Programming Interest Group

http://www.comp.hkbu.edu.hk/~chxw/pig/index.htm

Tutorial Four High-Precision Arithmetic

1

Machine Arithmetic

    32-bit machine  An integer is roughly in the range ± 2 31 = ± 2,147,483,648 64-bit machine  An integer is roughly in the range ± 2 63 = ± 9,223,372,036,854,775,808 Unsigned integer can double the upper limit Sometimes we need to operate on very large integers, especially in the area of cryptography  High-precision integers , or multiple precision integers 2

Integer Libraries

 C:  ,  C++:  GNU C++: Integer class  http://www.chemie.fu berlin.de/chemnet/use/info/libgpp/libgpp_20.html

 Java:  BigInteger class in java.math

 http://java.sun.com/j2se/1.4.2/docs/api/java/math/BigIn teger.html

3

Java BigInteger Example

 Factorial n (usually written n!

) is the product of all integers up to and including n (1x 2 x 3 x ... x n). import java.math.BigInteger; public class Factorial { public static void main(String[] args) { //-- BigInteger solution. BigInteger n = BigInteger.ONE; for (int i=1; i<=20; i++) { n = n.multiply(BigInteger.valueOf(i)); System.out.println(i + "! = " + n); } //-- int solution (BAD IDEA BECAUSE ONLY WORKS TO 12). int fact = 1; for (int i=1; i<=20; i++) { fact = fact * i; System.out.println(i + "! = " + fact); } } } 4

Java BigInteger Example (Cont.)

1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800 11! = 39916800 12! = 479001600 13! = 6227020800 14! = 87178291200 15! = 1307674368000 16! = 20922789888000 17! = 355687428096000 18! = 6402373705728000 19! = 121645100408832000 20! = 2432902008176640000 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800 11! = 39916800 12! = 479001600 13! = 1932053504 BAD 14! = 1278945280 BAD 15! = 2004310016 BAD 16! = 2004189184 BAD 17! = -288522240 18! = -898433024 BAD BAD 19! = 109641728 BAD 20! = -2102132736 BAD 5

What are we doing now?

  As a CS student, it’s not enough to just know how to use those high-precision integer libraries.

It’s better to know how to design and implement a high-precision integer arithmetic library.

 One student from our department has done this as his Honours Project.

 Addition, subtraction, multiplication, and division, modular, exponentiation, etc 6

Review of Number Systems

  Positional notation using

base b

(or

radix b

) is defined by the rule: (…a 3 a 2 a 1 a 0 .

a -1 a -2 …) b

Radix Point

= …+a 3 b 3 +a 2 b 2 +a 1 b 1 +a 0 +a -1 b -1 +a -2 b -2 +… Binary number system: b = 2  2 digits: 0 and 1   Decimal number system: b = 10  10 digits: 0, 1, 2, 3, …, 9 Hexadecimal number system: b = 16  16 digits: 0, 1, 2, …, 9, A, B, C, D, E, F 7

High-precision Integers

 How to represent enormous integers?

  Arrays of digits    The initial element of the array represents the least significant digits: An integer 1234567890 is stored as an array: {0, 9, 8, 7, 6, 5, 4, 3, 2, 1} Maintain a counter for the number of digits  Maintain the sign of the integer: positive or negative?

Linked list of digits   It can support real arbitrary precision arithmetic   But: Waste of memory (each node takes a pointer) The performance is not as good as arrays 8

An Example: Using Array

http://www.comp.hkbu.edu.hk/~chxw/pig/code/bignum.c

#define MAXDIGITS 100 #define PLUS 1 #define MINUS -1 typedef struct { char digits[MAXDIGITS]; int signbit; } bignum; int lastdigit; /*maximum length bignum */ /* positive sign bit */ /* negative sign bit */ /* represent the number */ /* PLUS or MINUS */ /* index of high-order digit */ Remarks: 1. Each digit (0-9) is represented using a single-byte character.

2. In fact, using higher numerical bases (e.g., 64, 128, 256) is more efficient.

9

Print a Bignum

} { void print_bignum(bignum *nPtr) int i; if (nPtr->signbit == MINUS) printf(“-”); for (i = nPtr->lastdigit; i >= 0; i--) printf(“%c”, ‘0’+nPtr->digits[i]); printf(“\n”); 10

}

Convert an Integer to Bignum

{ int_to_bignum(int s, bignum *nPtr) int i; int t; /* counter */ /* int to work with */ if (s >= 0) nPtr->signbit = PLUS; else nPtr->signbit = MINUS; for (i=0; idigits[i] = (char) 0; nPtr->lastdigit = -1; t = abs(s); while (t > 0) { nPtr->lastdigit ++; nPtr->digits[ nPtr->lastdigit ] = (t % 10); t = t / 10; } if (s == 0) nPtr->lastdigit = 0; 11

Initialize a Bignum

} { initialize_bignum(bignum *nPtr) int_to_bignum(0,nPtr); } { int max(int a, int b) if (a > b) return(a); else return(b); 12

Compare Two Bignums

} { compare_bignum(bignum *a, bignum *b) int i; /* counter */ if ((a->signbit == MINUS) && (b->signbit == PLUS)) return(PLUS); if ((a->signbit == PLUS) && (b->signbit == MINUS)) return(MINUS); if (b->lastdigit > a->lastdigit) return (PLUS * a->signbit); if (a->lastdigit > b->lastdigit) return (MINUS * a->signbit); } for (i = a->lastdigit; i>=0; i--) { if (a->digits[i] > b->digits[i]) return(MINUS * a->signbit); if (b->digits[i] > a->digits[i]) return(PLUS * a->signbit); return(0); 13

How to avoid leading zeros?

} { zero_justify(bignum *n) while ((n->lastdigit > 0) && (n->digits[ n->lastdigit ] == 0)) n->lastdigit --; if ((n->lastdigit == 0) && (n->digits[0] == 0)) n->signbit = PLUS; /* hack to avoid -0 */ 14

Addition of Two Bignums

{ add_bignum(bignum *a, bignum *b, bignum *c) int carry; int i; /* carry digit */ /* counter */ initialize_bignum(c); } if (a->signbit == b->signbit) c->signbit = a->signbit; else { if (a->signbit == MINUS) { a->signbit = PLUS; subtract_bignum(b,a,c); a->signbit = MINUS; } else { b->signbit = PLUS; subtract_bignum(a,b,c); b->signbit = MINUS; } return; 15

Addition of Two Bignums (cont.)

c->lastdigit = max(a->lastdigit,b->lastdigit)+1; carry = 0; } for (i=0; i<=(c->lastdigit); i++) { c->digits[i] = (char) (carry+a->digits[i]+b->digits[i]) % 10; carry = (carry + a->digits[i] + b->digits[i]) / 10; zero_justify(c); } 16

Subtraction of Two Bignums

{ subtract_bignum(bignum *a, bignum *b, bignum *c) int borrow; int v; int i; /* has anything been borrowed? */ /* placeholder digit */ /* counter */ initialize_bignum(c); } if ((a->signbit == MINUS) || (b->signbit == MINUS)) { b->signbit = -1 * b->signbit; add_bignum(a,b,c); b->signbit = -1 * b->signbit; return; } if (compare_bignum(a,b) == PLUS) { subtract_bignum(b,a,c); c->signbit = MINUS; return; 17

}

Subtraction of Two Bignums (Cont.)

c->lastdigit = max(a->lastdigit,b->lastdigit); borrow = 0; for (i=0; i<=(c->lastdigit); i++) { v = (a->digits[i] - borrow - b->digits[i]); if (a->digits[i] > 0) } if (v < 0) { borrow = 0; v = v + 10; borrow = 1; c->digits[i] = (char) v % 10; } zero_justify(c); 18

Digit Shift

} /* E.g., shift 123456 by 2 will get 12345600 In memory: {6,5,4,3,2,1}  {0,0,6,5,4,3,2,1} { */ digit_shift(bignum *nPtr, int d) int i; /* multiply *nPtr by 10^d */ /* counter */ if ((nPtr->lastdigit == 0) && (nPtr->digits[0] == 0)) return; for (i=nPtr->lastdigit; i>=0; i--) nPtr->digits[i+d] = nPtr->digits[i]; for (i=0; idigits[i] = 0; nPtr->lastdigit = nPtr->lastdigit + d; 19

}

Multiplication of Two Bignums

{ multiply_bignum(bignum *a, bignum *b, bignum *c) bignum row; bignum tmp; int i, j; /* represent shifted row */ /* placeholder bignum */ /* counters */ initialize_bignum(c); row = *a; } for (i=0; i <= b->lastdigit; i++) { for (j=1; j <= b->digits[i]; j++) { add_bignum(c,&row,&tmp); *c = tmp; } digit_shift(&row,1); c->signbit = a->signbit * b->signbit; zero_justify(c); 20

Division of Two Bignums

{ divide_bignum(bignum *a, bignum *b, bignum *c) bignum row; /* represent shifted row */ bignum tmp; int asign, bsign; int i, j; /* placeholder bignum */ /* temporary signs */ /* counters */ initialize_bignum(c); c->signbit = a->signbit * b->signbit; asign = a->signbit; bsign = b->signbit; a->signbit = PLUS; b->signbit = PLUS; 21

}

Division of Two Bignums (Cont.)

initialize_bignum(&row); initialize_bignum(&tmp); c->lastdigit = a->lastdigit; } for (i=a->lastdigit; i >= 0; i--) { digit_shift(&row,1); row.digits[0] = a->digits[i]; c->digits[i] = 0; while (compare_bignum(&row,b) != PLUS) { c->digits[i] ++; subtract_bignum(&row,b,&tmp); row = tmp; } zero_justify(c); a->signbit = asign; b->signbit = bsign; 22

For more information:

 Donald E. Knuth: The Art of Computer Programming, Volume 2.

 Seminumerical Algorithms, Chapter 4         Arithmetic Positional number systems Floating point arithmetic Multiple precision arithmetic Radix conversion Rational arithmetic Polynomial arithmetic Manipulation of power series 23

Practice I: Primary Arithmetic

    http://acm.uva.es/p/v100/10035.html

Children are taught to add multi-digit numbers from right-to left one digit at a time. Many find the "carry" operation - in which a 1 is carried from one digit position to be added to the next - to be a significant challenge. Your job is to count the number of carry operations for each of a set of addition problems so that educators may assess their difficulty.

Input

 Each line of input contains two unsigned integers less than 10 digits. The last line of input contains 0 0.

Output

 For each line of input except the last you should compute and print the number of carry operations that would result from adding the two numbers, in the format shown below. 24

Practice I: Primary Arithmetic

Sample Input

123 456 555 555 123 594 0 0

Sample Output

No carry operation. 3 carry operations. 1 carry operation.

Hint: Use an array of characters to store the digits of an integer 25

Practice II: Reverse and Add

   http://acm.uva.es/p/v100/10018.html

The "reverse and add" method is simple: choose a number, reverse its digits and add it to the original. If the sum is not a palindrome (which means, it is not the same number from left to right and right to left), repeat this procedure. For example: 195 Initial number 591 ---- 786 687 ---- 1473 3741 ---- 5214 4125 ---- 9339 Resulting palindrome In this particular case the palindrome 9339 appeared after the 4th addition. This method leads to palindromes in a few step for almost all of the integers. But there are interesting exceptions. 196 is the first number for which no palindrome has been found. It is not proven though, that there is no such a palindrome.

26

Practice II: Reverse and Add

 Task : You must write a program that gives the resulting palindrome and the number of iterations (additions) to compute the palindrome.  You might assume that all tests data on this problem: - will have an answer , - will be computable with less than 1000 iterations (additions), - will yield a palindrome that is not greater than 4,294,967,295. 

The Input

 The first line will have a number N with the number of test cases, the next N lines will have a number P to compute its palindrome. 

The Output

 For each of the N tests you will have to write a line with the following data : minimum number of iterations (additions) to get to the palindrome and the resulting palindrome itself separated by one space. 27

Practice II: Reverse and Add



Sample Input

3 195 265 750 

Sample Output

4 9339 5 45254 3 6666 28

Practice III: Archeologist’s Dilemma

     http://acm.uva.es/p/v7/701.html

An archeologist seeking proof of the presence of extraterrestrials in the Earth's past, stumbles upon a partially destroyed wall containing strange chains of numbers . The left-hand part of these lines of digits is always intact , but unfortunately the right-hand one the hypothesis that is often lost by erosion of the stone. However, she notices that all the numbers with all its digits intact are powers of 2, so that all of them are powers of 2 is obvious. To reinforce her belief, she selects a list of numbers on which it is apparent that the number of legible digits is strictly smaller than the number of lost ones , and asks you to find the smallest power of 2 (if any) whose first digits coincide with those of the list. Thus you must write a program such that given an integer, it determines (if it exists) the smallest exponent

E

such that the first digits of 2

E

coincide with the integer (remember that more than half of the digits are missing). 29

Practice III: Archeologist’s Dilemma

 

Input

 It is a set of lines with a positive integer

N

2147483648 in each of them. not bigger than

Output

 For every one of these integers a line containing the smallest positive integer

E

such that the first digits of 2

E

are precisely the digits of

N

, or, if there is no one, the sentence ``no power of 2". Sample Input: 1 2 10 Sample Output: 7 8 20 (2, 4, 8, 16 , 32, 64, 128=2 7 ) ( 2 , 4, 8, 16, 32, 64, 128, 256=2 8 ) (2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 , 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576=2 20 ) 30

Analysis of the problem

 The legible digits form a number

N

 Assume there are

k

missing digits  The we have the following inequalities: 

N

x10

k

<= 2

E

< (N+1)x10

k

 We perform the lg() operation:  lg(

N

x10

k

) <= lg(2

E

) < lg ((N+1)x10

k

), which leads to  lg

N

+

k

lg10 <=

E

< lg(

N

+1) +

k

lg10  We can search for the

k

such that, there is an integer in the middle 31

Sample Solution

{ void process(int n) int k, e; double d1, d2, d3, d4; double left, right; #include #include #include void process(int); int length(int); d4 = log(2); d3 = log(10) / d4; d1 = log(n) / d4; d2 = log(n+1) / d4; } { int main() int in; while( scanf("%u", &in) != EOF) process(in); return 0; } k = length(n)+1; left = d1+k*d3; right = d2+k*d3; e = (int)ceil(left); { while(e >= right) k++; left = d1+k*d3; right = d2+k*d3; e = (int)ceil(left); } printf("%d\n", e); } { int length(int n) int len = 0; if(n < 10) { return 1; while(n > 0) n = n/10; len++; } return len; 32

Practice IV: How Many Fibs?

 http://acm.uva.es/p/v101/10183.html

 Recall the definition of the Fibonacci numbers:   f 1 = 1, f 2 = 2, f n = f n-1 +f n-2 (n >= 3) Given two numbers

a

and

b

, calculate how many Fibonacci numbers are in the range [

a

,

b

]. 33

Practice IV: How Many Fibs?



Input Specification

 The input contains several test cases. Each test case consists of two non-negative integer numbers

a

and

b

. Input is terminated by

a=b=0

. Otherwise,

a<=b<=10 100

. The numbers

a

and

b

are given with no superfluous leading zeros. 

Output Specification

 For each test case output on a single line the number of Fibonacci numbers

f i

with

a<=f i <=b

. 34

Practice IV: How Many Fibs?



Sample Input

10 100  1234567890 9876543210 0 0

Sample Output

5 4 Hint: Use the bignum functions!

35

More practice on basic arithmetic

 http://acm.uva.es/p/v8/847.html

 http://acm.uva.es/p/v100/10077.html

 http://acm.uva.es/p/v101/10105.html

 http://acm.uva.es/p/v101/10127.html

 http://acm.uva.es/p/v102/10202.html

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