The Problem:

In this problem, we are given a linked list and we have to tell if the linked list forms a cycle. This problem is pretty similar to the last problem we did, the only difference being that this time if a cycle is formed, then we have to return the node where the cycle starts at. A cycle happens when the last node points to any of the previous node.

In the example shown above a cycle is formed as the tail of the linked list points to the second node. In this problem we would return the node 2, in the last problem we have to return True.

Before we begin trying to figure out a solution for this problem, let’s first revisit an important concepts Linked List and Recursion.

Understanding Recursion:

Recursion is essentially a function calling itself. It is frequently used to solve problems that can be divided into smaller, similar sub-problems and can be very useful in dynamic programming. To understand recursion, it can be helpful to break it down into 3 parts:

1. Base Condition: This condition prevents the recursive function from calling itself infinitely.
2. Recursive Call: This is responsible for calling the recursive function itself.
3. Small Calculation: This step, which can come before or after the Recursive Call, is responsible for performing some calculation that we need for the recursive function or for returning some result after a recursive call has been completed.

A recursive tree is a visual representation of the recursive calls made by a function. It can be very helpful in understanding how recursion works and in analyzing the time and space complexity of recursive algorithms.

Understanding Linked List:

A linked list is a linear data structure that consists of a sequence of nodes, each containing data and a reference to the next node in the list. It is often used as an alternative to arrays, as it can be more efficient in certain operations such as insertion and deletion.

To understand linked lists, it can be helpful to break it down into 3 parts:

1. Nodes: Each node in a linked list contains some data and a reference to the next node in the list.
2. Head: The head of a linked list is the first node in the list. It is used as the starting point for many operations on the list.
3. Traversal: Traversing a linked list involves starting at the head and following the references from one node to the next until the desired node is reached or the end of the list is reached.

Linked lists can be very useful in solving problems that involve dynamic data structures, as they allow for efficient insertion and deletion of elements. They can also be used in combination with other data structures, such as stacks and queues, to solve more complex problems.

Intuition & Algorithm:

With an understanding of how recursion and linked lists work, let’s try to break down the problem. We don’t need to create our own linked list data structure because it is already provided. This is a fairly straightforward problem, so we will begin by figuring out the base condition, which in this case would be when the recursion reaches the end of the linked list (head == None) This means that there are no cycles so we return Null.

We can then check the cache for the current list node (we don't have to worry about figuring out ways to hash the linked list node as each object has a unique object identifier and it can be used as a key in a hashMap.) If we find the current node in the hashMap then we simply return the node else we add the current node to the hashMap, update the node to the next node and call the function recursively till we hit the base condition or find a cycle.

This ensures that we will always be able to find cycles (if they exist) in a linked list.

Summary of Solution

Use recursion and a hash map to check if a linked list has a cycle. Store the current node in the hash map and call the function recursively on the next node. If the current node is already in the hash map, return the node. Otherwise, return None when the recursion reaches the end of the linked list (head == None).

Code

This can be translated to code in the following way:

# Definition for singly-linked list.
# class ListNode:
#     def __init__(self, x):
#         self.val = x
#         self.next = None

class Solution:
    def detectCycle(self, head: Optional[ListNode]) -> Optional[ListNode]:
        def check(head, hashMap):
            if head == None:
                return head
            elif head in hashMap:
                return head
            else:
                hashMap[head] = index
                return check(head.next, hashMap)
        hashMap = {}
        return check(head, hashMap)

• Time Complexity O(n)
• Space Complexity O(n)

The time complexity of this solution is O(n) This is because the function needs to traverse the linked lists to detect a cycle. Time taken for this operation would be proportional to the size of the linked list.

Space complexity is O(n) because of recursion call stack.

This approach is simple and efficient, making it a good solution to this problem.

I hope you enjoyed this post. Stay tuned for daily LeetCode blog posts!