item.
# Recursive call: returns the result of summing the numbers up to n-1 using
# term. All that's missing is term applied to the current value n.``````

Use Ok to test your code:

``python3 ok -q summation``

### Q3: GCD

The greatest common divisor of two positive integers `a` and `b` is the largest integer which evenly divides both numbers (with no remainder). Euclid, a Greek mathematician in 300 B.C., realized that the greatest common divisor of `a` and `b` is one of the following:

• the smaller value if it evenly divides the larger value, or
• the greatest common divisor of the smaller value and the remainder of the larger value divided by the smaller value

In other words, if `a` is greater than `b` and `a` is not divisible by `b`, then

``gcd(a, b) = gcd(b, a % b)``

Write the `gcd` function recursively using Euclid's algorithm.

``````def gcd(a, b):
"""Returns the greatest common divisor of a and b.
Should be implemented using recursion.

>>> gcd(34, 19)
1
>>> gcd(39, 91)
13
>>> gcd(20, 30)
10
>>> gcd(40, 40)
40
"""
a, b = max(a, b), min(a, b)
if a % b == 0:
return b
else:
return gcd(b, a % b)

# Iterative solution, if you're curious
def gcd_iter(a, b):
"""Returns the greatest common divisor of a and b, using iteration.

>>> gcd_iter(34, 19)
1
>>> gcd_iter(39, 91)
13
>>> gcd_iter(20, 30)
10
>>> gcd_iter(40, 40)
40
"""
if a < b:
return gcd_iter(b, a)
while a > b and not a % b == 0:
a, b = b, a % b
return b

# Video Walkthrough: https://youtu.be/yBhhfBObNxs``````

Use Ok to test your code:

``python3 ok -q gcd``

## Tree Recursion

### Q4: Insect Combinatorics

Consider an insect in an M by N grid. The insect starts at the bottom left corner, (0, 0), and wants to end up at the top right corner, (M-1, N-1). The insect is only capable of moving right or up. Write a function `paths` that takes a grid length and width and returns the number of different paths the insect can take from the start to the goal. (There is a closed-form solution to this problem, but try to answer it procedurally using recursion.) For example, the 2 by 2 grid has a total of two ways for the insect to move from the start to the goal. For the 3 by 3 grid, the insect has 6 diferent paths (only 3 are shown above).

``````def paths(m, n):
"""Return the number of paths from one corner of an
M by N grid to the opposite corner.

>>> paths(2, 2)
2
>>> paths(5, 7)
210
>>> paths(117, 1)
1
>>> paths(1, 157)
1
"""
if m == 1 or n == 1:
return 1
return paths(m - 1, n) + paths(m, n - 1)``````

Use Ok to test your code:

``python3 ok -q paths``

### Q5: Maximum Subsequence

A subsequence of a number is a series of (not necessarily contiguous) digits of the number. For example, 12345 has subsequences that include 123, 234, 124, 245, etc. Your task is to get the maximum subsequence below a certain length.

``````def max_subseq(n, l):
"""
Return the maximum subsequence of length at most l that can be found in the given number n.
For example, for n = 20125 and l = 3, we have that the subsequences are
2
0
1
2
5
20
21
22
25
01
02
05
12
15
25
201
202
205
212
215
225
012
015
025
125
and of these, the maxumum number is 225, so our answer is 225.

>>> max_subseq(20125, 3)
225
>>> max_subseq(20125, 5)
20125
>>> max_subseq(20125, 6) # note that 20125 == 020125
20125
>>> max_subseq(12345, 3)
345
>>> max_subseq(12345, 0) # 0 is of length 0
0
>>> max_subseq(12345, 1)
5
"""
if n == 0 or l == 0:
return 0
with_last = max_subseq(n // 10, l - 1) * 10 + n % 10
without_last = max_subseq(n // 10, l)
return max(with_last, without_last)``````

Use Ok to test your code:

``python3 ok -q max_subseq``
There are two key insights for this problem
• You need to split into the cases where the ones digit is used and the one where it is not. In the case where it is, we want to reduce `l` since we used one of the digits, and in the case where it isn't we do not.
• In the case where we are using the ones digit, you need to put the digit back onto the end, and the way to attach a digit `d` to the end of a number `n` is `10 * n + d`.

## Submit

Make sure to submit this assignment by running:

``python3 ok --submit``