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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 Combinations and Lucas Theorem Catalan Numbers Using Correspondence Josephus Problem 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 ?” 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. 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 Euclidean Algorithm • Assume m > n GCD(m,n) While n > 0 m = m mod n swap m and n Return m 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)…)) 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. Applications • Solve mx + ny = a for integers x and y • Can be solved if and only if a is divisible by gcd(m,n) 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) Least Common Multiple • What if we want to find the LCM of more than two numbers? • Apply LCM two-by-two? Extended Euclidean Algorithm • The table method • e.g. solve 93x + 27y = 6 Coeff of 93 Coeff of 27 (1) 93 1 0 (2) 27 0 1 (3): (1)-3(2) 12 1 -3 (4): (2)-2(3) 3 (gcd, 3*2=6) -2 (x = -4) 7 (y = 14) (5): (3)-4(4) 0 9 -31 • General form: • x = -4 + 9k, y = 14 - 31k, k integer 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 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 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)) 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 Modular Inverse • ab ≡ 1 (mod m) • ab + km = 1 • Extended Euclidean algorithm 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 Definition of Prime Numbers An integer p greater than 1 such that: 1. p has factors 1 and p only? 2. If p = ab, a b, then a = 1 and b=p? 3. If p divides ab, then p divides a or p divides b ? 4. p divides (p - 1)! + 1 ? 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?) Finding Prime Numbers • For each integer, check if it is a prime • Prime List • Sieve of Eratosthenes 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. Sieve of Eratosthenes • Stores an array of Boolean values Comp[i] which indicates whether i is a known composite number Sieve of Eratosthenes for i = 2 … n If not Comp[i] output i j = i *i while j n //why i*i? Comp[j] = true j=j+i Optimization • Consider odd numbers only • Do not forget to add 2, the only even prime 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 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? 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 Floating point arithmetic • Special values • Positive / Negative infinity • Not a number (NaN) • Checked in C++ by x!=x • Denormal number High Precision Arithmetic • 32-bit signed integer: -2147483648 … 2147483647 • 64-bit signed integer: -9223372036854775808 … 9223372036854775807 • How to store a 100 digit number? High Precision Arithmetic • Use an array to store the digits of the number • Operations: • • • • Comparison Addition / Subtraction Multiplication Division and remainder 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. 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 More on HPA • How to store • negative numbers? • fractions? • floating-point numbers? 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. 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 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] 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. Difference operation • Use an array d[i] to store the difference between a[i] and a[i1]. • 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] 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? 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 Comparison • Partial sum - Fast sum of range query • Difference - Fast range increment • Ordinary array - Fast query and increment on single element Runtime Comparison Range Query Single Query Single Update Range Update Partial sum Constant Constant Linear Linear Ordinary array Linear Constant Constant Linear Difference Linear Linear Constant Constant (Convert it back to the original array) (treat a single element as a range) (Have to update all sums involved) • 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. 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) = np|n(1-1/p) 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 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? 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) 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) 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 All the tedious works G(x) = F0 + F1 x + F2 x2 + F3 x3 +. . . xG(x) = F0 x + F1 x2 + F2 x3 +. . . x2G(x) = F0 x2 + F1 x3 +. . . G(x) - xG(x) - x2G(x) = F0 +F1 x - F0 x = x G(x) = x / (1 - x - x2) Let a = (-1 - sqrt(5)) / 2, b = (-1 + sqrt(5)) / 2 By Partial Fraction: G(x) = ((5 + sqrt(5)) / 10) / (a-x)+((5 - sqrt(5)) / 10) / (b-x) = -(sqrt(5) / 5) / (1- x/a) + (sqrt(5) / 5) / (1- x/b) • Note that 1 + rx + r2x2 +. . . = 1 / (1 - rx) G(x) = (sqrt(5) / 5)(-1-x/a-x2/a2-...+1+x/b+x2/b2+...) • By Uniqueness, Fn = (sqrt(5) / 5)(-1/an + 1/bn) • • • • • • • Shortcut • • • • Characteristic Equation Fn - Fn-1 - Fn-2 = 0 f(x) = x2 – x – 1 Then Fn = Aan + Bbn for some constants A, B where a, b are roots of f(x) However • How to compute ((-1-sqrt(5))/2)n ? • The result must be a whole number, but the intermediate values may not • Use floating point numbers • Precision problem? • If we are asked to find Fn mod m? What a programmer would do • Note that ( )( ) = ( ) 0 1 1 1 • Then ( 0 1 1 1 Fn Fn+1 Fn+1 Fn+2 F n )( 0 F1 )=( ) Fn Fn+1 • Matrix Exponential Just like fast exponential Twelvefold ways • Put n balls into m boxes • How many ways? • • • • Balls identical / distinct? Boxes identical / distinct? Allow empty boxes? Allow two balls in one boxes? Twelvefold ways Combinations • The number of ways to choose r objects among n different objects (without order) • nCr = n!/r!(n-r)! Combinations • • • • How to calculate nCr? Calculate n!, r!, (n-r)! ? Note nCr = n(n-1)...(n-r+1)/r! nCr = 1; for i = n-r+1 . . . n nCr *= i; for i = 1 . . . r nCr /= i; Combinations • Overflow problem? • Note nCr = (n/r)(n-1)C(r-1) • nCr = 1; //that is (n-r)C0 for i = 1 . . . r nCr *= (n - r + i); nCr /= i; Combinations • What if we are asked to find nCr mod p for very large n, r? • Lucas Theorem • Let n = nknk-1...n1n0 (base p) r = rkrk-1...r1r0 • Then nCr (nkCrk)(nk-1Crk-1)...(n0Cr0) (mod p) • Works only for prime p Combinations • When p is large (larger than r), computing niCri may be difficult • (nCr)(r!) = n(n-1)...(n-r+1) • nCrn(n-1)...(n-r+1)(r!)-1 (mod p) where (r!)((r!)-1) 1 (mod p) • Fermat Little Theorem gives (r!)-1 (r!)p-2 (mod p) Combinations • When we are asked to mod a square-free number, factorize it into primes • The situation becomes complicated when the number is not square-free • Store the prime factorization of numerator and denominator A drunken man • A drunken man was standing beside a wall. On each second he moves left or right by 1m at random. • What is the probability that he returns to the original position in k seconds without hitting the wall? Observations • If k is odd, then it is impossible Let k = 2n • If the man returns to original position, then there are n left and n right moves • Number of possible outcomes = 22n • We have to find the number of moving sequence such that the man returns to original position without hitting the wall Observations • If the wall is removed, the number of ways is 2n C n • Let An be the number of ways • If the first time the man returns to his original position is at the 2ith second • Note that the first move must be rightward, and the 2ith move must be leftward Observations • Also in the 2i - 2 seconds after the first, he cannot return to his original position • Think of an invisible wall on his original position • Ai-1 ways for the first 2i seconds • An-i ways for the remaining 2n-2i seconds (may return to original position again) • An = i = 1...nAi-1An-i Tedious works again • Again, generating function • g(x) = n=0,1,... Anxn • g(x)2 = n=0,1,...i=0...n Ai An-i xn = n=0,1,...An+1xn • g(x) = A0+ xn=1,... Anxn-1 = 1 + xg(x)2 • xg(x) = (1-sqrt(1-4x))/2 or (1+sqrt(1-4x))/2 (rejected since xg(x)=0 when x = 0) • Power series...... A much more elegant approach • We now remove the wall and note when the man arrives at the position 1m on the left of the original position (If the man never arrives at it, he won’t hit the wall) • When he arrives at the considered position on the first time, flip his remaining moves, i.e. left to right, right to left • He will now end at the position 2m on the left of the original position A much more elegant approach • Wall removed: 2n C n ways • End at 2m on the left of original position: 2n C n-1 ways • Therefore the number of ways that the man never arrives at the considered position is (2nCn) – (2nCn-1) = (2nCn)/(n+1) • This is called Catalan Number Applications of Catalan Numbers • The number of valid string consisting of n pairs of parenthesis ()()(), (())(), ()(()), (()()), ((())) • The number of binary trees with n nodes An evil game • N people in a circle • Kill the first person, skip the next k people, then kill the next, etc. • Josephus Problem N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 N = 8, k = 2 0 7 1 6 2 3 5 4 Josephus problem • Closed form for k = 2 Josephus problem • Naive simulation • O(nk) • With little technique in handling the index • O(n) • f(n, k) = (f(n-1, k) + k) mod n Josephus problem • With some more little technique in handling the index • O(k log n) Question?