Question

Given two lists of \(n\) integers cach, describe an \(O(n \log (n))\) algorithm for determining whether they have an element in common.

Short answer

Sort one list and binary search each element of the other list. Return "Yes" if any element is found; otherwise, return "No".

Step-by-step solution

Sorting One of the Lists

First, we sort one of the two given lists. Sorting can be done in\( O(n \log n) \) time using efficient sorting algorithms like Merge Sort or Quick Sort. Since we have two lists, let's call them List A and List B. We will sort List A.

Iterating Through the Second List

Next, we iterate through each element of List B. As we iterate, we perform a search for each element of List B in the now sorted List A.

Binary Search for Each Element

For each element from List B, we perform a binary search on the sorted List A. Since List A is already sorted, binary search will allow us to check for the presence of each element in \( O(\log n) \) time. Given that we have \( n \) elements in List B, this step will take \( O(n \log n) \) time overall.

Checking Commonality

During the binary search in Step 3, if an element from List B is found in List A, we conclude that the lists have a common element. We return "Yes" and terminate the process. If the loop completes without finding any common elements, we return "No".

Key concepts

Sorting Algorithms

Sorting algorithms are essential tools in computer science, especially when working with large sets of data. Their primary function is to reorder elements in a list so that they follow a particular sequence, usually numerical or lexicographical. This can greatly enhance the efficiency of other operations, such as searching.There are many sorting algorithms, each with its own benefits and drawbacks.

• Merge Sort: This algorithm divides the list into smaller sub-lists, sorts them, and then merges them back together. It is a stable sort, meaning it maintains the relative order of equal elements, and has a time complexity of \(O(n \log n)\).
• Quick Sort: Known for its efficiency, Quick Sort works by selecting a 'pivot' element and partitioning the other elements into two groups, according to whether they are less than or greater than the pivot. Like Merge Sort, it generally performs with a time complexity of \(O(n \log n)\), although in the worst-case scenario, it can degrade to \(O(n^2)\).

Understanding these algorithms helps us in optimizing the process of finding common elements in two lists, by ensuring that the list is sorted efficiently.

Binary Search

Binary search is an efficient algorithm designed to find an element in a sorted list. The name 'binary' comes from the way the list is divided into halves with each step, eliminating the half that is guaranteed not to contain the target element.Here's how it generally works:

• Start with the middle element of the list. If this matches the target, the search is complete.
• If the target element is less than the middle element, repeat the process on the left half of the list.
• If the target is greater, repeat on the right half.

Binary search operates with a time complexity of \(O(\log n)\), which is excellent for searching through large datasets quickly. In the context of our problem, once one list is sorted, binary search can help efficiently find if the other list contains any common elements.

Time Complexity

Time complexity is a critical concept in algorithm design as it gives us an idea of the runtime growth as the input size increases. It is expressed using Big O notation, which describes the upper bound of an algorithm's runtime.In this exercise, achieving a time complexity of \(O(n \log n)\) is crucial. By sorting one of the lists, we utilize efficient sorting algorithms like Merge Sort or Quick Sort, both of which have this desired time complexity. Once sorted, applying binary search to determine if elements from the second list are present ensures the entire process remains efficient.Efficient time complexity is not just about speed - it's about scalability. As the size of the lists increases, the ability to maintain lower time complexity ensures the algorithm will continue to perform well.

Data Structures

Data structures provide a way of organizing and storing data in a computer so that it can be accessed and modified effectively. They are foundational in algorithm design, impacting efficiency and complexity. For lists of data, commonly used data structures include arrays and linked lists and each has its respective advantages:

• Arrays: In an array, elements are stored at contiguous memory locations. This allows for quick access to elements through indexing, but resizing an array can be inefficient.
• Linked Lists: Consists of nodes where each node contains its own data and a reference (or link) to the next node in the sequence. They offer more efficient insertion and deletion at the cost of slower access times compared to arrays.

In the context of finding common elements between two lists, selecting the appropriate data structure can improve performance, especially when combined with the right algorithm, like sorting and searching techniques.