Algorithms and Data Structures with Basic Maths Concepts

Study Notes

The text is about India's most in-depth TS algo course with 455 modules, focusing on data structures and algorithms.
The speaker emphasizes starting with basic concepts before moving onto advanced topics.
Basic maths concepts will be taught first, starting with the digit concept.
The digit concept is important for solving problems in basic maths.
Extraction of digits means isolating each digit individually.
To extract a digit, perform a modulo operation with 10.
The remainder after division by 10 is the desired digit.
Repeat the process to extract each digit in reverse order.
Pseudo code can be written to extract all digits from a given number.
The given problem is to count the number of digits in a given number.
The example number is 156, and it has 3 digits.- The text discusses different methods to count the number of digits in a given number and reverse a number.
To count the number of digits, the speaker suggests extracting each digit one by one and keeping a counter variable that increases each time a digit is extracted.
The time complexity for counting digits is log base 10 of the number.
To reverse a number, the speaker suggests using the reverse digit extraction method, where you get the last digit and add it to the reverse number, then repeat the process with the next digit until the original number is empty.
The speaker also mentions a problem to check if a given number is a palindrome (reverse number is the same as the original number) but notes that the original number becomes 0 at the end of the extraction process, so a duplicate of the number must be stored first.
The text also touches upon Armstrong numbers, which are numbers where the sum of the cubes of digits equals the number itself.
To find Armstrong numbers, the speaker suggests taking the cubes of each digit and adding them up, then comparing the sum with the original number.
The speaker also mentions finding all divisors of a given number by looping from 1 to the number and checking if the number is completely divisible by the current number (no remainder).
The time complexity for finding divisors is n, as the loop runs from 1 to n.
The speaker suggests optimizing by observing that divisors are multiples of the square root of the number and only need to be calculated up to that point.- The text explains how to check for prime numbers and find the greatest common divisor (GCD) or highest common factor (HCF) of two numbers using different methods.
A prime number is defined as a number that has exactly two factors: 1 and itself.
Brute Force method: Check divisibility of each number from 1 to n to find factors and identify prime numbers. Time complexity: O(n).
Alternatively, check divisibility only up to the square root of the number since all factors will be identified within that range.
To find the GCD or HCF of two numbers, compare their common factors until only one remains, which will be the greatest common divisor.
Equivalence algorithm: A shortcut method to find the GCD of two numbers by subtracting smaller from larger and repeating the process until one of the numbers reaches zero.
GCD(N1, N2) = GCD(N1 - N2, N2)
Equivalence algorithm proof by induction: If GCD(N1, N2) = GCD(N1 - N2, N2), then GCD(N1, N2) = GCD(N1, N2 - (N1 mod N2)) = GCD(N1, N2 - (N2 mod N1))
The algorithm is simple to implement but might take a long time for larger numbers.
The given example demonstrates the application of the Brute Force method and the Equivalence algorithm to find the GCD of 20 and 15.
The text concludes by mentioning that Equivalence algorithm is more efficient than Brute Force method in most cases but still the worst-case scenario may not significantly improve the time complexity.

Description

Explore algorithms, data structures, and basic maths concepts including digit extraction, counting digits, reversing numbers, palindrome checking, Armstrong numbers, finding divisors, prime numbers, and GCD/HCF calculations. Learn about various methods and time complexities for these operations.