Big O Notation for the Runtime fo Reversing a Linked List Recursively in Python

Question

Using an ADT Linked List and the following code, how would I express the runtime in Big O Notation?:

def reverse (self):
 (1)if self._head is not None:
      (2)temp = self._head.item
      (3)self._head = self._head.next
      (4)self.reverse()
      (5)self.append(temp)

My thought process: Lines 1 - 3 were essentially constant as they are just setting and getting items from the beginning of the Linked List, and line 5 is theta(n) by definition. Each time the list becomes smaller, so I thought the function runs n(n-1)(n-2).... implying that it is theta(n!). Could I get some help?

Mateusz Dymczyk · Answer 1 · 2013-10-20T05:22:16.050

1

This is a recursive function therefore line 4 isn't theta(n) by definition.

This will actually run in O(n).

Basically this function's complexity is:

T(n) = T(n - 1) + O(1) // T(n - 1) for the recursive call on a list one element shorter, constant for other operations.

To solve this we use induction:

T(n) = n * O(1) + T(0) = n * O(1) + O(1) = O(n)

For the append, indeed in the worst case it might be O(n) but the amortized worst case scenario is O(1).

edited Oct 20 '13 at 05:22

answered Oct 20 '13 at 05:10

Mateusz Dymczyk

14,969
10
59
94

Sorry I meant line 5 is theta n by definition – Ryan Nemec Oct 20 '13 at 05:12
1

from what I remember the amortized complexity of append in Python is constant not n – Mateusz Dymczyk Oct 20 '13 at 05:15
That's what my textbook said in Worst Case running time – Ryan Nemec Oct 20 '13 at 05:16
If you want to assume `append()` is `O(n)` *all* the time - which it isn't - but try arguing with a textbook ;-) - then overall the algorithm is `O(n**2)`. But how could a textbook possibly make claims about the cost of `append()` in an ADT?! You need a concrete implementation to decide that. – Tim Peters Oct 20 '13 at 05:25
Textbook implications + inability to do computer science = Me failing my Abstract Data Structures class – Ryan Nemec Oct 20 '13 at 05:26
just get a better textbook ;-) – Mateusz Dymczyk Oct 20 '13 at 05:27
But seriously, you guys rock, don't ever change – Ryan Nemec Oct 20 '13 at 05:38

Big O Notation for the Runtime fo Reversing a Linked List Recursively in Python

1 Answers1