## Problem Description

Given a linked list, return the node where the cycle begins. If there is no cycle, return `null`.

There is a cycle in a linked list if there is some node in the list that can be reached again by continuously following the `next` pointer. Internally, `pos` is used to denote the index of the node that tail’s `next` pointer is connected to (0-indexed). It is `-1` if there is no cycle. Note that `pos` is not passed as a parameter.

Do not modify the linked list.

## Approach

To detect the starting node of a cycle in a linked list, we can use the Floyd’s cycle-finding algorithm. This algorithm involves using two pointers, `fast` and `slow`, initially pointing to the head of the linked list.

1. Initialize the `fast` and `slow` pointers to the head of the linked list.
2. Move the `fast` pointer by two steps and the `slow` pointer by one step at each iteration.
3. If there is a cycle in the linked list, the `fast` and `slow` pointers will eventually meet at the same node inside the cycle.
4. Once the `fast` and `slow` pointers meet, we reset one of the pointers (e.g., `slow`) to the head of the linked list.
5. Then, we move both the `fast` and `slow` pointers one step at a time until they meet again.
6. The node where they meet again is the starting point of the cycle.

``````public ListNode detectCycle(ListNode head) {
// Initialize fast and slow pointers

// Move the fast and slow pointers until they meet
while (fast != null && fast.next != null) {
slow = slow.next;
fast = fast.next.next;

// Check if the pointers have met inside the cycle
if (fast == slow) {
// Reset one pointer to the head and move both pointers one step at a time until they meet again
ListNode index1 = slow;

while (index1 != index2) {
index1 = index1.next;
index2 = index2.next;
}

// Return the starting node of the cycle
return index1;
}
}

// If no cycle is found, return null
return null;
}``````

## Complexity Analysis

• Time complexity: O(N), where N is the number of nodes in the linked list. We need to traverse the linked list at most twice: once to detect the cycle and once to find the starting node of the cycle.
• Space complexity: O(1), using only constant extra space.

The Floyd’s cycle-finding algorithm allows us to detect the starting node of a cycle in a linked list in an efficient manner. By manipulating the pointers correctly, we can identify the cycle and determine its starting node without using additional space beyond the original linked list.

Understanding this algorithm is crucial for solving related problems that involve detecting cycles in linked lists.

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