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Mathematics in OI

Mathematics in OI. Prepared by Ivan Li. Mathematics in OI A brief content. Greatest Common Divisor Modular Arithmetic Finding Primes Floating Point Arithmetic High Precision Arithmetic Partial Sum and Difference Euler Phi function Fibonacci Sequence and Recurrence Twelvefold ways

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Mathematics in OI

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  1. Mathematics in OI Prepared by Ivan Li

  2. Mathematics in OIA brief content • Greatest Common Divisor • Modular Arithmetic • Finding Primes • Floating Point Arithmetic • High Precision Arithmetic • Partial Sum and Difference • Euler Phi function • Fibonacci Sequence and Recurrence • Twelvefold ways • Combinations and Lucas Theorem • Catalan Numbers • Using Correspondence • Josephus Problem

  3. Greatest Common Divisor • Motivation • Sometimes we want “k divides m” and “k divides n” occur simultaneously. • And we want to merge the two statements into one equivalent statement: “k divides ?”

  4. Greatest Common Divisor • Definition • The greatest natural number dividing both n and m • A natural number k dividing both n and m, such that for each natural number h dividing both n and m, we have k divisible by h.

  5. Greatest Common Divisor • How to find it? • Check each natural number not greater than m and n if it divides both m and n. Then select the greatest one. • Euclidean Algorithm

  6. Euclidean Algorithm • Assume m > n GCD(m,n) While n > 0 m = m mod n swap m and n Return m

  7. Greatest Common Divisor • What if we want the greatest number which divides n1,n2, …, nm-1 and nm? • Apply GCD two-by-two • gcd(n1,n2, …,nm) = gcd(n1,gcd(n2,gcd(n3,…gcd(nm-1,nm)…))

  8. Applications • Simplifying a fraction m/n • If gcd(m,n) > 1, then the fraction can be simplified by dividing gcd(m,n) on the numerator and the denominator.

  9. Applications • Solve mx + ny = a for integers x and y • Can be solved if and only if a is divisible by gcd(m,n)

  10. Least Common Multiple • Definition • The least natural number divisible by both n and m • A natural number k divisible by both n and m, such that for each natural number h divisible by both n and m, we have k divides h. • Formula • lcm(m,n) = mn/gcd(m,n)

  11. Least Common Multiple • What if we want to find the LCM of more than two numbers? • Apply LCM two-by-two?

  12. Extended Euclidean Algorithm • The table method • e.g. solve 93x + 27y = 6 • General form: • x = -4 + 9k, y = 14 - 31k, k integer

  13. Modular Arithmetic • Divide 7 by 3 • Quotient = 2, Remainder = 1 • 7 ÷ 3 = 2...1 • In modular arithmetic • 7 ≡ 1 (mod 3) • a ≡ b (mod m) if a = km + b for an integer k

  14. Modular Arithmetic • Like the equal sign • Addition / subtraction on both sides • 7 ≡ 1 (mod 3) => 7+2≡1+2 (mod 3) • Multiplication on both side • 7 ≡ 1 (mod 3) => 7*2≡1*2 (mod 3) • We can multiply into m too • 7≡1 (mod 3)<=>7*2≡1*2 (mod 3*2) • Congruence in mod 6 is stronger than that in mod 3

  15. Modular Arithmetic • Division? • Careful: • 6≡4 (mod 2), but not 3≡2 (mod 2) • ac≡bc (mod m)<=>a≡b (mod m) when c, m coprime (gcd = 1) • Not coprime? • ac≡bc(mod cm)<=>a≡b(mod m) • ac ≡ bc(mod m)<=> a ≡ b (mod m/gcd(c,m))

  16. Modular Inverse • Given a, find b such that ab ≡ 1 (mod m) • Write b as a-1 • We can use it to do “division” • ax ≡ c (mod m)=> x ≡ a-1c (mod m) • Exist if and only if a and m are coprime (gcd = 1) • When m is prime, inverse exists for a not congruent to 0

  17. Modular Inverse • ab ≡ 1 (mod m) • ab + km = 1 • Extended Euclidean algorithm

  18. CAUTION!!! • The mod operator “%” does not always give a non-negative integer below the divisor • The answer is negative when the dividend is negative • Use ((a % b)+b)%b

  19. Definition of Prime Numbers An integer p greater than 1 such that: • p has factors 1 and p only? • If p = ab, a b, then a = 1 and b = p ? • If p divides ab, then p divides a or p divides b ? • p divides (p - 1)! + 1 ?

  20. Test for a prime number • By Property 1 • For each integer greater than 1 and less than p, check if it divides p • Actually we need only to check integers not greater than sqrt(p) (Why?)

  21. Finding Prime Numbers • For each integer, check if it is a prime • Prime List • Sieve of Eratosthenes

  22. Prime List • Stores a list of prime numbers found • For each integer, check if it is divisible by any of the prime numbers found • If not, then it is a prime. Add it to the list.

  23. Sieve of Eratosthenes • Stores an array of Boolean values Comp[i] which indicates whether i is a known composite number

  24. Sieve of Eratosthenes for i = 2 … n If not Comp[i] output i j = i *i //why i*i? while j n Comp[j] = true j = j + i

  25. Optimization • Consider odd numbers only • Do not forget to add 2, the only even prime

  26. Other usages of sieve • Prime factorization • When we mark an integer as composite, store the current prime divisor • For each integer, we can get a prime divisor instantly • Get the factorization recursively

  27. Floating point arithmetic • Pascal: real, single, double • C/C++: float, double • Sign, exponent, mantissa • Floating point error • 0.2 * 5.0 == 1.0 ? • 0.2 * 0.2 * 25.0 == 1.0?

  28. Floating point arithmetic • To tolerate some floating point • Introduce epsilon • EPS = 1e-8 to 1e-11 • a < b => a + EPS < b • a <= b => a <= b + EPS • a == b => abs(a - b) <= EPS

  29. Floating point arithmetic • Special values • Positive / Negative infinity • Not a number (NaN) • Checked in C++ by x!=x • Denormal number

  30. High Precision Arithmetic • 32-bit signed integer:-2147483648 … 2147483647 • 64-bit signed integer:-9223372036854775808 … 9223372036854775807 • How to store a 100 digit number?

  31. High Precision Arithmetic • Use an array to store the digits of the number • Operations: • Comparison • Addition / Subtraction • Multiplication • Division and remainder

  32. High Precision Division • Locate the position of the first digit of the quotient • For each digit of the quotient (starting from the first digit), find its value by binary search.

  33. High Precision Arithmetic • How to select the base? • Power of 2 : Saves memory • Power of 10 : Easier input / output • 1000 or 10000 for 16-bit integer array • Beware of carry

  34. More on HPA • How to store • negative numbers? • fractions? • floating-point numbers?

  35. Partial Sum • Motivation • How to find the sum of the 3rd to the 6th element of an array a[i] ? • a[3] + a[4] + a[5] + a[6] • How to find the sum of the 1000th to the 10000th element? • A for-loop will take much time • In order to find the sum of a range in an array efficiently, we need to do some preprocessing.

  36. Partial Sum • Use an array s[i] to store the sum of the first i elements. • s[i] = a[1] + a[2] + … + a[i] • The sum of the j th element to the k th element = s[k] – s[j-1] • We usually set s[0] = 0

  37. Partial Sum • How to compute s[i] ? • During input s[0] = 0 for i = 1 to n input a[i] s[i] = s[i-1] + a[i]

  38. Difference operation • Motivation • How to increment the 3rd to the 6th element of an array a[i] ? • a[3]++, a[4]++, a[5]++, a[6]++ • How to increment the 1000th to the 10000th element? • A for-loop will take much time • In order to increment(or add an arbitrary value to) a range of elements in an array efficiently, we will use a special method to store the array.

  39. Difference operation • Use an array d[i] to store the difference between a[i] and a[i-1]. • d[i] = a[i] - a[i-1] • When the the j th element to the k th element is incremented, • d[j] ++, d[k+1] - - • We usually set d[1] = a[1]

  40. Difference operation • Easy to compute d[i] • But how to convert it back to a[i]? • Before (or during) output a[0] = 0 for i = 1 to n a[i] = a[i-1] + d[i] output a[i] • Quite similar to partial sum, isn’t it?

  41. Relation between the two methods • They are “inverse” of each other • Denote the partial sum of a by a • Denote the difference of a by a • The difference operator • We have (a) = (a) = a

  42. Comparison • Partial sum - Fast sum of range query • Difference - Fast range increment • Ordinary array - Fast query and increment on single element

  43. Runtime Comparison

  44. Does there exist a method to perform range query and range update in constant time? • No, but there is a data structure that performs the two operations pretty fast.

  45. Euler Phi Function • (n) = number of integers in 1...n that are relatively prime to n • (p) = p-1 • (pn) = (p-1)pn-1 = pn(1-1/p) • Multiplicative property:(mn)=(m)(n) for (m,n)=1 • (n) = np|n(1-1/p)

  46. Euler Phi Function • Euler Theorem • a(n) 1 (mod n) for (a, n) = 1 • Then we can find modular inverse • aa(n)-1 1 (mod n) • a-1a(n)-1

  47. Fibonacci Number • F0 = 0, F1 = 1 • Fn = Fn-1 + Fn-2 for n > 1 • The number of rabbits • The number of ways to go upstairs • How to calculate Fn?

  48. What any half-wit can do • F0 = 0; F1 = 1;for i = 2 . . . n Fi = Fi-1 + Fi-2;return Fn; • Time Complexity: O(n) • Memory Complexity: O(n)

  49. What a normal person would do • a = 0; b = 1;for i = 2 . . . N t = b; b += a; a = t; return b; • Time Complexity: O(n) • Memory Complexity: O(1)

  50. What a Math student would do • Generating Function • G(x) = F0 + F1 x + F2 x2 +. . . • A generating function uniquely determines a sequence (if it exists) • Fn = dnG(x)/dxn (0) • A powerful (but tedious) tool in solving recurrence

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