Mathematical Algorithms

  • Last Updated : 26 Sep, 2023

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What are Mathematical Algorithms?

Mathematical algorithm can be defined as an algorithm or procedure which is utilized to solve a mathematical problem, or mathematical problem which can be solved using DSA.

In the field of DSA, mathematical algorithms refer to those algorithms which are generally used to solve problems related to the field of computer science.

What are some use cases of Mathematical Algorithms?

It is a very useful topic for solving problems of computer science and also has vast applications in competitive programming. Some of them are mentioned below:

1. Problem-Solving: It is mostly used in solving complex mathematical ideas in mathematics and computer science.

2. Competitive Programming: Algorithms like Sieve of Eratosthenes, Euclid algorithm, Binary exponentiation are some of the most common topics used in competitive programming.

3. Recurrence relation: A lot of problems in DSA require the usage of recursion. Using the knowledge of mathematics several recurrence relations can be turned into mathematical formulae which makes solving the problem quite an easy task.

4. Heavy calculation: Problems containing complex mathematical concepts and heavy calculations are easily done in comparatively less time using these algorithms instead of manual calculations.

5. Statistics: Mathematical algorithms are also important for data processing, i.e., for converting raw data into useful information and also for analyzing them.

Topics:

 

Divisibility & Large Numbers:

  1. Check if a large number is divisible by 3 or not
  2. Check if a large number is divisible by 4 or not
  3. Check if a large number is divisible by 6 or not
  4. Check divisibility by 7
  5. Check if a large number is divisible by 9 or not
  6. Check if a large number is divisible by 11 or not
  7. Divisibility by 12 for a large number
  8. Check if a large number is divisible by 13 or not
  9. Check if a large number is divisibility by 15
  10. Number is divisible by 29 or not

GCD and LCM:

  1. LCM of array
  2. GCD of array
  3. Basic and Extended Euclidean algorithms
  4. Stein’s Algorithm for finding GCD
  5. GCD, LCM and Distributive Property
  6. Count number of pairs (A <= N, B <= N) such that gcd (A, B) is B
  7. Program to find GCD of floating point numbers
  8. Series with largest GCD and sum equals to n
  9. Largest Subset with GCD 1
  10. Summation of GCD of all the pairs up to N

Series:

  1. Juggler Sequence
  2. Padovan Sequence
  3. Aliquot Sequence
  4. Moser-de Bruijn Sequence
  5. Stern-Brocot Sequence
  6. Newman-Conway Sequence
  7. Sylvester’s sequence
  8. Recaman’s sequence
  9. Sum of the sequence 2, 22, 222, ………
  10. Sum of series 1^2 + 3^2 + 5^2 + . . . + (2*n – 1)^2
  11. Sum of the series 0.6, 0.06, 0.006, 0.0006, …to n terms
  12. n-th term in series 2, 12, 36, 80, 150….

Number Digits:

  1. Minimum digits to remove to make a number Perfect Square
  2. Print first k digits of 1/n where n is a positive integer
  3. Check if a given number can be represented in given a no. of digits in any base
  4. Find element using minimum segments in Seven Segment Display
  5. Find next greater number with same set of digits
  6. Check if a number is jumbled or not
  7. Numbers having difference with digit sum more than s
  8. Total numbers with no repeated digits in a range
  9. K-th digit in ‘a’ raised to power ‘b’

Algebra:

  1. Program to add two polynomials
  2. Multiply two polynomials
  3. Find number of solutions of a linear equation of n variables
  4. Calculate the Discriminant Value
  5. Program for dot product and cross product of two vectors
  6. Iterated Logarithm log*(n)
  7. Program to find correlation coefficient
  8. Program for Muller Method
  9. Number of non-negative integral solutions of a + b + c = n
  10. Generate Pythagorean Triplets

Number System:

  1. Exponential notation of a decimal number
  2. Check if a number is power of k using base changing method
  3. Convert a binary number to hexadecimal number
  4. Program for decimal to hexadecimal conversion
  5. Converting a Real Number (between 0 and 1) to Binary String
  6. Convert from any base to decimal and vice versa
  7. Decimal to binary conversion without using arithmetic operators

Prime Numbers & Primality Tests:

  1. Prime Numbers
  2. Left-Truncatable Prime
  3. Mersenne Prime
  4. Super Prime
  5. Hardy-Ramanujan Theorem
  6. Rosser’s Theorem
  7. Fermat’s little theorem
  8. Introduction to Primality Test and School Method
  9. Vantieghems Theorem for Primality Test
  10. AKS Primality Test
  11. Lucas Primality Test

Prime Factorization & Divisors:

  1. Prime factors
  2. Smith Numbers
  3. Sphenic Number
  4. Hoax Number
  5. k-th prime factor of a given number
  6. Pollard’s Rho Algorithm for Prime Factorization
  7. Finding power of prime number p in n!
  8. Find all divisors of a natural number
  9. Find numbers with n-divisors in a given range

Modular Arithmetic:

  1. Modular Exponentiation (Power in Modular Arithmetic)
  2. Modular multiplicative inverse
  3. Modular Division
  4. Euler’s criterion (Check if square root under modulo p exists)
  5. Find sum of modulo K of first N natural number
  6. How to compute mod of a big number?
  7. Exponential Squaring (Fast Modulo Multiplication)
  8. Trick for modular division ( (x1 * x2 …. xn) / b ) mod (m)

Factorial:

  1. Program for factorial of a number
  2. Legendre’s formula (Given p and n, find the largest x such that p^x divides n!)
  3. Count trailing zeroes in factorial of a number
  4. Factorial of a large number
  5. Primorial of a number
  6. Find maximum power of a number that divides a factorial
  7. Largest power of k in n! (factorial) where k may not be prime
  8. Check if a number is a Krishnamurthy Number or not
  9. Last non-zero digit of a factorial
  10. Count digits in a factorial using Logarithm

Fibonacci Numbers:

  1. Fibonacci Numbers
  2. Interesting facts about Fibonacci numbers
  3. Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation)
  4. Finding nth Fibonacci Number using Golden Ratio
  5. Matrix Exponentiation
  6. Fibonacci Coding
  7. Cassini’s Identity
  8. Tail Recursion for Fibonacci

Catalan Numbers:

  1. Catalan numbers
  2. Applications of Catalan Numbers
  3. Dyck path
  4. Succinct Encoding of Binary Tree
  5. Find the number of valid parentheses expressions of given length

nCr Computations:

  1. Binomial Coefficient
  2. Introduction and Dynamic Programming solution to compute nCr%p
  3. Program to calculate value of nCr
  4. Rencontres Number (Counting partial derangements)
  5. Sum of squares of binomial coefficients
  6. Space and time efficient Binomial Coefficient
  7. Horner’s Method for Polynomial Evaluation

Set Theory:

  1. Power Set
  2. Minimize the absolute difference of sum of two subsets
  3. Sum of all subsets of a set formed by first n natural numbers
  4. Sum of average of all subsets
  5. Bell Numbers (Number of ways to Partition a Set)

Sieve Algorithms:

  1. Sieve of Eratosthenes
  2. Segmented Sieve
  3. Sieve of Atkin
  4. Sieve of Sundaram to print all primes smaller than n
  5. Sieve of Eratosthenes in 0(n) time complexity
  6. Prime Factorization using Sieve O(log n) for multiple queries

Euler Totient Function:

  1. Euler’s Totient Function
  2. Optimized Euler Totient Function for Multiple Evaluations
  3. Euler’s Totient function for all numbers smaller than or equal to n
  4. Primitive root of a prime number n modulo n
  5. Euler’s Four Square Identity

Chinese Remainder Theorem:

  1. Introduction to Chinese Remainder Theorem
  2. Implementation of Chinese Remainder theorem (Inverse Modulo based implementation)
  3. Cyclic Redundancy Check and Modulo-2 Division
  4. Using Chinese Remainder Theorem to Combine Modular equations

Some practice Problems:

  1. Interquartile Range (IQR)
  2. Simulated Annealing
  3. Pseudo Random Number Generator (PRNG)
  4. Square root of a number using log
  5. Find ways an Integer can be expressed as sum of n-th power of unique natural numbers
  6. N-th root of a number
  7. Fast Fourier Transformation for poynomial multiplication
  8. Find Harmonic mean using Arithmetic mean and Geometric mean
  9. Double Base Palindrome
  10. Program for Derivative of a Polynomial
  11. Sgn value of a polynomial
  12. Check if a number is a power of another number
  13. Program to evaluate simple expressions
  14. Make a fair coin from a biased coin
  15. Implement *, – and / operations using only + arithmetic operator

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