Question

Given two lists of \(n\) integers each, describe an \(O(n \log (n))\) algorithm for finding all elements that they have in common.

Short answer

Sort both lists and use two pointers to find common elements.

Step-by-step solution

Sort the Lists

First, sort both lists of integers. Sorting each list will arrange the elements in ascending order, which will facilitate efficient searching. Sorting can be achieved using algorithms like Merge Sort or Quick Sort, which have a time complexity of \(O(n \log n)\).

Initialize Pointers

Initialize two pointers, one for each sorted list. These pointers will start at the beginning of the respective lists and help in iterating through the lists to find common elements.

Iterate Through Lists to Find Common Elements

Using a loop, compare the elements at the current positions of the two pointers. If the elements are the same, record the element as common and move both pointers to the next element. If the element pointed to in the first list is smaller than that in the second list, move the pointer for the first list forward. Conversely, if the element in the second list is smaller, move the pointer for the second list forward. Continue this process until the end of one list is reached.

Key concepts

Sorting Algorithms

Sorting algorithms are essential in computer science for organizing data. They arrange data in a specific order, making it easier to perform tasks like searching. When dealing with lists, sorting can simplify the process of finding common elements between them.

• **Merge Sort** and **Quick Sort** are popular algorithms for sorting, both with a time complexity of \(O(n \log n)\).
• Merge Sort divides the list into smaller parts, sorts those parts, and then combines them.
• Quick Sort, on the other hand, picks a "pivot" and arranges the list into sublists below and above the pivot, sorting them recursively.

By sorting two lists with these efficient algorithms, subsequent operations like searching can be performed with ease. Sorting is the first step in many complex algorithms, as it prepares the data for easier access.

Time Complexity

Time complexity is a way to describe how fast an algorithm runs. It helps us understand how an algorithm's execution time increases as the input size grows. This is crucial when comparing different algorithms.

• An algorithm with a time complexity of \(O(n \log n)\) is more efficient than \(O(n^2)\); it means the processing time grows in relation to \(n \log n\), where \(n\) is the number of elements.
• This kind of time complexity often appears in efficient sorting algorithms like Merge Sort and Quick Sort.
• Knowing the time complexity allows you to choose the best algorithm for your problem, ensuring that your programs run efficiently.

For instance, sorting the lists with an \(O(n \log n)\) algorithm ensures that we can then quickly find common elements, making the entire process efficient.

Common Elements in Lists

Finding common elements in two lists is a common problem in computer science and can be solved efficiently using sorting and clever iteration.

• Once both lists are sorted, finding common elements is akin to merging the lists.
• Using two pointers, one for each list, traverse through comparing the elements.
• When the elements pointed at are equal, it denotes a common element, which you then store.
• If not equal, move the pointer of the smaller element forward, continuing the process until the end of a list is reached.

By sorting first, not only is the finding efficient, but it also ensures that each element is checked in an organized manner, reducing unnecessary comparisons and making the process quick and systematic.