Recursion¶
A recursive function is a function that calls itself.
Why would we want that?
To solve problems that can be broken down to smaller (easier) versions of the same problem
A recursive approach to problem solving has two main parts:
Base case(s). When the problem is so small, we solve it directly, without having to reduce it any further
Recursive step. Does the following things:
Performs an action that contributes to the solution
Reduces the problem to a smaller version of the same problem, and calls the function on this smaller subproblem
The recursive step is a form of “wishful thinking”: assuming the recursion fairy will take care of the smaller problem
In CS136/256, the recursion fairy will be introduced more formally as the induction (or inductive) hypothesis
From Recursive Ideas to Recursive Programs¶
Writing recursive functions is easy once you pin down the underlying recursive ideas! Try to spend some time thinking of the recursive definition of the program you’re trying to implement with pen and paper first before you try to write any code.
Example: \(a^n\)¶
The recursive definition:
\[\begin{split}\begin{align*}
a^0 &= 1 \\
a^n &= a * a^{n-1}
\end{align*}
\end{split}\]
which can be translated into the following recursive function.
def power(a, n):
"""
Returns a^n. Assumes n >= 0.
"""
if n == 0:
return 1
else:
return a * power(a, n-1)
print(power(5, 0))
print(power(5, 4))
1
625
# what happens when we're given an input n < 0?
print(power(5, -1))
---------------------------------------------------------------------------
RecursionError Traceback (most recent call last)
/var/folders/md/kwd9nc_d2ns0hw9wsvdrnt2c0000gn/T/ipykernel_93581/730912060.py in <module>
1 # what happens when we're given an input n < 0?
----> 2 print(power(5, -1))
/var/folders/md/kwd9nc_d2ns0hw9wsvdrnt2c0000gn/T/ipykernel_93581/577988384.py in power(a, n)
6 return 1
7 else:
----> 8 return a * power(a, n-1)
... last 1 frames repeated, from the frame below ...
/var/folders/md/kwd9nc_d2ns0hw9wsvdrnt2c0000gn/T/ipykernel_93581/577988384.py in power(a, n)
6 return 1
7 else:
----> 8 return a * power(a, n-1)
RecursionError: maximum recursion depth exceeded in comparison
Example: Fibonacci series¶
The recursive definition:
\[\begin{split}\begin{align*}
F_0 &= 0 \\
F_1 &= 1 \\
F^n &= F_{n-1} + F_{n-2}
\end{align*}
\end{split}\]
which can be translated into the following recursive function.
def fibonacci(n):
"""
Returns nth Fibonnaci number
"""
if n <= 1:
return n
else:
return fibonacci(n-1) + fibonacci(n-2)
print(fibonacci(5))
print(fibonacci(6))
print(fibonacci(7))
5
8
13
Recursion vs Iteration¶
Example 1:countDown : Write a function that prints integers from n down to 1
Iterative approach: Using loops
def countDownIterative(n):
for i in range(n, 0, -1):
print(i)
countDownIterative(5)
5
4
3
2
1
First recursion: countDown¶
Notice the repeated printing, despite the lack of an explicit for or while loop in the body of the function.
def countDown(n):
'''Prints numbers from n down to 1'''
if n < 1: # Base case
pass # Do nothing
else: # Recursive case: n >= 1:
print(n)
countDown(n-1)
countDown(5)
5
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2
1
countDown(-5)
countDown(10)
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2
1
Side-note: simplifying the code by changing the base case¶
If the base case does nothing, we can re-write the conditional. The reason we prefer an explicit base case (for now) in recursion is pedagogical: to ensure that students never forget about the base case.
def countDown(n):
'''Prints numbers from n down to 1'''
if n > 0:
print(n)
countDown(n-1)
countDown(6)
6
5
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3
2
1
Recursive countUp¶
Define a recursive function countUp that counts up from 1 to n rather than n to 1. countUp(5) should print:
1
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4
5
def countUp(n):
'''Prints out integers from 1 up to n'''
if n < 1:
pass # do nothing
else:
countUp(n-1)
print(n)
countUp(5)
1
2
3
4
5
countUp(-5)
countDownUp¶
How about a recursive function that counts down and up? This one is a bit more complex, but very elegant.
def countDownUp(n):
"""Prints integers from n down to 1
and then from 1 up to n."""
if n < 1:
pass # do nothing
else:
print(n)
countDownUp(n-1)
print(n)
countDownUp(4)
4
3
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1
1
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4
countDownUp(10)
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10
Gotcha-s in Recursion¶
Gotcha #1: Subproblem in not getting smaller¶
Below is an example of a common mistake when using recursion.
Can you figure out what is wrong in the code without running it?
def countDownGotcha(n):
'''Prints numbers from n down to 1'''
if n <= 0: # Base case
pass # Do nothing
else: # Recursive case: n>0:
print(n)
countDownGotcha(n)
countDownGotcha(10)
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---------------------------------------------------------------------------
RecursionError Traceback (most recent call last)
/var/folders/md/kwd9nc_d2ns0hw9wsvdrnt2c0000gn/T/ipykernel_93581/905308095.py in <module>
----> 1 countDownGotcha(10)
/var/folders/md/kwd9nc_d2ns0hw9wsvdrnt2c0000gn/T/ipykernel_93581/2020836085.py in countDownGotcha(n)
5 else: # Recursive case: n>0:
6 print(n)
----> 7 countDownGotcha(n)
... last 1 frames repeated, from the frame below ...
/var/folders/md/kwd9nc_d2ns0hw9wsvdrnt2c0000gn/T/ipykernel_93581/2020836085.py in countDownGotcha(n)
5 else: # Recursive case: n>0:
6 print(n)
----> 7 countDownGotcha(n)
RecursionError: maximum recursion depth exceeded while calling a Python object
If you run the line above, you should see toward the end of the output, the dreaded error:
RuntimeError: maximum recursion depth exceeded while calling a Python object
It means that the memory allocated to your program doesn’t have space anymore for all the opened execution frames that are opened while your function is recursively invoking itself, endlessly!
Restart the kernel if you ran this code and got the error.
Gotcha #2: Missing/Unreachable the base case¶
The following example will also lead to infinite recursion. Can you explain why?
How can you fix the problem?
def printHalvesGotcha(n):
"""Prints n, n/2, down to ... 1"""
if n > 0:
print(n)
printHalvesGotcha(n/2)
printHalvesGotcha(10)
10
5.0
2.5
1.25
0.625
0.3125
0.15625
0.078125
0.0390625
0.01953125
0.009765625
0.0048828125
0.00244140625
0.001220703125
0.0006103515625
0.00030517578125
0.000152587890625
7.62939453125e-05
3.814697265625e-05
1.9073486328125e-05
9.5367431640625e-06
4.76837158203125e-06
2.384185791015625e-06
1.1920928955078125e-06
5.960464477539062e-07
2.980232238769531e-07
1.4901161193847656e-07
7.450580596923828e-08
3.725290298461914e-08
1.862645149230957e-08
9.313225746154785e-09
4.6566128730773926e-09
2.3283064365386963e-09
1.1641532182693481e-09
5.820766091346741e-10
2.9103830456733704e-10
1.4551915228366852e-10
7.275957614183426e-11
3.637978807091713e-11
1.8189894035458565e-11
9.094947017729282e-12
4.547473508864641e-12
2.2737367544323206e-12
1.1368683772161603e-12
5.684341886080801e-13
2.8421709430404007e-13
1.4210854715202004e-13
7.105427357601002e-14
3.552713678800501e-14
1.7763568394002505e-14
8.881784197001252e-15
4.440892098500626e-15
2.220446049250313e-15
1.1102230246251565e-15
5.551115123125783e-16
2.7755575615628914e-16
1.3877787807814457e-16
6.938893903907228e-17
3.469446951953614e-17
1.734723475976807e-17
8.673617379884035e-18
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3.1861838222649046e-57
1.5930919111324523e-57
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3.982729777831131e-58
1.9913648889155653e-58
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4.978412222288913e-59
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1.2446030555722283e-59
6.223015277861142e-60
3.111507638930571e-60
1.5557538194652854e-60
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3.8893845486632136e-61
1.9446922743316068e-61
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4.861730685829017e-62
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1.2154326714572542e-62
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1.5192908393215678e-63
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3.7982270983039195e-64
1.8991135491519597e-64
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4.747783872879899e-65
2.3738919364399497e-65
1.1869459682199748e-65
5.934729841099874e-66
2.967364920549937e-66
1.4836824602749686e-66
7.418412301374843e-67
3.7092061506874214e-67
1.8546030753437107e-67
9.273015376718553e-68
4.636507688359277e-68
2.3182538441796384e-68
1.1591269220898192e-68
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