Time complexity regarding linked_list ( Recursion )

Question

let us say I have a recursive call for linked list: ( Singular )

def append(self, val):
    def append_rec(node, val):
        if node.next is None:
            node.next = Node(val)
        else:
            append_rec(node.next, val)
    if self.head is None:
        self.head = Node(val)
        self.len += 1
    else:
        append_rec(self.head, val)
        self.len += 1

what is the time complexity of a recursive + linked list altogether?
I am at a little of problem of knowing how to find time complexity.
Please explain to me, thanks.

Asked By: Ben Shaines

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Source

Answer 1

The number of calls of append_rec corresponds to the number of nodes that are already in the list.

All other steps in this code can be considered O(1) operations, so the total time complexity is linear to the number of nodes already in the list, i.e. O(n).

It may help to visualise the process. Let’s say we already have a list with 2 nodes with values 1 and 2:

 self
  ↓
┌─────────┐   ┌────────────┐   ┌────────────┐
│ head: ─────►│ val: 1     │   │ val: 2     │
│ len:  2 │   │ next: ────────►│ next: None │
└─────────┘   └────────────┘   └────────────┘

Now let’s say we call the method append with value 3, then first we get into the else block of append where append_rec(self.head, val) is called. Inside append_rec we have node referencing the first node now:

 self          node
  ↓             ↓
┌─────────┐   ┌────────────┐   ┌────────────┐
│ head: ─────►│ val: 1     │   │ val: 2     │
│ len:  2 │   │ next: ────────►│ next: None │
└─────────┘   └────────────┘   └────────────┘

The next action is a call of append_rec(node.next, val). So now we have a second execution of append_rec, where also a node name exists. Let’s give it an accent to distinguish it:

 self          node             node'
  ↓             ↓                ↓
┌─────────┐   ┌────────────┐   ┌────────────┐
│ head: ─────►│ val: 1     │   │ val: 2     │
│ len:  2 │   │ next: ────────►│ next: None │
└─────────┘   └────────────┘   └────────────┘

In this second execution of append_rec, there is no further recursive call made. Instead a node is created and the tail node’s next attribute gets a reference to it:

 self          node             node'
  ↓             ↓                ↓
┌─────────┐   ┌────────────┐   ┌────────────┐   ┌────────────┐
│ head: ─────►│ val: 1     │   │ val: 2     │   │ val: 3     │
│ len:  2 │   │ next: ────────►│ next: ────────►│ next: None │
└─────────┘   └────────────┘   └────────────┘   └────────────┘

Once this is done, the second execution instance of append_rec returns, and the first execution instance has nothing more to do, so it also returns. And finally the append execution updates the len attribute and returns too.

 self
  ↓
┌─────────┐   ┌────────────┐   ┌────────────┐   ┌────────────┐
│ head: ─────►│ val: 1     │   │ val: 2     │   │ val: 3     │
│ len:  3 │   │ next: ────────►│ next: ────────►│ next: None │
└─────────┘   └────────────┘   └────────────┘   └────────────┘

I hope this illustrates how append_rec is called as many times as there are nodes in the list, and how the other actions have a constant time complexity.

Answered By: trincot

Time complexity regarding linked_list ( Recursion )

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