How to apply Binary Search O(l...

Applying binary search on a sorted singly linked list is a challenging task due to the lack of random access in linked lists. Here is a detailed explanation of how to implement binary search on a singly linked list, including the necessary steps and considerations:

Key Concepts

1. Binary Search Basics: Binary search works by repeatedly dividing the search interval in half. It requires random access to the middle element of the list, which is straightforward in arrays but not in linked lists.

2. Linked List Characteristics: Linked lists do not support random access, meaning you cannot directly access the middle element without traversing the list from the head.

Steps to Implement Binary Search on a Singly Linked List

1. Finding the Middle Element:

   • Use the slow and fast pointer technique to find the middle element. The slow pointer moves one step at a time, while the fast pointer moves two steps at a time. When the fast pointer reaches the end, the slow pointer will be at the middle.
   • This operation takes $$O(n)$$ time.

2. Binary Search Algorithm:

   • Initialize two pointers, start and end, to represent the current search range.
   • Repeat the following steps until the search range is exhausted:
     • Find the middle element using the slow and fast pointer technique.
     • Compare the middle element with the target value:
       • If the middle element is equal to the target, return the middle element.
       • If the target is less than the middle element, update the end pointer to the middle element.
       • If the target is greater than the middle element, update the start pointer to the node after the middle element.
   • If the search range collapses and the element is not found, return NULL.

Time Complexity

• Finding the Middle Element: $$O(n)$$
• Binary Search: Since finding the middle element takes $$O(n)$$ and this operation is repeated $$O(\log n)$$ times, the overall time complexity is $$O(n \log n)$$.

Example Code

Here is a sample implementation in C++:

#include <iostream>
using namespace std;

struct Node {
    int data;
    Node* next;
};

Node* newNode(int data) {
    Node* node = new Node();
    node->data = data;
    node->next = NULL;
    return node;
}

Node* findMiddle(Node* start, Node* end) {
    if (start == NULL) return NULL;
    Node* slow = start;
    Node* fast = start->next;
    while (fast != end) {
        fast = fast->next;
        if (fast ...