Given a range of number [a,b], how to efficiently find Bitwise OR of all numbers in this range. Running a loop for range [a,b] and computing Bitwise OR of all the numbers individually is too much time consuming for a range which is very large, so this is not the option.
Any number of the form 2n-1 will be a bit pattern of n 1's. When you OR this with any number below it, you get 2n-1. So all the numbers below the highest 2n-1 in the range can be ignored.
The next number in the range will be a 1 followed by n 0s, and when you OR with this you'll get n+1 1s. Since we selected the above number as the maximum power of 2, we'll never get any more bits in the number.
So there's basically just 2 cases. If the top of the range is 2n-1, then the result is a number with n 1 bits. Otherwise it's n+1 1 bits.
The above assumes that the range includes a 2n-1 value. If not, just try the loop (there are probably some optimizations that can be made, but I can't think of them off the top of my head).
Instead of doing it for all numbers, you can do it for all positions. That would require you only log(n) steps.
So lets try to imagine - when will units place be 1? If either upper or lower is odd or if there is one number between them. So either lower % 2 == 1 or lower != upper.
Great we got units place. Now if remove the lower one bit from both upper and lower bits and repeat we get the other positions.
Only a special case if lower == upper. In that case we return the lower itself.
Following is the code -
unsigned int bitwiseor(unsigned int a, unsigned int b){
if (a==b)
return a;
unsigned final = 0;
unsigned rev = 0;
while(b){
final*=2;
if (a%2==1 || a != b)
final++;
a/=2;
b/=2;
}
while(final){
rev *= 2;
rev += final % 2;
final/=2;
}
return rev;
}
The second loop is to just reserve the bit sequence.
Demo here - https://ideone.com/MCIugW
Thank you #Meixner for the driver code.
#include <stdio.h>
#include <stdlib.h>
int dumb(int a, int b)
{
int z=0;
while(a<=b) z|=a++;
return z;
}
int smart(int a, int b)
{
int d,z;
if(a>b) return 0;
d=b-a+1;
z=0;
while(d>1) { z=(z<<1)|1; d>>=1; }
d=z;
z|=a;
a+=d;
while(a<=b) z|=a++;
return z;
}
int main(int argc, char *argv[])
{
int a,b;
for(a=0;a<1000;a++) {
for(b=a;b<1000;b++) {
int z1=dumb(a,b);
int z2=smart(a,b);
if(z1!=z2) {
printf("fail %d %d\n",a,b);
}
}
}
return 0;
}
Related
Hello guys i am trying to implement a program which is finding the happy numbers were between two numbers A and B.
Summing the squares of all the digits of the number, we replace the number with the outcome, and repeat the process. If after some steps the result is equal to 1 (and stay there), then we say that the number N is **<happy>**. Conversely, if the process is repeated indefinitely without ever showing the number 1, then we say that the number N is **<sad>**.
For example, the number 7 is happy because the procedure described above leads to the following steps: 7, 49, 97, 130, 10, 1, 1, 1 ... Conversely, the number 42 is sad because the process leads to a infinite sequence 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37 ...
I try this right down but i am getting either segm faults or no results.
Thanks in advance.
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <stdlib.h>
void happy( char * A, int n);
int numPlaces (long n);
int main(void)
{
long A,B;
int npA;
char *Ap;
printf("Give 2 Numbers\n");
scanf("%li %li",&A,&B);
npA = numPlaces(A);
Ap = malloc(npA);
printf("%ld %d\n",A,npA);
//Search for happy numbers from A to B
do{
sprintf(Ap, "%ld", A);
happy(Ap,npA);
A++;
if ( npA < numPlaces(A) )
{
npA++;
Ap = realloc(Ap, npA);
}
}while( A <= B);
}
//Finds happy numbers
void happy( char * A, int n)
{
//Basic Condition
if ( n == 1)
{
if (A[0] == 1 || A[0] == 7)
printf("%c\n",A[0]);
printf("%s\n",A);
return;
}
long sum = 0 ;
char * sumA;
int nsum;
int Ai;
//Sum the squares of the current number
for(int i = 0 ; i < n;i++)
{
Ai = atoi(&A[i]);
sum = sum + (Ai*Ai);
}
nsum = numPlaces (sum);
sumA = malloc(nsum);
sprintf(sumA, "%li", sum);
happy(sumA,nsum);
free(sumA);
}
//Count digits of a number
int numPlaces (long n)
{
if (n < 0) return 0;
if (n < 10) return 1;
return 1 + numPlaces (n / 10);
}
Thanks for your time.
by the definition of your program sad numbers will cause your program to run forever
Conversely, if the process is repeated indefinitely
You need to add a stopping condition, like if I have looped for 1000 times, or if you hit a well known non terminating number (like 4) (is there a definite list of these? I dont know)
I find this solution tested and working..
Thanks for your time and I am sorry for my vagueness.
Every advice about this solution would be welcome
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <stdlib.h>
void happy( char * A, int n);
int numPlaces (long n);
int happynum = 0;
int main(void)
{
long A,B;
int npA;
char *Ap;
printf("Give 2 Numbers\n");
scanf("%li %li",&A,&B);
npA = numPlaces(A);
Ap = malloc(npA);
//Search for happy numbers from A to B
do{
sprintf(Ap, "%ld", A);
happy(Ap,npA);
if (happynum ==1)
printf("%s\n",Ap);
A++;
if ( npA < numPlaces(A) )
{
npA++;
Ap = realloc(Ap, npA);
}
}while( A <= B);
}
//Finds happy numbers
void happy( char * A, int n)
{
//Basic Condition
if ( n == 1)
{
if (A[0] == '3' || A[0] == '6' || A[0] == '9')
{
happynum = 0;
}
else
{
happynum = 1;
}
return;
}
long sum = 0;
char * sumA;
int nsum;
int Ai;
//Sum the squares of the current number
for(int i = 0 ; i < n;i++)
{
Ai = (int)(A[i]-48);
sum = sum + (Ai*Ai);
}
nsum = numPlaces (sum);
sumA = malloc(nsum);
sprintf(sumA, "%li", sum);
happy(sumA,nsum);
free(sumA);
}
//Count digits of a number
int numPlaces (long n)
{
if (n < 0) return 0;
if (n < 10) return 1;
return 1 + numPlaces (n / 10);
}
Your code uses some questionable practices. Yoe may be misguided because you are concerned about performance and memory usage.
When you allocate memory for the string, you forget to allocate one character for the null terminator. But you shouldn't be allocating, re-allocating and freeing constantly anyway. Dynamic memory allocation is expensive compared to your other operations.
Your limits are long, which may be a 32-bit or 64-bit signed integer, depending on your platform. The maximum number that can be represented with e 64-bit signed integer is 9,223,372,036,854,775,807. This is a number with 19 digits. Add one for the null terminator and one for a possible minus sign, so that overflow won't hurt, you and use a buffer of 21 chars on the stack.
You probably shouldn't be using strings inthe first place. Use the basic code to extract the digits: Split off the digit by taking the remainder of a division by 10. Then divide by 10 until you get zero. (And if you use strings with a fixed buffer size, as described above, you don't have to calculate the difits separately: sprintf returns the number of characters written to the string.
Your functions shouldn't be recursive. A loop is enough. As pm100 has noted, you need a termination criterion: You must keep track of the numbers that you have already visited. Each recursive call creates a new state; it is easier to keep an array, that can be repeatedly looked at in a loop. When you see a number that you have already seen (other than 1, of course), your number is sad.
Happy and sad numbers have this property that when your sum of squares is a number with a known happiness, the original number has this happiness, too. If you visit a known das number, the original number is sad. If you visit a known happy number, the original number is happy.
The limits of your ranges may ba large, but the sum of square digits is not large; it can be at most the number of digits times 81. In particular:
type max. number number of max. square sum dss
int 2,147,483,647 1,999,999,999 730
uint 4,294,967,295 3,999,999,999 738
long 9,223,372,036,854,775,807 8,999,999,999,999,999,999 1522
ulong 18,446,744,073,709,55,1616 9,999,999,999,999,999,999 1539
That means that when you take the sum of digit squares of an unsigned long, you will get a number that is smaller than 1540. Create an array of 1540 entries and mark all known happy numbers with 1. Then you can reduce your problem to taking the sum of digit squares once and then looking up the happiness of the number in this array.
(You can do the precalculation of the array once when you start the program.)
I am working in a math software with different features one of them to be to find all Carmichael numbers in a given interval [a,b)
This is my code, but I don't know if I have done it correctly or not cause I can't test it since the smallest Carmichael number is 560 which is too big for my pc to process.
#include <stdio.h>
int main() {
unsigned int begin, end;
printf("Write an int (begin):\n");
scanf("%d", &begin);
printf("Write an int (end):\n");
scanf("%d", &end);
int i;
for( int i=begin; i<end; i++ ) {
long unsigned int a_nr = i-1;
int a[a_nr];
for( int j=0; j<a_nr; j++ ) {
a[j] = j;
}
unsigned long c_nr[a_nr];
for( int k=0; k<a_nr; k++ ) {
unsigned long current_c_nr;
int mod;
for( int l=0; l<i; l++ ) {
current_c_nr= current_c_nr * a[k];
}
mod = current_c_nr%i;
if( mod==a[k] && mod!=a[k] ) {
c_nr[k] = i;
}
}
}
return 0;
}
If it is not correct, where is the mistake?
Thank you
P.S Overflow should be prevented.
When you say "This is my code, but I don't know if I have done it correctly or not cause I can't test it since the smallest Carmichael number is 560 which is too big for my pc to process" then the conclusion is -- you haven't done it correctly. You should be able to process 561 (560 must be a typo) in a small fraction of a second. Even if your algorithm is in principle correct, if it can't handle the smallest Carmichael number then it is useless.
n is Carmichael if and only if it is composite and, for all a with 1 < a < n which are relatively prime to n, the congruence a^(n-1) = 1 (mod n) holds. To use this definition directly, you need:
1) An efficient way to test if a and n are relatively prime
2) An efficient way to compute a^(n-1) (mod n)
For the first -- use the Euclidean algorithm for greatest common divisors. It is most efficiently computed in a loop, but can also be defined via the simple recurrence gcd(a,b) = gcd(b,a%b) with basis gcd(a,0) = a. In C this is just:
unsigned int gcd(unsigned int a, unsigned int b){
return b == 0? a : gcd(b, a%b);
}
For the second point -- almost the worst possible thing you can do when computing a^k (mod n) is to first compute a^k via repeated multiplication and to then mod the result by n. Instead -- use exponentiation by squaring, taking the remainder (mod n) at intermediate stages. It is a divide-and-conquer algorithm based on the observation that e.g. a^10 = (a^5)^2 and a^11 = (a^5)^2 * a. A simple C implementation is:
unsigned int modexp(unsigned int a, unsigned int p, unsigned int n){
unsigned long long b;
switch(p){
case 0:
return 1;
case 1:
return a%n;
default:
b = modexp(a,p/2,n);
b = (b*b) % n;
if(p%2 == 1) b = (b*a) % n;
return b;
}
}
Note the use of unsigned long long to guard against overflow in the calculation of b*b.
To test if n is Carmichael, you might as well first test if n is even and return 0 in that case. Otherwise, step through numbers, a, in the range 2 to n-1. First check if gcd(a,n) == 1 Note that if n is composite then you must have at least one a before you reach the square root of n with gcd(a,n) > 1). Keep a Boolean flag which keeps track of whether or not such an a has been encountered and if you exceed the square root without finding such an a, return 0. For those a with gcd(a,n) == 1, compute the modular exponentiation a^(n-1) (mod n). If this is ever different from 1, return 0. If your loop finishes checking all a below n without returning 0, then the number is Carmichael, so return 1. An implementation is:
int is_carmichael(unsigned int n){
int a,s;
int factor_found = 0;
if (n%2 == 0) return 0;
//else:
s = sqrt(n);
a = 2;
while(a < n){
if(a > s && !factor_found){
return 0;
}
if(gcd(a,n) > 1){
factor_found = 1;
}
else{
if(modexp(a,n-1,n) != 1){
return 0;
}
}
a++;
}
return 1; //anything that survives to here is a carmichael
}
A simple driver program:
int main(void){
unsigned int n;
for(n = 2; n < 100000; n ++){
if(is_carmichael(n)) printf("%u\n",n);
}
return 0;
}
output:
C:\Programs>gcc carmichael.c
C:\Programs>a
561
1105
1729
2465
2821
6601
8911
10585
15841
29341
41041
46657
52633
62745
63973
75361
This only takes about 2 seconds to run and matches the initial part of this list.
This is probably a somewhat practical method for checking if numbers up to a million or so are Carmichael numbers. For larger numbers, you should probably get yourself a good factoring algorithm and use Korseldt's criterion as described in the Wikipedia entry on Carmichael numbers.
The assignment is :
Write a program that calculates the sum of the divisors of a number from input.
A number is considered perfect if the sum of it's divisiors equal the number (ex: 6 = 1+2+3 ;28 = 1 + 2 + 4 + 7 +14).
Another definition:
a perfect number is a number that is half the sum of all of its positive divisors (including itself)
Generate the first k perfect numbers (k<150).
The main problem with this is that it's confusing the two asking points don't really relate.
In this program i calculated the sum of divisors of an entered number, but i don't know how to relate it with the second point (Generate the first k perfect numbers (k<150)).
#include <stdio.h>
#include <stdlib.h>
main()
{
int x,i,y,div,suma,k;
printf("Introduceti numarul\n"); \\enter the number
scanf("%d",&x);
suma=0; \\sum is 0
for(i=1;i<=x;i++)
{
if(x%i==0)
suma=suma+i; \\sum=sum+i;
}
printf("Suma divizorilor naturali este: %d\n",suma); \\the sum of the divisors is
for(k=1;k<150;k++) \\ bad part
{
if (x==suma)
printf("%d",k);
}
}
Suppose you have a function which can tell whether a given integer is perfect or not:
int isPerfect(int);
(function body not shown)
Now your main program will look like:
int candidate;
int perfectNumbers;
for(candidate = 1, perfectNumbers = 0; perfectNumbers < 150; candidate++) {
if (isPerfect(candidate)) {
printf("Number %d is perfect\n", candidate);
perfectNumbers++;
}
}
EDIT
For the same program without functions:
int candidate;
int perfectNumbers;
for(candidate = 1, perfectNumbers = 0; perfectNumbers < 150; candidate++) {
[... here your algorithm to compute the sum of the divisors of "candidate" ...]
if (candidate*2 == sum_of_divisors) {
printf("Number %d is perfect\n", candidate);
perfectNumbers++;
}
}
EDIT2: Just a note on perfect numbers
As noted in the comments section below, perfect numbers are very rare, only 48th of them are known as of 2014. The sequence (A000396) also grows very fast: using 64-bit integers you'll be able to compute up to the 8th perfect number (which happen to be 2,305,843,008,139,952,128). In this case the variable candidate will wrap around and start "finding" "new" perfect numbers from the beginning (until 150 of them are found: actually 19 repetitions of the only 8 findable in 64-bit integers). Note though that your algorithm must not choke on a candidate equals to 0 or to negative numbers (only to 0 if you declare candidate as unsigned int).
I am interpreting the question to mean generate all numbers under 150 that could are perfect numbers.
Therefore, if your program works for calculating perfect numbers, you keep calculating them until the starting number is >= 150.
Hope that makes sense.
Well, here's my solution ..
First, you have to make a reliable way of getting divisors.Here's a function I made for that:
size_t
getdivisors(num, divisors)
long long num;
long long *divisors;
{
size_t divs = 0;
for(long long i = num; i > 0; --i)
if (num%i == 0)
divisors[divs++] = i;
return divs;
}
Second, you need to check if the number's divisors match the perfect number's divisors properties (the sum of them is half the number).
Here's a second function for that:
bool
isperfect(num)
long long num;
{
long long divisors[num/2+1];
size_t divs = getdivisors(num, divisors);
if (divs == 0)
return false;
long long n = 0;
for(int i = 1; i < divs; ++i)
n += divisors[i];
return (n == num);
}
Now, from your question, I think you need to print all perfect numbers less than 150, right ?
See this:
int
main(argc, argv)
int argc;
char ** argv;
{
for(int i = 1; i < 150; ++i)
if (isperfect(i))
printf("%d is perfect.\n", i);
return 0;
}
I hope that answers your question ..
I'm building a program in C that can get powers of 2. The user inputs the value of n, and the program calculates 2^n.
Here's the code.
The problem comes when I input 100
What I am getting:
1,267,650,600,228,229,400,000,000,000,000
What I should get
1,267,650,600,228,229,401,496,703,205,376
It has to be coded entirely in ANSI C. Any ideas on how to increase the precision? The maximum value of N has to be 256 (256 bits, I imagine, which means the maximum output should be 2^256).
What I'm lacking here is precision, and I don't know how to fix that. Any ideas?
I think it's easiest if you work in base 10 from the start. This is because while calculating powers of 2 in binary is trivial, the conversion back to base 10 is a lot harder.
If you have an array of base 10 digits1, you only need to implement base 10 addition with carry to be able to multiply by 2 (by adding the number to itself). Do that n times in a loop and you have your answer.
If you wish to support higher exponents, you can also look into implementing exponentiation by squaring, but that's harder, since you'll need general multiplication, not just by 2 for that.
1 Tip: It's more convenient if you store the digits in reverse order.
Here is my quick and dirty implementation of hammar's approach., storing the decimal number as a C string with the digits in reverse order.
Run the code on ideone
void doubleDecimal(char * decimal)
{
char buffer[256] = "";
char c;
unsigned char d, carry = 0;
int i = 0;
while (c = decimal[i])
{
d = 2 * (c - '0') + carry;
buffer[i] = (d % 10) + '0';
carry = d / 10;
i++;
}
if (carry > 0)
buffer[i++] = (carry % 10) + '0';
buffer[i] = '\0';
strncpy(decimal, buffer, 256);
}
void reverse(char * str)
{
int i = 0;
int j = strlen(str) - 1;
while (j > i)
{
char tmp = str[i];
str[i] = str[j];
str[j] = tmp;
i++;
j--;
}
}
int main(void)
{
char decimal[256] = "1";
int i;
for (i = 0; i < 100; i++)
doubleDecimal(decimal);
reverse(decimal);
printf("%s", decimal);
return 0;
}
Output:
1267650600228229401496703205376
double is a (probably) 64bit value. You can't store 256 bits of precision in 64 bits. The reason that you are getting a number that is sort of close is because floating point numbers are stored with varying precision -- not all sequential numbers can be represented, but you can represent very large numbers. Pretty useless in this case.
What you want is either to use an arbitrary precision library or, since this is probably homework, you are expected to write your own.
A typical double, using 64-bit IEEE 754, has about 51 bits precision, IIRC.
Most probably the point of supporting exponents up to 256 is to exceed that precision, and also the precision of a long double or long long, so that you have to do things yourself.
As a homework exercise, then,
Store decimal digit values in an array + a digit count
Implement doubling of the value in such array + count
Start with 1 and double value appropriate number of times.
A few things you'll want to think about to solve this:
You are only dealing with integers so you should use an integer
representation (you will need to roll your own because you can't use
long long which is "only" 64 bits long).
Powers of 2 you say -how convenient - computers store numbers using powers of 2 (you'll
only need to use shift operations and bit fiddling .... no
multiplications will be needed).
How can you convert a base 2 number to a base 10 number for display purposes (think of division and outputting one number at a time (think about what a hardware divisor does in order to get the bit manipulations correct).
You can't the store 256 bits of precision in 64 bits. Reason that you are getting a number to close is because floating point numbers are stored with varying precision. To all sequential numbers can be represented, but you can represent very large numbers. Pretty useless in this case.
#include <conio.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
//constants
#define MAX_DIGITS 1000
//big integer number struct
struct bigint {
char Digits[MAX_DIGITS];
};
//assign a value
void assign(struct bigint* Number,int Value) {
if (Value!=1) {
printf("Can not assign value other than 1\n");
exit(0);
}
memset(Number,0,sizeof(bigint));
Number->Digits[0] = Value;
}
//multiply the big integer number with value
void multiply(struct bigint* Number,int Value) {
int Digit,New_Digit;
int Carry = 0;
for (int Index=0; Index<MAX_DIGITS; Index++) {
Digit = Number->Digits[Index];
New_Digit = Digit*Value%10;
if (New_Digit+Carry<10) {
New_Digit = New_Digit+Carry;
Carry = Digit*Value/10;
}
else {
New_Digit = (New_Digit+Carry)%10;
Carry = (Digit*Value/10)+1;
}
//set the new digit
Number->Digits[Index] = New_Digit;
}//for loop
}
//print out the value of big integer type
void print(struct bigint* Number) {
int Index = MAX_DIGITS-1;
while (Number->Digits[Index]==0 && Index>=0)
Index--;
//the big integer value is zero
if (Index==-1) {
printf("0");
return;
}
while (Index>=0) {
printf("%u",Number->Digits[Index]);
Index--;
}
}
//main programme entry point
int main(int Argc,char** Args) {
int Power = 100;
struct bigint Number;
//assign the initial value
assign(&Number,1);
//do the multiplication
for (int Index=0; Index<Power; Index++)
multiply(&Number,2);
//print result
print(&Number);
getch();
}
//END-OF-FILE
#include <stdio.h>
long factorial(int num)
{
int counter;
int fact = 1;
for (counter = num; counter > 0; counter--) fact *= counter;
return fact;
}
float combinations(int n, int k)
{
int numerator = factorial(n);
int denominator = factorial(k) * factorial(n-k);
float fraction = numerator/denominator;
return fraction;
}
int main()
{
printf("How many rows of Pascal\'s triangle should I print?\t");
int rows = GetInteger();
int counter;
int counter2;
for (counter = 1; counter <= rows; counter++)
{
int y = rows-counter;
for (; y > 0; y--) printf(" ");
for (counter2 = 0; counter2 <= counter; counter2++)
printf("%6.0lu", (long) combinations(counter, counter2));
printf("\n");
}
}
Every time I go past twelve rows, the numbers start to decrease. What am i doing wrong?
And, GetInteger() is just a scanf() with a few touch ups. I am 100% sure it works perfectly.
After 12th row factorial and so pascal triangle elements become too large so int type cannot hold them - so you get overflow (most probably values you get are wrapped around maximum int value).
P.S. why do you use 3 different types in your code (long, int, float)? As k!*(n-k)! always divides n! you do not need float value (you use integer division and cast result to long anyway). Just use the biggest integer type you can, or some custom BigInt type that can hold integer numbers of arbitrary length - so you can show correct values for large row numbers.
Don't start from factorials. Start from the following facts about Pascal's triangle:
the nth row of the triangle has n elements (if we start counting from 1)
the first and last elements of each row are 1
each element aside from the first and last one is the sum of the two elements diagonally above it (if the triangle is written in a symmetric way)
You will of course be limited by the size of the data type you are holding results in, but not any sooner than necessary (by intermediate results such as factorials).
INT_MAX is usually 2,147,483,647
12! is 479,001,600
13! is 6,227,020,800 but your function factorial(13) returns 1,932,053,504 (= 6,227,020,800 - 4,294,967,296)