Python supports recursive functions; functions that call themselves in order to loop. Recursion is an advanced and somewhat uncommon practice in Python as other looping patterns are often simpler, and faster. That said, recursive functions are very useful for traversing unknown shapes and structures.
You will be able to:
- Understand and use the concept of a recursive function
- Understand scope in the context of recursive functions
- Understand and compare depth first versus breadth first searches
Let's take a look at a simple case to start.
def mysum(L):
if not L:
return 0
else:
return L[0] + mysum(L[1:])
mysum([1,2,3,4,5,6,7,8,9])45
This function takes a list of numbers and recursively calls itself to add the next item to a summation. The first condition if not L allows us to terminate the recursive function; when the list is empty we add 0 to our sum and terminate, as no more recursive calls are made. To better understand this, let's add a couple of print statements to our function.
def mysum(L):
print(L)
if not L:
return 0
else:
return L[0] + mysum(L[1:])
mysum([1,2,3,4,5,6,7,8,9])[1, 2, 3, 4, 5, 6, 7, 8, 9]
[2, 3, 4, 5, 6, 7, 8, 9]
[3, 4, 5, 6, 7, 8, 9]
[4, 5, 6, 7, 8, 9]
[5, 6, 7, 8, 9]
[6, 7, 8, 9]
[7, 8, 9]
[8, 9]
[9]
[]
45
Each new call to your function creates a new local scope. In other words, this creates a new namespace for the variables within that function call. This can be seen above as our print(L) statement continually shows L within the local context of that particular function call.
In this case, a summation problem could be easily replaced by using a while loop. This eliminates the need to define function that then makes recursive calls to itself (which creates new scopes). In practice, this can also increase execution performance time.
L = [1,2,3,4,5,6,7,8,9]
sum = 0
while L:
sum += L[0]
L = L[1:]
sum45
We could also use a fairly simple for loop:
L = [1,2,3,4,5,6,7,8,9]
sum = 0
for x in L:
sum += x
sum45
Let's pretend we want our mysum() function to sum nested lists, or arbitrary shape and depth. In other words, our list might contain numbers, lists of numbers, lists of lists of numbers, etc. If we want to write such a function, our simple while or for loop will no longer suffice. Observe:
L = [1,[2,3,4],5,[6,[7,8],9]]
sum = 0
for x in L:
sum += x
sum---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-5-f1a29f127e30> in <module>()
2 sum = 0
3 for x in L:
----> 4 sum += x
5 sum
TypeError: unsupported operand type(s) for +=: 'int' and 'list'
We get an error because when we try to add the second item, python is unable to add a int and a list; the two are seperate data types. Our recursive function however, does just as good a job by successively iterating through our nested data:
def mysum(L):
print(L)
if not L:
return 0
elif type(L[0]) == list:
return mysum(L[0]) + mysum(L[1:])
else:
return L[0] + mysum(L[1:])
L = [1,[2,3,4],5,[6,[7,8],9]]
mysum(L)[1, [2, 3, 4], 5, [6, [7, 8], 9]]
[[2, 3, 4], 5, [6, [7, 8], 9]]
[2, 3, 4]
[3, 4]
[4]
[]
[5, [6, [7, 8], 9]]
[[6, [7, 8], 9]]
[6, [7, 8], 9]
[[7, 8], 9]
[7, 8]
[8]
[]
[9]
[]
[]
45
In general, recursive functions can also often be replaced with stacks which determine a queue for successive function calls. See the resources below which adds some more context on this.
Notice how we added a second branch in our conditional logic to further recurse on nested lists. You can also see how this works in practice by the print statements. Notice how the scope of L is redefined when we reach the second item at index 1; the nested list [2,3,4]. Once we reach that, we recurse to the mysum of this list and successively see printouts for [2,3,4], [3,4]], [4] and [] as the mysum() function sums this list within our master list.
Another important concept to touch upon and address at this point is the difference between depth and breadth first search trees. In our above example, we have searched in a depth first approach; when we hit the second item which was a nested list, we continued all the way down the rabbit hole before moving on. A depth first search would be if we processed everything at the first level, putting nested lists on hold till later. Here's the same function rewritten as a breadth first traversal:
#We want to process everything layer by layer.
#Thus we process everything in the list, and put off items for later processing if they are nested data structures.
def mysum(L, tot=0):
print(L)
deeper_data = [] #Initialize a container for nested data we come across
#Process the current depth layer
for i in L:
if type(i) != list:
tot += i #If its not a list, its a number! Add it up!
else:
[deeper_data.append(j) for j in i] #Add nested items to our list of things to process later on
#Notice we're digging in a layer here by iterating
#through i which we know is a nested data structure from our conditional
if deeper_data != []:
return mysum(deeper_data, tot=tot)
#If we have deeper_data we got to iterate!
else:
return tot
#If we don't have nested data, we're done!
L = [1,[2,3,4],5,[6,[7,8],9]]
mysum(L)[1, [2, 3, 4], 5, [6, [7, 8], 9]]
[2, 3, 4, 6, [7, 8], 9]
[7, 8]
45
Check out the link below for a further discussion about depth versus breadth first searches!
https://brilliant.org/wiki/depth-first-search-dfs/
Recursive functions make calls to themselves. Each call defines new scopes for variables and functions within these recursive calls. While useful for traversing unknown data structures, standard for and while loops will have faster execution times when .
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