Summary
Highlights
This unit covers iterative algorithms, their analysis, and examples. The curriculum includes basic algorithms (GCD, factorial, Fibonacci), searching algorithms (linear search), and sorting algorithms (bubble, selection, and insertion sort). The video focuses on how these algorithms work, their examples, and their time and space complexity.
The GCD, also known as the Highest Common Factor (HCF), is the largest number that divides two given numbers without leaving a remainder. The video explains the Euclidean algorithm for GCD using an example (21 and 6), demonstrating how to find the remainder iteratively until it reaches zero. The GCD is the value of the non-zero number at that point. The time complexity is O(N), where N is the minimum of the two numbers, and the space complexity is O(1) because it uses a constant number of variables.
The Fibonacci series starts with 0 and 1, and each subsequent number is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3, 5...). The algorithm initializes the first two terms as 0 and 1, then iteratively calculates terms by adding the previous two. The video explains how to find the Nth term using a loop. The time complexity is O(N) because it iterates N-2 times, and the space complexity is O(1) as it uses a fixed number of variables (temp, first, second).
The factorial of a non-negative integer N is the product of all positive integers less than or equal to N (e.g., 5! = 5 x 4 x 3 x 2 x 1). The algorithm takes an integer N, initializes a result variable to 1, and then iterates from 1 to N, multiplying the result by each number. For N=5, the result will be 120. The time complexity is O(N) as it performs N multiplications, and the space complexity is O(1) because it uses a constant number of variables.
Linear search is a simple searching algorithm that sequentially checks each element of a list until a match is found or the entire list has been searched. The video illustrates this by comparing a search element with each item in the list until it finds a match. The best case is finding the element at the first position, and the worst case is searching the entire list or when the element is not present. The time complexity is O(N) in the worst case, and the space complexity is O(1).
Bubble sort is a simple sorting algorithm that repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order. The video demonstrates this by showing how the largest element 'bubbles' to the end of the list in each pass. For a list of N elements, there are N-1 passes, and in each pass, the number of comparisons decreases. The time complexity is O(N^2) due to nested loops, and the space complexity is O(1).
Selection sort works by repeatedly finding the minimum element from the unsorted part of the array and putting it at the beginning. The video explains how to find the smallest element in the unsorted portion and swap it with the element at the current position. For N elements, it performs N-1 passes. Similar to bubble sort, it involves nested comparisons. The time complexity is O(N^2), and the space complexity is O(1).
Insertion sort builds the final sorted array one item at a time. It iterates through the input array and removes one element at a time, finding the place where it belongs within the already sorted part and inserting it there. The video shows an example where elements are picked and inserted into their correct sorted position relative to the elements already processed. The time complexity is O(N^2) in the worst case, and the space complexity is O(1).