Twitter OA prepare: Rational Sum - neverlandly - 博客园
In mathematics, a rational number is any number that can be expressed in the form of a fraction p/q , where p & q are two integers, and the denominator q is not equal to zero. Hence, all integers are rational numbers where denominator, in the most reduced form, is equal to 1. You are given a list of N rational number, {a1/b1, a2/b2, ..., aN/bN}. Print the sum ( = a1/b1 + a2/b2 + ... + aN/bN = num/den) in the most reduced form. Input The first line of input contains an integer, N, the number of rational numbers. N lines follow. ithline contains two space separated integers, ai bi, where aiis the numerator and bi is the denominator for the ith rational number. Output You have to print two space separated integers, num den, where num and den are numerator and denominator of the sum respectively. Constraints 1 <= N <= 15 1 <= ai <= 10 1 <= bi <= 10 Notes Make sure the sum displayed as output is in the most reduced form. If sum is an integer, you have to print 1 as denominator. Sample Input 4 4 2 2 4 2 4 2 3 Sample Output 11 3 Explanation Sum is 4/2 + 2/4 + 2/4 + 2/3 = (24 + 6 + 6 + 8)/12 = 44/12 = 11/3. So you have to print "11 3", which is the most reduced form.
Below is the syntax highlighted version of Rational.java from §9.2 Symbolic Methods. 摘自http://introcs.cs.princeton.edu/java/92symbolic/Rational.java.html
5 * Immutable ADT for Rational numbers. 6 * 7 * Invariants 8 * ----------- 9 * - gcd(num, den) = 1, i.e, the rational number is in reduced form 10 * - den >= 1, the denominator is always a positive integer 11 * - 0/1 is the unique representation of 0 12 * 13 * We employ some tricks to stave of overflow, but if you 14 * need arbitrary precision rationals, use BigRational.java. 15 * 16 *************************************************************************/ 17 18 public class Rational implements Comparable<Rational> { 19 private static Rational zero = new Rational(0, 1); 20 21 private int num; // the numerator 22 private int den; // the denominator 23 24 // create and initialize a new Rational object 25 public Rational(int numerator, int denominator) { 26 27 // deal with x/0 28 //if (denominator == 0) { 29 // throw new RuntimeException("Denominator is zero"); 30 //} 31 32 // reduce fraction 33 int g = gcd(numerator, denominator); 34 num = numerator / g; 35 den = denominator / g; 36 37 // only needed for negative numbers 38 if (den < 0) { den = -den; num = -num; } 39 } 40 41 // return the numerator and denominator of (this) 42 public int numerator() { return num; } 43 public int denominator() { return den; } 44 45 // return double precision representation of (this) 46 public double toDouble() { 47 return (double) num / den; 48 } 49 50 // return string representation of (this) 51 public String toString() { 52 if (den == 1) return num + ""; 53 else return num + "/" + den; 54 } 55 56 // return { -1, 0, +1 } if a < b, a = b, or a > b 57 public int compareTo(Rational b) { 58 Rational a = this; 59 int lhs = a.num * b.den; 60 int rhs = a.den * b.num; 61 if (lhs < rhs) return -1; 62 if (lhs > rhs) return +1; 63 return 0; 64 } 65 66 // is this Rational object equal to y? 67 public boolean equals(Object y) { 68 if (y == null) return false; 69 if (y.getClass() != this.getClass()) return false; 70 Rational b = (Rational) y; 71 return compareTo(b) == 0; 72 } 73 74 // hashCode consistent with equals() and compareTo() 75 public int hashCode() { 76 return this.toString().hashCode(); 77 } 78 79 80 // create and return a new rational (r.num + s.num) / (r.den + s.den) 81 public static Rational mediant(Rational r, Rational s) { 82 return new Rational(r.num + s.num, r.den + s.den); 83 } 84 85 // return gcd(|m|, |n|) 86 private static int gcd(int m, int n) { 87 if (m < 0) m = -m; 88 if (n < 0) n = -n; 89 if (0 == n) return m; 90 else return gcd(n, m % n); 91 } 92 93 // return lcm(|m|, |n|) 94 private static int lcm(int m, int n) { 95 if (m < 0) m = -m; 96 if (n < 0) n = -n; 97 return m * (n / gcd(m, n)); // parentheses important to avoid overflow 98 } 99 100 // return a * b, staving off overflow as much as possible by cross-cancellation 101 public Rational times(Rational b) { 102 Rational a = this; 103 104 // reduce p1/q2 and p2/q1, then multiply, where a = p1/q1 and b = p2/q2 105 Rational c = new Rational(a.num, b.den); 106 Rational d = new Rational(b.num, a.den); 107 return new Rational(c.num * d.num, c.den * d.den); 108 } 109 110 111 // return a + b, staving off overflow 112 public Rational plus(Rational b) { 113 Rational a = this; 114 115 // special cases 116 if (a.compareTo(zero) == 0) return b; 117 if (b.compareTo(zero) == 0) return a; 118 119 // Find gcd of numerators and denominators 120 int f = gcd(a.num, b.num); 121 int g = gcd(a.den, b.den); 122 123 // add cross-product terms for numerator 124 Rational s = new Rational((a.num / f) * (b.den / g) + (b.num / f) * (a.den / g), 125 lcm(a.den, b.den)); 126 127 // multiply back in 128 s.num *= f; 129 return s; 130 } 131 132 // return -a 133 public Rational negate() { 134 return new Rational(-num, den); 135 } 136 137 // return a - b 138 public Rational minus(Rational b) { 139 Rational a = this; 140 return a.plus(b.negate()); 141 } 142 143 144 public Rational reciprocal() { return new Rational(den, num); } 145 146 // return a / b 147 public Rational divides(Rational b) { 148 Rational a = this; 149 return a.times(b.reciprocal()); 150 } 151 152 153 // test client 154 public static void main(String[] args) { 155 Rational x, y, z; 156 157 // 1/2 + 1/3 = 5/6 158 x = new Rational(1, 2); 159 y = new Rational(1, 3); 160 z = x.plus(y); 161 System.out.println(z); 162 163 // 8/9 + 1/9 = 1 164 x = new Rational(8, 9); 165 y = new Rational(1, 9); 166 z = x.plus(y); 167 System.out.println(z); 168 169 // 1/200000000 + 1/300000000 = 1/120000000 170 x = new Rational(1, 200000000); 171 y = new Rational(1, 300000000); 172 z = x.plus(y); 173 System.out.println(z); 174 175 // 1073741789/20 + 1073741789/30 = 1073741789/12 176 x = new Rational(1073741789, 20); 177 y = new Rational(1073741789, 30); 178 z = x.plus(y); 179 System.out.println(z); 180 181 // 4/17 * 17/4 = 1 182 x = new Rational(4, 17); 183 y = new Rational(17, 4); 184 z = x.times(y); 185 System.out.println(z); 186 187 // 3037141/3247033 * 3037547/3246599 = 841/961 188 x = new Rational(3037141, 3247033); 189 y = new Rational(3037547, 3246599); 190 z = x.times(y); 191 System.out.println(z); 192 193 // 1/6 - -4/-8 = -1/3 194 x = new Rational( 1, 6); 195 y = new Rational(-4, -8); 196 z = x.minus(y); 197 System.out.println(z); 198 } 199Read full article from Twitter OA prepare: Rational Sum - neverlandly - 博客园
