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Implementing operations on linked lists is a staple of programming classes and technical interviews.

I resist them because it is unlikely that you will ever have to implement a linked list in your professional work. And if you do, someone has made a bad decision.

However, they can be good études, that is, pieces that you practice in order to learn, but never perform.

For many of these problems, there are several possible solutions, depending on the requirements:

• Are you allowed to modify an existing list, or do you have to create a new one?

• If you modify an existing structure, are you also supposed to return a reference to it?

• Are you allowed to allocate temporary structures, or do you have to perform all operations in place?

And for all of these problems, you could write a solution iteratively or recursively. So there are many possible solutions for each.

As we consider alternatives, some of the factors to keep in mind are:

• Performance in terms of time and space.

• Readability and demonstrably correctness.

In general, performance should be asymptotically efficient; for example, if there is a constant time solution, a linear solution would not be acceptable. But we might be willing to pay some overhead to achieve bulletproof correctness.

Here’s the class we’ll use to represent the nodes in a list.

class Node:
def __init__(self, data, next=None):
self.data = data
self.next = next

def __repr__(self):
return f'Node({self.data}, {repr(self.next)})'


We can create nodes like this:

node1 = Node(1)
node2 = Node(2)
node3 = Node(3)

node1

Node(1, None)


And then link them up, like this:

node1.next = node2
node2.next = node3

node1

Node(1, Node(2, Node(3, None)))


There are two ways to think about what node1 is:

• It is “just” a node object, which happens to contain a link to another node.

• It is the first node is a linked list of nodes.

When we pass a node as a parameter, sometimes we think of it as a node and sometimes we think of it as the beginning of a list.

For some operations, it will be convenient to have another object that represents the whole list (as opposed to one of its nodes).

Here’s the class definition.

class LinkedList:

def __repr__(self):


If we create a LinkedList with a reference to node1, we can think of the result as a list with three elements.

t = LinkedList(node1)
t

LinkedList(Node(1, Node(2, Node(3, None))))


## Push and Pop¶

Adding and removing elements from the left side of a linked list is relatively easy:

def lpush(t, value):

t = LinkedList()
lpush(t, 3)
lpush(t, 2)
lpush(t, 1)
t

LinkedList(Node(1, Node(2, Node(3, None))))

def lpop(t):
if t.head is None:
raise ValueError('Tried to pop from empty LinkedList')
return node.data

lpop(t), lpop(t), lpop(t)

(1, 2, 3)

t

LinkedList(None)


Adding and removing from the end right side take longer because we have to traverse the list.

Exercise: Write rpush and rpop.

You can use the following example to test your code.

t = LinkedList()
rpush(t, 1)
t

LinkedList(Node(1, None))

rpush(t, 2)
t

LinkedList(Node(1, Node(2, None)))

rpop(t)

2

rpop(t)

1

try:
rpop(t)
except ValueError as e:
print(e)

Tried to rpop from an empty list


## Reverse¶

Reversing a linked list is a classic interview question, although at this point it is so classic you will probably never encounter it.

But it is still a good exercise, in part because there are so many ways to do it. My solutions here are based on this tutorial.

If you are allowed to make a new list, you can traverse the old list and lpush the elements onto the new list:

def reverse(t):
while node:
lpush(t2, node.data)
node = node.next

return t2

t = LinkedList(Node(1, Node(2, Node(3, None))))
reverse(t)

LinkedList(Node(3, Node(2, Node(1, None))))


Here’s a recursive version that doesn’t allocate anything

def reverse(t):

def reverse_rec(node):

# if there are 0 or 1 nodes
if node is None or node.next is None:
return node

# reverse the rest LinkedList
rest = reverse_rec(node.next)

# Put first element at the end
node.next.next = node
node.next = None

return rest

t = LinkedList(Node(1, Node(2, Node(3, None))))
reverse(t)
t

LinkedList(Node(3, Node(2, Node(1, None))))


And finally an iterative version that doesn’t allocate anything.

def reverse(t):
prev = None
while current :
next = current.next
current.next = prev
prev = current
current = next

t = LinkedList(Node(1, Node(2, Node(3, None))))
reverse(t)
t

LinkedList(Node(3, Node(2, Node(1, None))))


## Remove¶

One of the advantages of a linked list (compared to an array list) is that we can add and remove elements from the middle of the list in constant time.

For example, the following function takes a node and removes the node that follows it.

def remove_after(node):
removed = node.next
node.next = node.next.next
return removed.data


Here’s an example:

t = LinkedList(Node(1, Node(2, Node(3, None))))
t

LinkedList(Node(1, Node(3, None)))


Exercise: Write a function called remove that takes a LinkedList and a target value. It should remove the first node that contains the value, or raise a ValueError if it is not found.

Hint: This one is a little tricky.

You can use this example to test your code.

t = LinkedList(Node(1, Node(2, Node(3, None))))
remove(t, 2)
t

LinkedList(Node(1, Node(3, None)))

remove(t, 1)
t

LinkedList(Node(3, None))

try:
remove(t, 4)
except ValueError as e:
print(e)

Value not found

remove(t, 3)
t

LinkedList(None)

try:
remove(t, 5)
except ValueError as e:
print(e)

Value not found (empty list)


Although remove_after is constant time, remove is not. Because we have to iterate through the nodes to find the target, remove takes linear time.

## Insert Sorted¶

Similarly, you can insert an element into the middle of a linked list in constant time.

The following function inserts data after the given node in a list.

def insert_after(node, data):
node.next = Node(data, node.next)

t = LinkedList(Node(1, Node(2, Node(3, None))))
t

LinkedList(Node(1, Node(5, Node(2, Node(3, None)))))


Exercise: Write a function called insert_sorted (also known as insort) that takes a linked list and a value and inserts the value in the list in the first place where it will be in increasing sorted order, that is, with the smallest element at the beginning.

def insert_sorted(t, data):
if t.head is None or t.head.data > data:
lpush(t, data)
return

while node.next:
if node.next.data > data:
insert_after(node, data)
return
node = node.next

insert_after(node, data)


You can use the following example to test your code.

t = LinkedList()
insert_sorted(t, 1)
t

LinkedList(Node(1, None))

insert_sorted(t, 3)
t

LinkedList(Node(1, Node(3, None)))

insert_sorted(t, 0)
t

LinkedList(Node(0, Node(1, Node(3, None))))

insert_sorted(t, 2)
t

LinkedList(Node(0, Node(1, Node(2, Node(3, None)))))


Although insert_after is constant time, insert_sorted is not. Because we have to iterate through the nodes to find the insertion point, insert_sorted takes linear time.

Data Structures and Information Retrieval in Python

Copyright 2021 Allen Downey