Find pairs with given product in a sorted Doubly Linked List

Given a sorted doubly linked list of positive distinct elements, the task is to find pairs in the doubly linked list whose product is equal to given value x, without using any extra space.

Examples: 

Input : List = 1 <=> 2 <=> 4 <=> 5 <=> 6 <=> 8 <=> 9
        x = 8
Output: (1, 8), (2, 4)

Input : List = 1 <=> 2 <=> 3 <=> 4 <=> 5 <=> 6 <=> 7
        x = 6
Output: (1, 6), (2, 3)

A Simple Approach for this problem is to traverse the linked list using two nested loops and find all pairs and check for the pairs with product equals to x. Time complexity for this problem will be O(n^2), n is total number of nodes in doubly linked list.

An Efficient Solution for this problem is same as this article. Here is the algorithm : 

• Initialize two pointer variables to find the candidate elements in the sorted doubly linked list. Initialize first with start of doubly linked list i.e; first=head and initialize second with last node of doubly linked list i.e; second=last_node.
• We initialize first and second pointers as first and last nodes. Here we don’t have random access, so to find second pointer, we traverse the list to initialize second.
• If current product of first and second is less than x, then we move first in forward direction. If current product of first and second element is greater than x, then we move second in backward direction.
• Loop termination conditions are also different from arrays. The loop terminates when either of two pointers become NULL, or they cross each other (second->next = first), or they become same (first == second)

Below is the implementation of the above approach: 

C++

// C++ program to find a pair with
// given product x in sorted Doubly
// Linked List
#include <bits/stdc++.h>
using namespace std;

// Doubly Linked List Node
struct Node {
    int data;
    struct Node *next, *prev;
};

// Function to find pair whose product
// equal to given value x
void pairProduct(struct Node* head, int x)
{
    // Set two pointers,
    // first to the beginning of DLL
    // and second to the end of DLL.
    struct Node* first = head;
    struct Node* second = head;
    while (second->next != NULL)
        second = second->next;

    // To track if we find a pair or not
    bool found = false;

    // The loop terminates when either of two pointers
    // become NULL, or they cross each other (second->next
    // == first), or they become same (first == second)
    while (first != NULL && second != NULL && first != second
           && second->next != first) {
        // pair found
        if ((first->data * second->data) == x) {
            found = true;
            cout << "(" << first->data << ", "
                 << second->data << ")" << endl;

            // move first in forward direction
            first = first->next;

            // move second in backward direction
            second = second->prev;
        }
        else {
            if ((first->data * second->data) < x)
                first = first->next;
            else
                second = second->prev;
        }
    }

    // if pair is not present
    if (found == false)
        cout << "No pair found";
}

// A utility function to insert a new node at the
// beginning of doubly linked list
void insert(struct Node** head, int data)
{
    struct Node* temp = new Node;
    temp->data = data;
    temp->next = temp->prev = NULL;
    if (!(*head))
        (*head) = temp;
    else {
        temp->next = *head;
        (*head)->prev = temp;
        (*head) = temp;
    }
}

// Driver Code
int main()
{
    // Create Doubly Linked List
    struct Node* head = NULL;
    insert(&head, 9);
    insert(&head, 8);
    insert(&head, 6);
    insert(&head, 5);
    insert(&head, 4);
    insert(&head, 2);
    insert(&head, 1);

    int x = 8;

    pairProduct(head, x);

    return 0;
}

Java

// Java program to find a pair with
// given product x in sorted Doubly
// Linked List

class GFG {

    // Doubly Linked List Node
    static class Node {
        int data;
        Node next, prev;
    };

    // Function to find pair whose product
    // equal to given value x
    static void pairProduct(Node head, int x)
    {
        // Set two pointers,
        // first to the beginning of DLL
        // and second to the end of DLL.
        Node first = head;
        Node second = head;
        while (second.next != null)
            second = second.next;

        // To track if we find a pair or not
        boolean found = false;

        // The loop terminates when either of two pointers
        // become null, or they cross each other (second.next
        // == first), or they become same (first == second)
        while (first != null && second != null && first != second
               && second.next != first) {
            // pair found
            if ((first.data * second.data) == x) {
                found = true;
                System.out.println("(" + first.data + ", "
                                   + second.data + ")");

                // move first in forward direction
                first = first.next;

                // move second in backward direction
                second = second.prev;
            }
            else {
                if ((first.data * second.data) < x)
                    first = first.next;
                else
                    second = second.prev;
            }
        }

        // if pair is not present
        if (found == false)
            System.out.println("No pair found");
    }

    // A utility function to insert a new node at the
    // beginning of doubly linked list
    static Node insert(Node head, int data)
    {
        Node temp = new Node();
        temp.data = data;
        temp.next = temp.prev = null;
        if ((head) == null)
            (head) = temp;
        else {
            temp.next = head;
            (head).prev = temp;
            (head) = temp;
        }
        return head;
    }

    // Driver Code
    public static void main(String args[])
    {
        // Create Doubly Linked List
        Node head = null;
        head = insert(head, 9);
        head = insert(head, 8);
        head = insert(head, 6);
        head = insert(head, 5);
        head = insert(head, 4);
        head = insert(head, 2);
        head = insert(head, 1);

        int x = 8;

        pairProduct(head, x);
    }
}

// This code is contributed by Arnab Kundu

Python3

# Python3 program to find a pair with
# given product x in sorted Doubly
# Linked List

# Node of the doubly linked list
class Node:

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

# Function to find pair whose product
# equal to given value x
def pairProduct(head, x):

    # Set two pointers,
    # first to the beginning of DLL
    # and second to the end of DLL.
    first = head
    second = head
    while (second.next != None):
        second = second.next

    # To track if we find a pair or not
    found = False

    # The loop terminates when either of two pointers
    # become None, or they cross each other (second.next
    # == first), or they become same (first == second)
    while (first != None and second != None and
           first != second and second.next != first) :
        # pair found
        if ((first.data * second.data) == x) :
            found = True
            print("(", first.data, ", ", second.data, ")")

            # move first in forward direction
            first = first.next

            # move second in backward direction
            second = second.prev

        else :
            if ((first.data * second.data) < x):
                first = first.next
            else:
                second = second.prev

    # if pair is not present
    if (found == False):
        print( "No pair found")

# A utility function to insert a new node at the
# beginning of doubly linked list
def insert(head, data):

    temp = Node(0)
    temp.data = data
    temp.next = temp.prev = None
    if (head == None):
        (head) = temp
    else :
        temp.next = head
        (head).prev = temp
        (head) = temp
    return head

# Driver Code
if __name__ == "__main__":

    # Create Doubly Linked List
    head = None
    head = insert(head, 9)
    head = insert(head, 8)
    head = insert(head, 6)
    head = insert(head, 5)
    head = insert(head, 4)
    head = insert(head, 2)
    head = insert(head, 1)

    x = 8

    pairProduct(head, x)

# This code is contributed by Arnab Kundu

C#

// C# program to find a pair with
// given product x in sorted Doubly
// Linked List
using System;

class GFG {

    // Doubly Linked List Node
    public class Node {
        public int data;
        public Node next, prev;
    };

    // Function to find pair whose product
    // equal to given value x
    static void pairProduct(Node head, int x)
    {
        // Set two pointers,
        // first to the beginning of DLL
        // and second to the end of DLL.
        Node first = head;
        Node second = head;
        while (second.next != null)
            second = second.next;

        // To track if we find a pair or not
        bool found = false;

        // The loop terminates when either of two pointers
        // become null, or they cross each other (second.next
        // == first), or they become same (first == second)
        while (first != null && second != null && first != second
               && second.next != first) {
            // pair found
            if ((first.data * second.data) == x) {
                found = true;
                Console.WriteLine("(" + first.data + ", "
                                  + second.data + ")");

                // move first in forward direction
                first = first.next;

                // move second in backward direction
                second = second.prev;
            }
            else {
                if ((first.data * second.data) < x)
                    first = first.next;
                else
                    second = second.prev;
            }
        }

        // if pair is not present
        if (found == false)
            Console.WriteLine("No pair found");
    }

    // A utility function to insert a new node at the
    // beginning of doubly linked list
    static Node insert(Node head, int data)
    {
        Node temp = new Node();
        temp.data = data;
        temp.next = temp.prev = null;
        if ((head) == null)
            (head) = temp;
        else {
            temp.next = head;
            (head).prev = temp;
            (head) = temp;
        }
        return head;
    }

    // Driver Code
    public static void Main(String[] args)
    {
        // Create Doubly Linked List
        Node head = null;
        head = insert(head, 9);
        head = insert(head, 8);
        head = insert(head, 6);
        head = insert(head, 5);
        head = insert(head, 4);
        head = insert(head, 2);
        head = insert(head, 1);

        int x = 8;

        pairProduct(head, x);
    }
}

// This code contributed by Rajput-Ji

Javascript

<script>
// javascript program to find a pair with
// given product x in sorted Doubly
// Linked List
    // Doubly Linked List Node
class Node {
    constructor() {
        this.data = 0;
        this.prev = null;
        this.next = null;
    }
}



    // Function to find pair whose product
    // equal to given value x
    function pairProduct( head , x) {
        // Set two pointers,
        // first to the beginning of DLL
        // and second to the end of DLL.
         first = head;
         second = head;
        while (second.next != null)
            second = second.next;

        // To track if we find a pair or not
        var found = false;

        // The loop terminates when either of two pointers
        // become null, or they cross each other (second.next
        // == first), or they become same (first == second)
        while (first != null && second != null && first != second && second.next != first) {
            // pair found
            if ((first.data * second.data) == x) {
                found = true;
                document.write("(" + first.data + ", " + second.data + ")<br/>");

                // move first in forward direction
                first = first.next;

                // move second in backward direction
                second = second.prev;
            } else {
                if ((first.data * second.data) < x)
                    first = first.next;
                else
                    second = second.prev;
            }
        }

        // if pair is not present
        if (found == false)
            document.write("No pair found");
    }

    // A utility function to insert a new node at the
    // beginning of doubly linked list
    function insert( head , data) {
         temp = new Node();
        temp.data = data;
        temp.next = temp.prev = null;
        if ((head) == null)
            (head) = temp;
        else {
            temp.next = head;
            (head).prev = temp;
            (head) = temp;
        }
        return head;
    }

    // Driver Code

        // Create Doubly Linked List
         head = null;
        head = insert(head, 9);
        head = insert(head, 8);
        head = insert(head, 6);
        head = insert(head, 5);
        head = insert(head, 4);
        head = insert(head, 2);
        head = insert(head, 1);

        var x = 8;

        pairProduct(head, x);

// This code contributed by aashish1995
</script>

Output

(1, 8)
(2, 4)

Time complexity: O(n)

Auxiliary Space: O(1)