In a linked list of size n, where n is even, the ith node (0-indexed) of the linked list is known as the twin of the (n-1-i)th node, if 0 <= i <= (n / 2) - 1.

• For example, if n = 4, then node 0 is the twin of node 3, and node 1 is the twin of node 2. These are the only nodes with twins for n = 4.

The twin sum is defined as the sum of a node and its twin.

Given the head of a linked list with even length, return the maximum twin sum of the linked list.

Example 1:

Input: head = [5,4,2,1]
Output: 6
Explanation:
Nodes 0 and 1 are the twins of nodes 3 and 2, respectively. All have twin sum = 6.
There are no other nodes with twins in the linked list.
Thus, the maximum twin sum of the linked list is 6.

Example 2:

Input: head = [4,2,2,3]
Output: 7
Explanation:
The nodes with twins present in this linked list are:
- Node 0 is the twin of node 3 having a twin sum of 4 + 3 = 7.
- Node 1 is the twin of node 2 having a twin sum of 2 + 2 = 4.
Thus, the maximum twin sum of the linked list is max(7, 4) = 7.

Example 3:

Input: head = [1,100000]
Output: 100001
Explanation:
There is only one node with a twin in the linked list having twin sum of 1 + 100000 = 100001.

Constraints:

• The number of nodes in the list is an even integer in the range [2, 105].
• 1 <= Node.val <= 105

Explanation

Imagine you have a line of numbers connected together, like a train. You want to find the biggest sum you can get by adding the first number to the last, the second number to the second-to-last, and so on.

🚅Find the Middle:

• Walk along the train with two people: one walking normally (slow), and one walking twice as fast (fast).
• When the fast walker reaches the end, the slow walker will be in the middle of the train.

🚅Reverse the Second Half:

• Take the second half of the train (from the middle to the end) and turn it around, so the last car becomes the first.

🚅Calculate Twin Sums:

• Now, walk along the first half of the train and the reversed second half at the same time.
• Add the number in the current car of the first half to the number in the current car of the reversed second half.
• Keep track of the biggest sum you find.

🚅Return the Biggest Sum:

• Once you’ve added all the pairs, tell me the biggest sum you found.

Simplified Example

Let’s say the train is: [5, 4, 2, 1]

🚅Find the Middle: The middle is between 4 and 2.

🚅Reverse the Second Half: The second half [2, 1] becomes [1, 2].

🚅Calculate Twin Sums:

• 5 + 1 = 6
• 4 + 2 = 6

Return the Biggest Sum: The biggest sum is 6.

Implementations

Python

# Definition for singly-linked list.
# class ListNode:
#     def __init__(self, val=0, next=None):
#         self.val = val
#         self.next = next
class Solution:
    def pairSum(self, head: Optional[ListNode]) -> int:
        """
        Finds the maximum twin sum in a linked list with even length.

        Args:
            head: The head of the linked list.

        Returns:
            The maximum twin sum.
        """
        if not head or not head.next:
            return 0

        # 1. Find the middle of the linked list
        slow = head
        fast = head
        while fast and fast.next:
            slow = slow.next
            fast = fast.next.next

        # 2. Reverse the second half of the linked list
        prev = None
        curr = slow
        while curr:
            next_node = curr.next
            curr.next = prev
            prev = curr
            curr = next_node

        # 3. Calculate twin sums and find the maximum
        max_sum = 0
        left = head
        right = prev
        while right:
            max_sum = max(max_sum, left.val + right.val)
            left = left.next
            right = right.next

        return max_sum

Typescript

/**
 * Definition for singly-linked list.
 * class ListNode {
 * val: number
 * next: ListNode | null
 * constructor(val?: number, next?: ListNode | null) {
 * this.val = (val===undefined ? 0 : val)
 * this.next = (next===undefined ? null : next)
 * }
 * }
 */

function pairSum(head: ListNode | null): number {
    if (!head || !head.next) {
        return 0;
    }

    // 1. Find the middle of the linked list
    let slow: ListNode | null = head;
    let fast: ListNode | null = head;
    while (fast && fast.next) {
        if (slow) {
            slow = slow.next;
        }
        fast = fast.next.next;
    }

    // 2. Reverse the second half of the linked list
    let prev: ListNode | null = null;
    let curr: ListNode | null = slow;
    while (curr) {
        const nextNode: ListNode | null = curr.next;
        curr.next = prev;
        prev = curr;
        curr = nextNode;
    }

    // 3. Calculate twin sums and find the maximum
    let maxSum: number = 0;
    let left: ListNode | null = head;
    let right: ListNode | null = prev;
    while (right) {
        if (left && right) {
            maxSum = Math.max(maxSum, left.val + right.val);
            left = left.next;
            right = right.next;
        }
    }

    return maxSum;
}

Go

/**
 * Definition for singly-linked list.
 * type ListNode struct {
 * Val int
 * Next *ListNode
 * }
 */
func pairSum(head *ListNode) int {
    if head == nil || head.Next == nil {
        return 0
    }

    // 1. Find the middle of the linked list
    slow := head
    fast := head
    for fast != nil && fast.Next != nil {
        slow = slow.Next
        fast = fast.Next.Next
    }

    // 2. Reverse the second half of the linked list
    var prev *ListNode
    curr := slow
    for curr != nil {
        next := curr.Next
        curr.Next = prev
        prev = curr
        curr = next
    }

    // 3. Calculate twin sums and find the maximum
    maxSum := 0
    left := head
    right := prev
    for right != nil {
        sum := left.Val + right.Val
        if sum > maxSum {
            maxSum = sum
        }
        left = left.Next
        right = right.Next
    }

    return maxSum
}

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